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<TITLE>Hardy–Weinberg principle - Wikipedia</TITLE>

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			<H1 id="firstHeading" class="firstHeading" lang="en">Hardy–Weinberg principle</H1>
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Hardy–Weinberg proportions for two <A href="/wiki/Allele" title="Allele">alleles</A>: the horizontal axis shows the two <A href="/wiki/Allele_frequency" title="Allele frequency">allele frequencies</A> <I>p</I> and <I>q</I> and the vertical axis shows the expected <A href="/wiki/Genotype_frequencies" class="mw-redirect" title="Genotype frequencies">genotype frequencies</A>. Each line shows one of the three possible genotypes.</DIV>
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<P>The <B>Hardy–Weinberg principle</B>, also known as the <B>Hardy–Weinberg equilibrium, model, theorem</B>, or <B>law</B>, states that <A href="/wiki/Allele_frequency" title="Allele frequency">allele</A> and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences. These influences include <I><A href="/wiki/Mate_choice" title="Mate choice">mate choice</A></I>, <I><A href="/wiki/Mutation" title="Mutation">mutation</A></I>, <I><A href="/wiki/Selection_(biology)" class="mw-redirect" title="Selection (biology)">selection</A></I>, <I><A href="/wiki/Genetic_drift" title="Genetic drift">genetic drift</A></I>, <I><A href="/wiki/Gene_flow" title="Gene flow">gene flow</A></I> and <I><A href="/wiki/Meiotic_drive" title="Meiotic drive">meiotic drive</A></I>.</P>
<P>In the simplest case of a single locus with two <A href="/wiki/Allele" title="Allele">alleles</A> denoted A and a with frequencies <SPAN class="texhtml"><VAR>f</VAR>(A) = <I>p</I></SPAN> and <SPAN class="texhtml"><VAR>f</VAR>(a) = <I>q</I></SPAN>, respectively, the expected genotype frequencies under random mating are <SPAN class="texhtml"><VAR>f</VAR>(AA) = <I>p</I><SUP>2</SUP></SPAN> for the AA <A href="/wiki/Homozygote" class="mw-redirect" title="Homozygote">homozygotes</A>, <SPAN class="texhtml"><VAR>f</VAR>(aa) = <I>q</I><SUP>2</SUP></SPAN> for the aa homozygotes, and <SPAN class="texhtml"><VAR>f</VAR>(Aa) = 2<I>pq</I></SPAN> for the <A href="/wiki/Heterozygote" class="mw-redirect" title="Heterozygote">heterozygotes</A>. In the absence of selection, mutation, genetic drift, or other forces, allele frequencies <I>p</I> and <I>q</I> are constant between generations, so equilibrium is reached.</P>
<P>The principle is named after <A href="/wiki/G._H._Hardy" title="G. H. Hardy">G. H. Hardy</A> and <A href="/wiki/Wilhelm_Weinberg" title="Wilhelm Weinberg">Wilhelm Weinberg</A>, who first demonstrated it mathematically. Hardy's paper was focused on debunking the then-commonly held view that a <A href="/wiki/Dominance_(genetics)" title="Dominance (genetics)">dominant</A> allele would automatically tend to increase in frequency; today, confusion between dominance and selection is less common. Today, tests for Hardy-Weinberg genotype frequencies are used primarily to test for <A href="/wiki/Population_stratification" title="Population stratification">population stratification</A> and other forms of non-random mating.</P>
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<DIV id="toc" class="toc">
<DIV class="toctitle">
<H2>Contents</H2>
<SPAN class="toctoggle"> [<A role="button" tabindex="0" class="togglelink">hide</A>] </SPAN></DIV>
<UL>
<LI class="toclevel-1 tocsection-1"><A href="#Derivation"><SPAN class="tocnumber">1</SPAN> <SPAN class="toctext">Derivation</SPAN></A></LI>
<LI class="toclevel-1 tocsection-2"><A href="#Deviations_from_Hardy.E2.80.93Weinberg_equilibrium"><SPAN class="tocnumber">2</SPAN> <SPAN class="toctext">Deviations from Hardy–Weinberg equilibrium</SPAN></A></LI>
<LI class="toclevel-1 tocsection-3"><A href="#Sex_linkage"><SPAN class="tocnumber">3</SPAN> <SPAN class="toctext">Sex linkage</SPAN></A></LI>
<LI class="toclevel-1 tocsection-4"><A href="#Generalizations"><SPAN class="tocnumber">4</SPAN> <SPAN class="toctext">Generalizations</SPAN></A>
<UL>
<LI class="toclevel-2 tocsection-5"><A href="#Generalization_for_more_than_two_alleles"><SPAN class="tocnumber">4.1</SPAN> <SPAN class="toctext">Generalization for more than two alleles</SPAN></A></LI>
<LI class="toclevel-2 tocsection-6"><A href="#Generalization_for_polyploidy"><SPAN class="tocnumber">4.2</SPAN> <SPAN class="toctext">Generalization for polyploidy</SPAN></A></LI>
<LI class="toclevel-2 tocsection-7"><A href="#Complete_generalization"><SPAN class="tocnumber">4.3</SPAN> <SPAN class="toctext">Complete generalization</SPAN></A></LI>
</UL>
</LI>
<LI class="toclevel-1 tocsection-8"><A href="#Applications"><SPAN class="tocnumber">5</SPAN> <SPAN class="toctext">Applications</SPAN></A>
<UL>
<LI class="toclevel-2 tocsection-9"><A href="#Application_to_cases_of_complete_dominance"><SPAN class="tocnumber">5.1</SPAN> <SPAN class="toctext">Application to cases of complete dominance</SPAN></A></LI>
</UL>
</LI>
<LI class="toclevel-1 tocsection-10"><A href="#Significance_tests_for_deviation"><SPAN class="tocnumber">6</SPAN> <SPAN class="toctext">Significance tests for deviation</SPAN></A>
<UL>
<LI class="toclevel-2 tocsection-11"><A href="#Example_.7F.27.22.60UNIQ--postMath-0000001E-QINU.60.22.27.7F_test_for_deviation"><SPAN class="tocnumber">6.1</SPAN> <SPAN class="toctext">Example <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle \chi ^{2}}</annotation>
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<LI class="toclevel-2 tocsection-12"><A href="#Fisher.27s_exact_test_.28probability_test.29"><SPAN class="tocnumber">6.2</SPAN> <SPAN class="toctext">Fisher's exact test (probability test)</SPAN></A></LI>
</UL>
</LI>
<LI class="toclevel-1 tocsection-13"><A href="#Inbreeding_coefficient"><SPAN class="tocnumber">7</SPAN> <SPAN class="toctext">Inbreeding coefficient</SPAN></A></LI>
<LI class="toclevel-1 tocsection-14"><A href="#History"><SPAN class="tocnumber">8</SPAN> <SPAN class="toctext">History</SPAN></A>
<UL>
<LI class="toclevel-2 tocsection-15"><A href="#Derivation_of_Hardy.27s_equations"><SPAN class="tocnumber">8.1</SPAN> <SPAN class="toctext">Derivation of Hardy's equations</SPAN></A></LI>
<LI class="toclevel-2 tocsection-16"><A href="#Numerical_example"><SPAN class="tocnumber">8.2</SPAN> <SPAN class="toctext">Numerical example</SPAN></A></LI>
</UL>
</LI>
<LI class="toclevel-1 tocsection-17"><A href="#Graphical_representation"><SPAN class="tocnumber">9</SPAN> <SPAN class="toctext">Graphical representation</SPAN></A></LI>
<LI class="toclevel-1 tocsection-18"><A href="#See_also"><SPAN class="tocnumber">10</SPAN> <SPAN class="toctext">See also</SPAN></A></LI>
<LI class="toclevel-1 tocsection-19"><A href="#References"><SPAN class="tocnumber">11</SPAN> <SPAN class="toctext">References</SPAN></A></LI>
<LI class="toclevel-1 tocsection-20"><A href="#Bibliography"><SPAN class="tocnumber">12</SPAN> <SPAN class="toctext">Bibliography</SPAN></A></LI>
<LI class="toclevel-1 tocsection-21"><A href="#External_links"><SPAN class="tocnumber">13</SPAN> <SPAN class="toctext">External links</SPAN></A></LI>
</UL>
</DIV>
<P></P>
<H2><SPAN class="mw-headline" id="Derivation">Derivation</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=1" title="Edit section: Derivation">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P>Consider a population of <A href="/wiki/Monoecious" class="mw-redirect" title="Monoecious">monoecious</A> <A href="/wiki/Diploids" class="mw-redirect" title="Diploids">diploids</A>, where each organism produces male and female gametes at equal frequency, and has two alleles at each gene locus. Organisms reproduce by random union of gametes (the “gene pool” population model). A locus in this population has two alleles, A and a, that occur with initial frequencies <SPAN class="texhtml"><I>f</I><SUB>0</SUB>(A) = <I>p</I></SPAN> and <SPAN class="texhtml"><I>f</I><SUB>0</SUB>(a) = <I>q</I></SPAN>, respectively.<SUP id="cite_ref-1" class="reference"><A href="#cite_note-1">[1]</A></SUP> The allele frequencies at each generation are obtained by pooling together the alleles from each <A href="/wiki/Genotype" title="Genotype">genotype</A> of the same generation according to the expected contribution from the homozygote and heterozygote genotypes, which are 1 and 1/2, respectively:</P>
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    <annotation encoding="application/x-tex">{\displaystyle f_{t}({\text{A}})=f_{t}({\text{AA}})+{\tfrac {1}{2}}f_{t}({\text{Aa}})}</annotation>
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<P style="margin:0pt;"><B>(<SPAN id="math_1" class="reference nourlexpansion" style="font-weight:bold; font-style:italic;">1</SPAN>)</B></P>
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          <mrow class="MJX-TeXAtom-ORD">
            <mi>t</mi>
          </mrow>
        </msub>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>a</mtext>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo>=</mo>
        <msub>
          <mi>f</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>t</mi>
          </mrow>
        </msub>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>aa</mtext>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mstyle displaystyle="false" scriptlevel="0">
            <mfrac>
              <mn>1</mn>
              <mn>2</mn>
            </mfrac>
          </mstyle>
        </mrow>
        <msub>
          <mi>f</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>t</mi>
          </mrow>
        </msub>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>Aa</mtext>
        </mrow>
        <mo stretchy="false">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle f_{t}({\text{a}})=f_{t}({\text{aa}})+{\tfrac {1}{2}}f_{t}({\text{Aa}})}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7b283c64441b70a92b55579d28d5dafad86e937" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; width:25.51ex; height:3.509ex;" alt="{\displaystyle f_{t}({\text{a}})=f_{t}({\text{aa}})+{\tfrac {1}{2}}f_{t}({\text{Aa}})}" /></SPAN></P>
</TD>
<TD style="vertical-align:middle; width:99%; border:none; padding:0.08em;">
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<P style="margin:0; font-size:1pt;"> </P>
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</TBODY></TABLE>
</TD>
<TD style="vertical-align:middle; border:none; padding:0.08em;" class="nowrap">
<P style="margin:0pt;"><B>(<SPAN id="math_2" class="reference nourlexpansion" style="font-weight:bold; font-style:italic;">2</SPAN>)</B></P>
</TD>
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</DD>
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<DIV class="thumb tright">
<DIV class="thumbinner" style="width:262px;"><A href="/wiki/File:Hardy%E2%80%93Weinberg_law_-_Punnett_square.svg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Hardy%E2%80%93Weinberg_law_-_Punnett_square.svg/260px-Hardy%E2%80%93Weinberg_law_-_Punnett_square.svg.png" width="260" height="255" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Hardy%E2%80%93Weinberg_law_-_Punnett_square.svg/390px-Hardy%E2%80%93Weinberg_law_-_Punnett_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Hardy%E2%80%93Weinberg_law_-_Punnett_square.svg/520px-Hardy%E2%80%93Weinberg_law_-_Punnett_square.svg.png 2x" data-file-width="642" data-file-height="630" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Hardy%E2%80%93Weinberg_law_-_Punnett_square.svg" class="internal" title="Enlarge"></A></DIV>
Length of <SPAN class="texhtml"><I>p</I>, <I>q</I></SPAN> corresponds to allele frequencies (here <SPAN class="texhtml"><I>p</I> = 0.6, <I>q</I> = 0.4</SPAN>). Then area of rectangle represents genotype frequencies (thus <SPAN class="texhtml">AA : Aa : aa = 0.36 : 0.48 : 0.16</SPAN>).</DIV>
</DIV>
</DIV>
<P>The different ways to form genotypes for the next generation can be shown in a <A href="/wiki/Punnett_square" title="Punnett square">Punnett square</A>, where the proportion of each genotype is equal to the product of the row and column allele frequencies from the current generation.</P>
<TABLE class="wikitable">
<CAPTION>Table 1: Punnett square for Hardy–Weinberg</CAPTION>
<TBODY><TR>
<TH colspan="2" rowspan="2"></TH>
<TH colspan="2">Females</TH>
</TR>
<TR>
<TH>A (<I>p</I>)</TH>
<TH>a (<I>q</I>)</TH>
</TR>
<TR>
<TH rowspan="2">Males</TH>
<TH>A (<I>p</I>)</TH>
<TD>AA (<I>p</I><SUP>2</SUP>)</TD>
<TD>Aa (<I>pq</I>)</TD>
</TR>
<TR>
<TH>a (<I>q</I>)</TH>
<TD>Aa (<I>qp</I>)</TD>
<TD>aa (<I>q</I><SUP>2</SUP>)</TD>
</TR>
</TBODY></TABLE>
<P>The sum of the entries is <SPAN class="texhtml"><I>p</I><SUP>2</SUP> + 2<I>pq</I> + <I>q</I><SUP>2</SUP> = 1</SPAN>, as the genotype frequencies must sum to one.</P>
<P>Note again that as <SPAN class="texhtml"><I>p</I> + <I>q</I> = 1</SPAN>, the binomial expansion of <SPAN class="texhtml">(<I>p</I> + <I>q</I>)<SUP>2</SUP> = <I>p</I><SUP>2</SUP> + 2<I>pq</I> + <I>q</I><SUP>2</SUP> = 1</SPAN> gives the same relationships.</P>
<P>Summing the elements of the Punnett square or the binomial expansion, we obtain the expected genotype proportions among the offspring after a single generation:</P>
<DL>
<DD>
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<TBODY><TR>
<TD style="vertical-align:middle; border:none; padding:0.08em;" class="nowrap">
<P style="margin:0;"><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>f</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>AA</mtext>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo>=</mo>
        <msup>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>=</mo>
        <msub>
          <mi>f</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>0</mn>
          </mrow>
        </msub>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>A</mtext>
        </mrow>
        <msup>
          <mo stretchy="false">)</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
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      </mstyle>
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    <annotation encoding="application/x-tex">{\displaystyle f_{1}({\text{AA}})=p^{2}=f_{0}({\text{A}})^{2}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4935b478d0196424ac04aa0cadee2fd4fde83c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:22.865ex; height:3.176ex;" alt="f_{1}({\text{AA}})=p^{2}=f_{0}({\text{A}})^{2}" /></SPAN></P>
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<P style="margin:0; font-size:4pt;"> </P>
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<P style="margin:0pt;"><B>(<SPAN id="math_3" class="reference nourlexpansion" style="font-weight:bold; font-style:italic;">3</SPAN>)</B></P>
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<P style="margin:0;"><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>f</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>Aa</mtext>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo>=</mo>
        <mi>p</mi>
        <mi>q</mi>
        <mo>+</mo>
        <mi>q</mi>
        <mi>p</mi>
        <mo>=</mo>
        <mn>2</mn>
        <mi>p</mi>
        <mi>q</mi>
        <mo>=</mo>
        <mn>2</mn>
        <msub>
          <mi>f</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>0</mn>
          </mrow>
        </msub>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>A</mtext>
        </mrow>
        <mo stretchy="false">)</mo>
        <msub>
          <mi>f</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>0</mn>
          </mrow>
        </msub>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>a</mtext>
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        <mo stretchy="false">)</mo>
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    <annotation encoding="application/x-tex">{\displaystyle f_{1}({\text{Aa}})=pq+qp=2pq=2f_{0}({\text{A}})f_{0}({\text{a}})}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c2a6d969adb260c55771e429d810e4063591e29" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:39.281ex; height:2.843ex;" alt="f_{1}({\text{Aa}})=pq+qp=2pq=2f_{0}({\text{A}})f_{0}({\text{a}})" /></SPAN></P>
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</TBODY></TABLE>
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<TD style="vertical-align:middle; border:none; padding:0.08em;" class="nowrap">
<P style="margin:0pt;"><B>(<SPAN id="math_4" class="reference nourlexpansion" style="font-weight:bold; font-style:italic;">4</SPAN>)</B></P>
</TD>
</TR>
</TBODY></TABLE>
</DD>
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<DL>
<DD>
<TABLE style="border-collapse:collapse; background:none; margin:0; border:none;">
<TBODY><TR>
<TD style="vertical-align:middle; border:none; padding:0.08em;" class="nowrap">
<P style="margin:0;"><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>f</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>aa</mtext>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo>=</mo>
        <msup>
          <mi>q</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>=</mo>
        <msub>
          <mi>f</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>0</mn>
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        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>a</mtext>
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        <msup>
          <mo stretchy="false">)</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
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    <annotation encoding="application/x-tex">{\displaystyle f_{1}({\text{aa}})=q^{2}=f_{0}({\text{a}})^{2}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1bcb9d7a8f34ed7d864b7a6571a57a2d7b5c700" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:21.033ex; height:3.176ex;" alt="f_{1}({\text{aa}})=q^{2}=f_{0}({\text{a}})^{2}" /></SPAN></P>
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<P style="margin:0;"></P>
<TABLE style="border-collapse:collapse; background:none; margin:0; border:none; width:99%;">
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<P style="margin:0; font-size:4pt;"> </P>
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<TD style="vertical-align:middle; border:none; padding:0.08em;" class="nowrap">
<P style="margin:0pt;"><B>(<SPAN id="math_5" class="reference nourlexpansion" style="font-weight:bold; font-style:italic;">5</SPAN>)</B></P>
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<P>These frequencies define the Hardy–Weinberg equilibrium. It should be mentioned that the genotype frequencies after the first generation need not equal the genotype frequencies from the initial generation, e.g. <SPAN class="texhtml"><I>f</I><SUB>1</SUB>(AA) ≠ <I>f</I><SUB>0</SUB>(AA)</SPAN>. However, the genotype frequencies for all <I>future</I> times will equal the Hardy–Weinberg frequencies, e.g. <SPAN class="texhtml"><I>f<SUB>t</SUB></I>(AA) = <I>f</I><SUB>1</SUB>(AA)</SPAN> for <SPAN class="texhtml"><I>t</I> > 1</SPAN>. This follows since the genotype frequencies of the next generation depend only on the allele frequencies of the current generation which, as calculated by equations (<B><A href="#math_1">1</A></B>) and (<B><A href="#math_2">2</A></B>), are preserved from the initial generation:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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            <mtr>
              <mtd>
                <msub>
                  <mi>f</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>1</mn>
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                </msub>
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                <mo stretchy="false">(</mo>
                <mrow class="MJX-TeXAtom-ORD">
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                  <mi>f</mi>
                  <mrow class="MJX-TeXAtom-ORD">
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                <mi>p</mi>
                <mo stretchy="false">(</mo>
                <mi>p</mi>
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                <mi>q</mi>
                <mo stretchy="false">)</mo>
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                <mi>p</mi>
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                  <mi>f</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>1</mn>
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                <mo stretchy="false">(</mo>
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                <mi></mi>
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                <msub>
                  <mi>f</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>1</mn>
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                <mo stretchy="false">(</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mtext>aa</mtext>
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                <mo stretchy="false">)</mo>
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                      <mn>1</mn>
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                  <mi>f</mi>
                  <mrow class="MJX-TeXAtom-ORD">
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                <mo stretchy="false">(</mo>
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                  <mi>q</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>2</mn>
                  </mrow>
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                <mo>+</mo>
                <mi>p</mi>
                <mi>q</mi>
                <mo>=</mo>
                <mi>q</mi>
                <mo stretchy="false">(</mo>
                <mi>p</mi>
                <mo>+</mo>
                <mi>q</mi>
                <mo stretchy="false">)</mo>
                <mo>=</mo>
                <mi>q</mi>
                <mo>=</mo>
                <msub>
                  <mi>f</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>0</mn>
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                <mo stretchy="false">(</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mtext>a</mtext>
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                <mo stretchy="false">)</mo>
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    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f_{1}({\text{A}})&=f_{1}({\text{AA}})+{\tfrac {1}{2}}f_{1}({\text{Aa}})=p^{2}+pq=p(p+q)=p=f_{0}({\text{A}})\\f_{1}({\text{a}})&=f_{1}({\text{aa}})+{\tfrac {1}{2}}f_{1}({\text{Aa}})=q^{2}+pq=q(p+q)=q=f_{0}({\text{a}})\end{aligned}}}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07e67c45267c671d9a282d2d308df507d910f1a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:63.572ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}f_{1}({\text{A}})&=f_{1}({\text{AA}})+{\tfrac {1}{2}}f_{1}({\text{Aa}})=p^{2}+pq=p(p+q)=p=f_{0}({\text{A}})\\f_{1}({\text{a}})&=f_{1}({\text{aa}})+{\tfrac {1}{2}}f_{1}({\text{Aa}})=q^{2}+pq=q(p+q)=q=f_{0}({\text{a}})\end{aligned}}}" /></SPAN></DD>
</DL>
<P>For the more general case of <A href="/wiki/Dioecious" class="mw-redirect" title="Dioecious">dioecious</A> <A href="/wiki/Diploids" class="mw-redirect" title="Diploids">diploids</A> [organisms are either male or female] that reproduce by random mating of individuals, it is necessary to calculate the genotype frequencies from the nine possible matings between each parental genotype (<I>AA</I>, <I>Aa</I>, and <I>aa</I>) in either sex, weighted by the expected genotype contributions of each such mating.<SUP id="cite_ref-2" class="reference"><A href="#cite_note-2">[2]</A></SUP> Equivalently, one considers the six unique diploid-diploid combinations:</P>
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    <annotation encoding="application/x-tex">{\displaystyle \left[({\text{AA}},{\text{AA}}),({\text{AA}},{\text{Aa}}),({\text{AA}},{\text{aa}}),({\text{Aa}},{\text{Aa}}),({\text{Aa}},{\text{aa}}),({\text{aa}},{\text{aa}})\right]}</annotation>
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</DL>
<P>and constructs a Punnett square for each, so as to calculate its contribution to the next generation's genotypes. These contributions are weighted according to the probability of each diploid-diploid combination, which follows a <A href="/wiki/Multinomial_distribution" title="Multinomial distribution">multinomial distribution</A> with <SPAN class="texhtml"><I>k</I> = 3</SPAN>. For example, the probability of the mating combination <SPAN class="texhtml">(AA,aa)</SPAN> is <SPAN class="texhtml">2 <I>f</I><SUB><I>t</I></SUB>(AA)<I>f</I><SUB><I>t</I></SUB>(aa)</SPAN> and it can only result in the <SPAN class="texhtml">Aa</SPAN> genotype: <SPAN class="texhtml">[0,1,0]</SPAN>. Overall, the resulting genotype frequencies are calculated as:</P>
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    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\left[f_{t+1}({\text{AA}}),f_{t+1}({\text{Aa}}),f_{t+1}({\text{aa}})\right]=\\&\qquad =f_{t}({\text{AA}})f_{t}({\text{AA}})\left[1,0,0\right]+2f_{t}({\text{AA}})f_{t}({\text{Aa}})\left[{\tfrac {1}{2}},{\tfrac {1}{2}},0\right]+2f_{t}({\text{AA}})f_{t}({\text{aa}})\left[0,1,0\right]\\&\qquad \qquad +f_{t}({\text{Aa}})f_{t}({\text{Aa}})\left[{\tfrac {1}{4}},{\tfrac {1}{2}},{\tfrac {1}{4}}\right]+2f_{t}({\text{Aa}})f_{t}({\text{aa}})\left[0,{\tfrac {1}{2}},{\tfrac {1}{2}}\right]+f_{t}({\text{aa}})f_{t}({\text{aa}})\left[0,0,1\right]\\&\qquad =\left[\left(f_{t}({\text{AA}})+{\tfrac {1}{2}}f_{t}({\text{Aa}})\right)^{2},2\left(f_{t}({\text{AA}})+{\tfrac {1}{2}}f_{t}({\text{Aa}})\right)\left(f_{t}({\text{aa}})+{\tfrac {1}{2}}f_{t}({\text{Aa}})\right),\left(f_{t}({\text{aa}})+{\tfrac {1}{2}}f_{t}({\text{Aa}})\right)^{2}\right]\\&\qquad =\left[f_{t}({\text{A}})^{2},2f_{t}({\text{A}})f_{t}({\text{a}}),f_{t}({\text{a}})^{2}\right]\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32ff07f926253f3d0f6094c227d5da522836777c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -10.838ex; width:100.696ex; height:22.843ex;" alt="{\displaystyle {\begin{aligned}&\left[f_{t+1}({\text{AA}}),f_{t+1}({\text{Aa}}),f_{t+1}({\text{aa}})\right]=\\&\qquad =f_{t}({\text{AA}})f_{t}({\text{AA}})\left[1,0,0\right]+2f_{t}({\text{AA}})f_{t}({\text{Aa}})\left[{\tfrac {1}{2}},{\tfrac {1}{2}},0\right]+2f_{t}({\text{AA}})f_{t}({\text{aa}})\left[0,1,0\right]\\&\qquad \qquad +f_{t}({\text{Aa}})f_{t}({\text{Aa}})\left[{\tfrac {1}{4}},{\tfrac {1}{2}},{\tfrac {1}{4}}\right]+2f_{t}({\text{Aa}})f_{t}({\text{aa}})\left[0,{\tfrac {1}{2}},{\tfrac {1}{2}}\right]+f_{t}({\text{aa}})f_{t}({\text{aa}})\left[0,0,1\right]\\&\qquad =\left[\left(f_{t}({\text{AA}})+{\tfrac {1}{2}}f_{t}({\text{Aa}})\right)^{2},2\left(f_{t}({\text{AA}})+{\tfrac {1}{2}}f_{t}({\text{Aa}})\right)\left(f_{t}({\text{aa}})+{\tfrac {1}{2}}f_{t}({\text{Aa}})\right),\left(f_{t}({\text{aa}})+{\tfrac {1}{2}}f_{t}({\text{Aa}})\right)^{2}\right]\\&\qquad =\left[f_{t}({\text{A}})^{2},2f_{t}({\text{A}})f_{t}({\text{a}}),f_{t}({\text{a}})^{2}\right]\end{aligned}}}" /></SPAN></DD>
</DL>
<P>As before, one can show that the allele frequencies at time <SPAN class="texhtml"><I>t</I>+1</SPAN> equal those at time <SPAN class="texhtml"><I>t</I></SPAN>, and so, are constant in time. Similarly, the genotype frequencies depend only on the allele frequencies, and so, after time <SPAN class="texhtml"><I>t</I>=1</SPAN> are also constant in time.</P>
<P>If in either <A href="/wiki/Monoecious" class="mw-redirect" title="Monoecious">monoecious</A> or <A href="/wiki/Dioecious" class="mw-redirect" title="Dioecious">dioecious</A> organisms, either the allele or genotype proportions are initially unequal in either sex, it can be shown that constant proportions are obtained after one generation of random mating. If <A href="/wiki/Dioecious" class="mw-redirect" title="Dioecious">dioecious</A> organisms are <A href="/wiki/Heterogametic" class="mw-redirect" title="Heterogametic">heterogametic</A> and the gene locus is located on the <A href="/wiki/X_chromosome" title="X chromosome">X chromosome</A>, it can be shown that if the allele frequencies are initially unequal in the two sexes [<I>e.g</I>., XX females and XY males, as in humans], <SPAN class="texhtml"><VAR>f</VAR>′(a)</SPAN> in the <A href="/wiki/Heterogametic" class="mw-redirect" title="Heterogametic">heterogametic</A> sex ‘chases’ <SPAN class="texhtml"><VAR>f</VAR>(a)</SPAN> in the <A href="/wiki/Homogametic" class="mw-redirect" title="Homogametic">homogametic</A> sex of the previous generation, until an equilibrium is reached at the weighted average of the two initial frequencies.</P>
<H2><SPAN id="Deviations_from_Hardy–Weinberg_equilibrium"></SPAN><SPAN class="mw-headline" id="Deviations_from_Hardy.E2.80.93Weinberg_equilibrium">Deviations from Hardy–Weinberg equilibrium</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=2" title="Edit section: Deviations from Hardy–Weinberg equilibrium">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P>The seven assumptions underlying Hardy–Weinberg equilibrium are as follows:<SUP id="cite_ref-Hartl2007_3-0" class="reference"><A href="#cite_note-Hartl2007-3">[3]</A></SUP></P>
<UL>
<LI>organisms are diploid</LI>
<LI>only sexual reproduction occurs</LI>
<LI>generations are nonoverlapping</LI>
<LI>mating is random</LI>
<LI>population size is infinitely large</LI>
<LI>allele frequencies are equal in the sexes</LI>
<LI>there is no migration, mutation or selection</LI>
</UL>
<P>Violations of the Hardy–Weinberg assumptions can cause deviations from <A href="/wiki/Expected_value" title="Expected value">expectation</A>. How this affects the population depends on the assumptions that are violated.</P>
<UL>
<LI><A href="/wiki/Random_mating" class="mw-redirect" title="Random mating">Random mating</A>. The HWP states the population will have the given genotypic frequencies (called Hardy–Weinberg proportions) after a single generation of random mating within the population. When the random mating assumption is violated, the population will not have Hardy–Weinberg proportions. A common cause of non-random mating is <A href="/wiki/Inbreeding" title="Inbreeding">inbreeding</A>, which causes an increase in <A href="/wiki/Zygosity" title="Zygosity">homozygosity</A> for all genes.</LI>
</UL>
<P>If a population violates one of the following four assumptions, the population may continue to have Hardy–Weinberg proportions each generation, but the allele frequencies will change over time.</P>
<UL>
<LI><A href="/wiki/Natural_selection" title="Natural selection">Selection</A>, in general, causes allele frequencies to change, often quite rapidly. While <A href="/wiki/Directional_selection" title="Directional selection">directional selection</A> eventually leads to the loss of all alleles except the favored one (unless one allele is dominant, in which case recessive alleles can survive at low frequencies), some forms of selection, such as <A href="/wiki/Balancing_selection" title="Balancing selection">balancing selection</A>, lead to equilibrium without loss of alleles.</LI>
<LI><A href="/wiki/Mutation" title="Mutation">Mutation</A> will have a very subtle effect on allele frequencies. Mutation rates are of the order 10<SUP>−4</SUP> to 10<SUP>−8</SUP>, and the change in allele frequency will be, at most, the same order. Recurrent mutation will maintain alleles in the population, even if there is strong selection against them.</LI>
<LI>Migration genetically links two or more populations together. In general, allele frequencies will become more homogeneous among the populations. Some models for migration inherently include nonrandom mating (<A href="/wiki/Wahlund_effect" title="Wahlund effect">Wahlund effect</A>, for example). For those models, the Hardy–Weinberg proportions will normally not be valid.</LI>
<LI><A href="/wiki/Small_population_size" title="Small population size">Small population size</A> can cause a random change in allele frequencies. This is due to a sampling effect, and is called <A href="/wiki/Genetic_drift" title="Genetic drift">genetic drift</A>. Sampling effects are most important when the allele is present in a small number of copies.</LI>
</UL>
<H2><SPAN class="mw-headline" id="Sex_linkage">Sex linkage</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=3" title="Edit section: Sex linkage">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P>Where the A gene is <A href="/wiki/Sex_linkage" title="Sex linkage">sex linked</A>, the heterogametic sex (<I>e.g.</I>, mammalian males; avian females) have only one copy of the gene (and are termed hemizygous), while the homogametic sex (<I>e.g.</I>, <A href="/wiki/Human" title="Human">human</A> females) have two copies. The genotype frequencies at equilibrium are <I>p</I> and <I>q</I> for the heterogametic sex but <I>p</I><SUP>2</SUP>, 2<I>pq</I> and <I>q</I><SUP>2</SUP> for the homogametic sex.</P>
<P>For example, in humans <A href="/wiki/Red%E2%80%93green_colorblindness" class="mw-redirect" title="Red–green colorblindness">red–green colorblindness</A> is an X-linked recessive trait. In western European males, the trait affects about 1 in 12, (<I>q</I> = 0.083) whereas it affects about 1 in 200 females (0.005, compared to <I>q</I><SUP>2</SUP> = 0.007), very close to Hardy–Weinberg proportions.</P>
<P>If a population is brought together with males and females with a different allele frequency in each subpopulation (males or females), the allele frequency of the male population in the next generation will follow that of the female population because each son receives its X chromosome from its mother. The population converges on equilibrium very quickly.</P>
<H2><SPAN class="mw-headline" id="Generalizations">Generalizations</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=4" title="Edit section: Generalizations">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P>The simple derivation above can be generalized for more than two alleles and <A href="/wiki/Polyploidy" class="mw-redirect" title="Polyploidy">polyploidy</A>.</P>
<H3><SPAN class="mw-headline" id="Generalization_for_more_than_two_alleles">Generalization for more than two alleles</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=5" title="Edit section: Generalization for more than two alleles">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:222px;"><A href="/wiki/File:Hardy%E2%80%93Weinberg_law_-_Punnett_square2.svg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Hardy%E2%80%93Weinberg_law_-_Punnett_square2.svg/220px-Hardy%E2%80%93Weinberg_law_-_Punnett_square2.svg.png" width="220" height="118" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Hardy%E2%80%93Weinberg_law_-_Punnett_square2.svg/330px-Hardy%E2%80%93Weinberg_law_-_Punnett_square2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/25/Hardy%E2%80%93Weinberg_law_-_Punnett_square2.svg/440px-Hardy%E2%80%93Weinberg_law_-_Punnett_square2.svg.png 2x" data-file-width="1057" data-file-height="567" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Hardy%E2%80%93Weinberg_law_-_Punnett_square2.svg" class="internal" title="Enlarge"></A></DIV>
Punnett square for three-allele case (left) and four-allele case (right). White areas are homozygotes. Colored areas are heterozygotes.</DIV>
</DIV>
</DIV>
<P>Consider an extra allele frequency, <I>r</I>. The two-allele case is the <A href="/wiki/Binomial_theorem" title="Binomial theorem">binomial expansion</A> of (<I>p</I> + <I>q</I>)<SUP>2</SUP>, and thus the three-allele case is the trinomial expansion of (<I>p</I> + <I>q</I>+ <I>r</I>)<SUP>2</SUP>.</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mo stretchy="false">(</mo>
        <mi>p</mi>
        <mo>+</mo>
        <mi>q</mi>
        <mo>+</mo>
        <mi>r</mi>
        <msup>
          <mo stretchy="false">)</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>=</mo>
        <msup>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
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        </msup>
        <mo>+</mo>
        <msup>
          <mi>q</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>+</mo>
        <msup>
          <mi>r</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>+</mo>
        <mn>2</mn>
        <mi>p</mi>
        <mi>q</mi>
        <mo>+</mo>
        <mn>2</mn>
        <mi>p</mi>
        <mi>r</mi>
        <mo>+</mo>
        <mn>2</mn>
        <mi>q</mi>
        <mi>r</mi>
        <mspace width="thinmathspace"></mspace>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle (p+q+r)^{2}=p^{2}+q^{2}+r^{2}+2pq+2pr+2qr\,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64c82520d2533a5f2586c8eba68341cdb597704a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:46.333ex; height:3.176ex;" alt="(p+q+r)^{2}=p^{2}+q^{2}+r^{2}+2pq+2pr+2qr\," /></SPAN></DD>
</DL>
<P>More generally, consider the alleles A<SUB>1</SUB>, ..., A<SUB><I>n</I></SUB> given by the allele frequencies <I>p</I><SUB>1</SUB> to <I>p</I><SUB><I>n</I></SUB>;</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mo stretchy="false">(</mo>
        <msub>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo>+</mo>
        <mo>⋯<!-- ⋯ --></mo>
        <mo>+</mo>
        <msub>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>n</mi>
          </mrow>
        </msub>
        <msup>
          <mo stretchy="false">)</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mspace width="thinmathspace"></mspace>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle (p_{1}+\cdots +p_{n})^{2}\,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6505d35a6461b80c0803b7f7bd3dce076d423267" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:16.362ex; height:3.176ex;" alt="(p_{1}+\cdots +p_{n})^{2}\," /></SPAN></DD>
</DL>
<P>giving for all <A href="/wiki/Zygosity" title="Zygosity">homozygotes</A>:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>f</mi>
        <mo stretchy="false">(</mo>
        <msub>
          <mi>A</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>i</mi>
          </mrow>
        </msub>
        <msub>
          <mi>A</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>i</mi>
          </mrow>
        </msub>
        <mo stretchy="false">)</mo>
        <mo>=</mo>
        <msubsup>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>i</mi>
          </mrow>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msubsup>
        <mspace width="thinmathspace"></mspace>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle f(A_{i}A_{i})=p_{i}^{2}\,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65d3215d85f544a8e5fe1d25414a16b43e0a7742" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:13.978ex; height:3.176ex;" alt="f(A_{i}A_{i})=p_{i}^{2}\," /></SPAN></DD>
</DL>
<P>and for all <A href="/wiki/Zygosity" title="Zygosity">heterozygotes</A>:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>f</mi>
        <mo stretchy="false">(</mo>
        <msub>
          <mi>A</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>i</mi>
          </mrow>
        </msub>
        <msub>
          <mi>A</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>j</mi>
          </mrow>
        </msub>
        <mo stretchy="false">)</mo>
        <mo>=</mo>
        <mn>2</mn>
        <msub>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>i</mi>
          </mrow>
        </msub>
        <msub>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>j</mi>
          </mrow>
        </msub>
        <mspace width="thinmathspace"></mspace>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle f(A_{i}A_{j})=2p_{i}p_{j}\,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88cd7cd0811d4dfc4e9dcaddacd075fb925604a0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:17.103ex; height:3.009ex;" alt="f(A_{i}A_{j})=2p_{i}p_{j}\," /></SPAN></DD>
</DL>
<H3><SPAN class="mw-headline" id="Generalization_for_polyploidy">Generalization for polyploidy</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=6" title="Edit section: Generalization for polyploidy">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>The Hardy–Weinberg principle may also be generalized to <A href="/wiki/Polyploid" title="Polyploid">polyploid</A> systems, that is, for organisms that have more than two copies of each chromosome. Consider again only two alleles. The diploid case is the <A href="/wiki/Binomial_theorem" title="Binomial theorem">binomial expansion</A> of:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mo stretchy="false">(</mo>
        <mi>p</mi>
        <mo>+</mo>
        <mi>q</mi>
        <msup>
          <mo stretchy="false">)</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mspace width="thinmathspace"></mspace>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle (p+q)^{2}\,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a800aa7e9ed1093374cd7408d596d3f8b96a0c8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.39ex; height:3.176ex;" alt="(p+q)^{2}\," /></SPAN></DD>
</DL>
<P>and therefore the polyploid case is the <A href="/wiki/Polynomial_expansion" title="Polynomial expansion">polynomial expansion</A> of:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mo stretchy="false">(</mo>
        <mi>p</mi>
        <mo>+</mo>
        <mi>q</mi>
        <msup>
          <mo stretchy="false">)</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>c</mi>
          </mrow>
        </msup>
        <mspace width="thinmathspace"></mspace>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle (p+q)^{c}\,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e21233666d184fd8713fbf56242922e5b36afe24" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.28ex; height:2.843ex;" alt="(p+q)^{c}\," /></SPAN></DD>
</DL>
<P>where <I>c</I> is the <A href="/wiki/Ploidy" title="Ploidy">ploidy</A>, for example with tetraploid (<I>c</I> = 4):</P>
<TABLE class="wikitable">
<CAPTION>Table 2: Expected genotype frequencies for tetraploidy</CAPTION>
<TBODY><TR>
<TH>Genotype</TH>
<TH>Frequency</TH>
</TR>
<TR>
<TD>AAAA</TD>
<TD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msup>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>4</mn>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle p^{4}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d25f38cfd691b87f67bbf2a4c2ad98c4893320" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.079ex; width:2.32ex; height:3.009ex;" alt="p^{4}" /></SPAN></TD>
</TR>
<TR>
<TD>AAAa</TD>
<TD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>4</mn>
        <msup>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>3</mn>
          </mrow>
        </msup>
        <mi>q</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 4p^{3}q}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dcf3fc8cdcf09211a74a1c2b3195325eb046abe" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.494ex; height:3.009ex;" alt="4p^{3}q" /></SPAN></TD>
</TR>
<TR>
<TD>AAaa</TD>
<TD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>6</mn>
        <msup>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <msup>
          <mi>q</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 6p^{2}q^{2}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddf51cf6f702ca68fc2c2e42a5cec8c4c0abb6d3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:5.566ex; height:3.009ex;" alt="6p^{2}q^{2}" /></SPAN></TD>
</TR>
<TR>
<TD>Aaaa</TD>
<TD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>4</mn>
        <mi>p</mi>
        <msup>
          <mi>q</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>3</mn>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 4pq^{3}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6de43160964d1223ba8054872cb30e372a050c2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.504ex; height:3.009ex;" alt="4pq^{3}" /></SPAN></TD>
</TR>
<TR>
<TD>aaaa</TD>
<TD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msup>
          <mi>q</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>4</mn>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle q^{4}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57b23fe1fcef55aa4fa4406b52cb19c9746bc531" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.152ex; height:3.009ex;" alt="q^{4}" /></SPAN></TD>
</TR>
</TBODY></TABLE>
<P>Depending on whether the organism is a 'true' tetraploid or an amphidiploid will determine how long it will take for the population to reach Hardy–Weinberg equilibrium.</P>
<H3><SPAN class="mw-headline" id="Complete_generalization">Complete generalization</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=7" title="Edit section: Complete generalization">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>For <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>n</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle n}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.405ex; height:1.676ex;" alt="n" /></SPAN> distinct alleles in <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>c</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle c}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.017ex; height:1.676ex;" alt="c" /></SPAN>-ploids, the genotype frequencies in the Hardy–Weinberg equilibrium are given by individual terms in the <A href="/wiki/Multinomial_theorem" title="Multinomial theorem">multinomial expansion</A> of <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mo stretchy="false">(</mo>
        <msub>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo>+</mo>
        <mo>⋯<!-- ⋯ --></mo>
        <mo>+</mo>
        <msub>
          <mi>p</mi>
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            <mi>n</mi>
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          <mo stretchy="false">)</mo>
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            <mi>c</mi>
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      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle (p_{1}+\cdots +p_{n})^{c}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546f3897b414bfb6bd0f55a69a7bc614eb06ff7c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:15.864ex; height:2.843ex;" alt="(p_{1}+\cdots +p_{n})^{c}" /></SPAN>:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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          <mo>∑<!-- ∑ --></mo>
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            <mo>,</mo>
            <msub>
              <mi>k</mi>
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                <mi>n</mi>
              </mrow>
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            <mo>∈<!-- ∈ --></mo>
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              <mi mathvariant="double-struck">N</mi>
            </mrow>
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          <mi>p</mi>
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          </mrow>
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    <annotation encoding="application/x-tex">{\displaystyle (p_{1}+\cdots +p_{n})^{c}=\sum _{k_{1},\ldots ,k_{n}\ \in \mathbb {N} :k_{1}+\cdots +k_{n}=c}{c \choose k_{1},\ldots ,k_{n}}p_{1}^{k_{1}}\cdots p_{n}^{k_{n}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56355b9e505b81f4721cad0f3b232a745855cc52" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.505ex; width:62.399ex; height:7.176ex;" alt="(p_{1}+\cdots +p_{n})^{c}=\sum _{{k_{1},\ldots ,k_{n}\ \in {\mathbb  {N}}:k_{1}+\cdots +k_{n}=c}}{c \choose k_{1},\ldots ,k_{n}}p_{1}^{{k_{1}}}\cdots p_{n}^{{k_{n}}}" /></SPAN></DD>
</DL>
<H2><SPAN class="mw-headline" id="Applications">Applications</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=8" title="Edit section: Applications">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P>The Hardy–Weinberg principle may be applied in two ways, either a population is assumed to be in Hardy–Weinberg proportions, in which the genotype frequencies can be calculated, or if the genotype frequencies of all three genotypes are known, they can be tested for deviations that are <A href="/wiki/Statistically_significant" class="mw-redirect" title="Statistically significant">statistically significant</A>.</P>
<H3><SPAN class="mw-headline" id="Application_to_cases_of_complete_dominance">Application to cases of complete dominance</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=9" title="Edit section: Application to cases of complete dominance">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>Suppose that the phenotypes of AA and Aa are indistinguishable, <I>i.e.</I>, there is complete <A href="/wiki/Dominance_relationship" class="mw-redirect" title="Dominance relationship">dominance</A>. Assuming that the Hardy–Weinberg principle applies to the population, then <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
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        <mi>q</mi>
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  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.08ex; height:2.009ex;" alt="q" /></SPAN> can still be calculated from <I>f</I>(aa):</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
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          <msqrt>
            <mi>f</mi>
            <mo stretchy="false">(</mo>
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              <mtext>aa</mtext>
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            <mo stretchy="false">)</mo>
          </msqrt>
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    <annotation encoding="application/x-tex">{\displaystyle q={\sqrt {f({\text{aa}})}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de40bebee61cc75a5c7b2787ec9111e69d102425" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:11.988ex; height:4.843ex;" alt="q={\sqrt  {f({\text{aa}})}}" /></SPAN></DD>
</DL>
<P>and <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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        <mi>p</mi>
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    <annotation encoding="application/x-tex">{\displaystyle p}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.079ex; width:1.259ex; height:2.009ex;" alt="p" /></SPAN> can be calculated from <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>q</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle q}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.08ex; height:2.009ex;" alt="q" /></SPAN>. And thus an estimate of <I>f</I>(AA) and <I>f</I>(Aa) derived from <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msup>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle p^{2}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef685027b97072ee63a8c738f395cd40f63767e1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.079ex; width:2.32ex; height:3.009ex;" alt="p^{2}" /></SPAN> and <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>2</mn>
        <mi>p</mi>
        <mi>q</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 2pq}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78517b9af890ce1ef8e703f2bb77d8dcda4c22f4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.433ex; height:2.509ex;" alt="2pq" /></SPAN> respectively. Note however, such a population cannot be tested for equilibrium using the significance tests below because it is assumed <I><A href="/wiki/A_priori_and_a_posteriori" title="A priori and a posteriori">a priori</A></I>.</P>
<H2><SPAN class="mw-headline" id="Significance_tests_for_deviation">Significance tests for deviation</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=10" title="Edit section: Significance tests for deviation">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P>Testing deviation from the HWP is generally performed using <A href="/wiki/Karl_Pearson" title="Karl Pearson">Pearson's</A> <A href="/wiki/Pearson%27s_chi-squared_test" title="Pearson's chi-squared test">chi-squared test</A>, using the observed genotype frequencies obtained from the data and the expected genotype frequencies obtained using the HWP. For systems where there are large numbers of alleles, this may result in data with many empty possible genotypes and low genotype counts, because there are often not enough individuals present in the sample to adequately represent all genotype classes. If this is the case, then the <A href="/wiki/Asymptotic" class="mw-redirect" title="Asymptotic">asymptotic</A> assumption of the <A href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</A>, will no longer hold, and it may be necessary to use a form of <A href="/wiki/Ronald_Fisher" title="Ronald Fisher">Fisher's</A> <A href="/wiki/Fisher%27s_exact_test" title="Fisher's exact test">exact test</A>, which requires a <A href="/wiki/Computer" title="Computer">computer</A> to solve. More recently a number of <A href="/wiki/Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">MCMC</A> methods of testing for deviations from HWP have been proposed (Guo & Thompson, 1992; Wigginton <I>et al.</I> 2005)</P>
<H3><SPAN id="Example_'"`UNIQ--postMath-0000001E-QINU`"'_test_for_deviation"></SPAN><SPAN class="mw-headline" id="Example_.7F.27.22.60UNIQ--postMath-0000001E-QINU.60.22.27.7F_test_for_deviation">Example <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle \chi ^{2}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c0cc9237ec72a1da6d18bc8e7fb24cdda43a49a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.527ex; height:3.009ex;" alt="\chi ^{2}" /></SPAN> test for deviation</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=11" title="Edit section: Example  $\chi ^{2} test for deviation">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>This data is from <A href="/wiki/E.B._Ford" class="mw-redirect" title="E.B. Ford">E.B. Ford</A> (1971) on the <A href="/wiki/Scarlet_tiger_moth" title="Scarlet tiger moth">Scarlet tiger moth</A>, for which the <A href="/wiki/Phenotype" title="Phenotype">phenotypes</A> of a sample of the population were recorded. <A href="/wiki/Genotype-phenotype_distinction" class="mw-redirect" title="Genotype-phenotype distinction">Genotype-phenotype distinction</A> is assumed to be negligibly small. The <A href="/wiki/Null_hypothesis" title="Null hypothesis">null hypothesis</A> is that the population is in Hardy–Weinberg proportions, and the <A href="/wiki/Alternative_hypothesis" title="Alternative hypothesis">alternative hypothesis</A> is that the population is not in Hardy–Weinberg proportions.</P>
<TABLE class="wikitable">
<CAPTION>Table 3: Example Hardy–Weinberg principle calculation</CAPTION>
<TBODY><TR>
<TH>Phenotype</TH>
<TD>White-spotted (AA)</TD>
<TD>Intermediate (Aa)</TD>
<TD>Little spotting (aa)</TD>
<TH>Total</TH>
</TR>
<TR>
<TH>Number</TH>
<TD>1469</TD>
<TD>138</TD>
<TD>5</TD>
<TD>1612</TD>
</TR>
</TBODY></TABLE>
<P>From this, allele frequencies can be calculated:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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            <mtr>
              <mtd>
                <mi>p</mi>
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              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mfrac>
                    <mrow>
                      <mn>2</mn>
                      <mo>×<!-- × --></mo>
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                        <mi mathvariant="normal">o</mi>
                        <mi mathvariant="normal">b</mi>
                        <mi mathvariant="normal">s</mi>
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                      <mo stretchy="false">(</mo>
                      <mrow class="MJX-TeXAtom-ORD">
                        <mtext>AA</mtext>
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                        <mi mathvariant="normal">o</mi>
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                      <mo stretchy="false">(</mo>
                      <mrow class="MJX-TeXAtom-ORD">
                        <mtext>Aa</mtext>
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                      <mo stretchy="false">)</mo>
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                    <mrow>
                      <mn>2</mn>
                      <mo>×<!-- × --></mo>
                      <mo stretchy="false">(</mo>
                      <mrow class="MJX-TeXAtom-ORD">
                        <mi mathvariant="normal">o</mi>
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                      <mo stretchy="false">(</mo>
                      <mrow class="MJX-TeXAtom-ORD">
                        <mtext>AA</mtext>
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                      <mo stretchy="false">)</mo>
                      <mo>+</mo>
                      <mrow class="MJX-TeXAtom-ORD">
                        <mi mathvariant="normal">o</mi>
                        <mi mathvariant="normal">b</mi>
                        <mi mathvariant="normal">s</mi>
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                      <mo stretchy="false">(</mo>
                      <mrow class="MJX-TeXAtom-ORD">
                        <mtext>Aa</mtext>
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                      <mo stretchy="false">)</mo>
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                        <mi mathvariant="normal">o</mi>
                        <mi mathvariant="normal">b</mi>
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                      <mo stretchy="false">(</mo>
                      <mrow class="MJX-TeXAtom-ORD">
                        <mtext>aa</mtext>
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                      <mo stretchy="false">)</mo>
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                </mrow>
              </mtd>
            </mtr>
            <mtr>
              <mtd></mtd>
            </mtr>
            <mtr>
              <mtd></mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mfrac>
                    <mrow>
                      <mn>1469</mn>
                      <mo>×<!-- × --></mo>
                      <mn>2</mn>
                      <mo>+</mo>
                      <mn>138</mn>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                      <mo>×<!-- × --></mo>
                      <mo stretchy="false">(</mo>
                      <mn>1469</mn>
                      <mo>+</mo>
                      <mn>138</mn>
                      <mo>+</mo>
                      <mn>5</mn>
                      <mo stretchy="false">)</mo>
                    </mrow>
                  </mfrac>
                </mrow>
              </mtd>
            </mtr>
            <mtr>
              <mtd></mtd>
            </mtr>
            <mtr>
              <mtd></mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mfrac>
                    <mn>3076</mn>
                    <mn>3224</mn>
                  </mfrac>
                </mrow>
              </mtd>
            </mtr>
            <mtr>
              <mtd></mtd>
            </mtr>
            <mtr>
              <mtd></mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mn>0.954</mn>
              </mtd>
            </mtr>
          </mtable>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p&={2\times \mathrm {obs} ({\text{AA}})+\mathrm {obs} ({\text{Aa}}) \over 2\times (\mathrm {obs} ({\text{AA}})+\mathrm {obs} ({\text{Aa}})+\mathrm {obs} ({\text{aa}}))}\\\\&={1469\times 2+138 \over 2\times (1469+138+5)}\\\\&={3076 \over 3224}\\\\&=0.954\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/340d32ea06aa830911f744e1ede67ab6d02d2e58" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -14.671ex; width:41.91ex; height:30.509ex;" alt="{\begin{aligned}p&={2\times {\mathrm  {obs}}({\text{AA}})+{\mathrm  {obs}}({\text{Aa}}) \over 2\times ({\mathrm  {obs}}({\text{AA}})+{\mathrm  {obs}}({\text{Aa}})+{\mathrm  {obs}}({\text{aa}}))}\\\\&={1469\times 2+138 \over 2\times (1469+138+5)}\\\\&={3076 \over 3224}\\\\&=0.954\end{aligned}}" /></SPAN></DD>
</DL>
<P>and</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
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        <mrow class="MJX-TeXAtom-ORD">
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            <mtr>
              <mtd>
                <mi>q</mi>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mn>1</mn>
                <mo>−<!-- − --></mo>
                <mi>p</mi>
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            <mtr>
              <mtd></mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
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                <mo>−<!-- − --></mo>
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            <mtr>
              <mtd></mtd>
              <mtd>
                <mi></mi>
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                <mn>0.046</mn>
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    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}q&=1-p\\&=1-0.954\\&=0.046\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c8bc5c240ae76c2e06f11ade5309a5ed323924" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.838ex; width:14.313ex; height:8.843ex;" alt="{\begin{aligned}q&=1-p\\&=1-0.954\\&=0.046\end{aligned}}" /></SPAN></DD>
</DL>
<P>So the Hardy–Weinberg <A href="/wiki/Expected_value" title="Expected value">expectation</A> is:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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            <mtr>
              <mtd>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="normal">E</mi>
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              <mtd>
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                  <mi mathvariant="normal">E</mi>
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                <mo>=</mo>
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                <mn>1612</mn>
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              <mtd>
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                  <mrow class="MJX-TeXAtom-ORD">
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                <mo>=</mo>
                <mn>3.4</mn>
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            </mtr>
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {Exp} ({\text{AA}})&=p^{2}n=0.954^{2}\times 1612=1467.4\\\mathrm {Exp} ({\text{Aa}})&=2pqn=2\times 0.954\times 0.046\times 1612=141.2\\\mathrm {Exp} ({\text{aa}})&=q^{2}n=0.046^{2}\times 1612=3.4\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d2e2e61519c92259035edfa62c6bcde414cbc7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -4.338ex; width:54.852ex; height:9.843ex;" alt="{\begin{aligned}{\mathrm  {Exp}}({\text{AA}})&=p^{2}n=0.954^{2}\times 1612=1467.4\\{\mathrm  {Exp}}({\text{Aa}})&=2pqn=2\times 0.954\times 0.046\times 1612=141.2\\{\mathrm  {Exp}}({\text{aa}})&=q^{2}n=0.046^{2}\times 1612=3.4\end{aligned}}" /></SPAN></DD>
</DL>
<P><A href="/wiki/Pearson%27s_chi-squared_test" title="Pearson's chi-squared test">Pearson's chi-squared test</A> states:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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              <mtd>
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              <mtd></mtd>
              <mtd>
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                      <msup>
                        <mo stretchy="false">)</mo>
                        <mrow class="MJX-TeXAtom-ORD">
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                      <msup>
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                        <mrow class="MJX-TeXAtom-ORD">
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                      <msup>
                        <mo stretchy="false">)</mo>
                        <mrow class="MJX-TeXAtom-ORD">
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            </mtr>
            <mtr>
              <mtd></mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mn>0.001</mn>
                <mo>+</mo>
                <mn>0.073</mn>
                <mo>+</mo>
                <mn>0.756</mn>
              </mtd>
            </mtr>
            <mtr>
              <mtd></mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mn>0.83</mn>
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            </mtr>
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      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\chi ^{2}&=\sum {(O-E)^{2} \over E}\\&={(1469-1467.4)^{2} \over 1467.4}+{(138-141.2)^{2} \over 141.2}+{(5-3.4)^{2} \over 3.4}\\&=0.001+0.073+0.756\\&=0.83\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d596480c9b056e6073c24f66ef4a98ce2ff6478" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -8.505ex; width:56.083ex; height:18.176ex;" alt="{\begin{aligned}\chi ^{2}&=\sum {(O-E)^{2} \over E}\\&={(1469-1467.4)^{2} \over 1467.4}+{(138-141.2)^{2} \over 141.2}+{(5-3.4)^{2} \over 3.4}\\&=0.001+0.073+0.756\\&=0.83\end{aligned}}" /></SPAN></DD>
</DL>
<P>There is 1 <A href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">degree of freedom</A> (degrees of freedom for test for Hardy–Weinberg proportions are # genotypes − # alleles). The 5% <A href="/wiki/Significance_level" class="mw-redirect" title="Significance level">significance level</A> for 1 degree of freedom is 3.84, and since the χ<SUP>2</SUP> value is less than this, the <A href="/wiki/Null_hypothesis" title="Null hypothesis">null hypothesis</A> that the population is in Hardy–Weinberg frequencies is <B>not</B> rejected.</P>
<H3><SPAN id="Fisher's_exact_test_(probability_test)"></SPAN><SPAN class="mw-headline" id="Fisher.27s_exact_test_.28probability_test.29">Fisher's exact test (probability test)</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=12" title="Edit section: Fisher's exact test (probability test)">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P><A href="/wiki/Fisher%27s_exact_test" title="Fisher's exact test">Fisher's exact test</A> can be applied to testing for Hardy–Weinberg proportions. Since the test is conditional on the allele frequencies, <I>p</I> and <I>q</I>, the problem can be viewed as testing for the proper number of heterozygotes. In this way, the hypothesis of Hardy–Weinberg proportions is rejected if the number of heterozygotes is too large or too small. The conditional probabilities for the heterozygote, given the allele frequencies are given in Emigh (1980) as</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle \operatorname {prob} [n_{12}|n_{1}]={\frac {\binom {n}{n_{11},n_{12},n_{22}}}{\binom {2n}{n_{1},n_{2}}}}2^{n_{12}},}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2bd892cd424f7a6791bed592abf1e100469f34" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.671ex; width:31.542ex; height:8.343ex;" alt="\operatorname {prob}[n_{{12}}|n_{1}]={\frac  {{\binom  {n}{n_{{11}},n_{{12}},n_{{22}}}}}{{\binom  {2n}{n_{1},n_{2}}}}}2^{{n_{{12}}}}," /></SPAN></DD>
</DL>
<P>where <I>n</I><SUB>11</SUB>, <I>n</I><SUB>12</SUB>, <I>n</I><SUB>22</SUB> are the observed numbers of the three genotypes, AA, Aa, and aa, respectively, and <I>n</I><SUB>1</SUB> is the number of A alleles, where <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>n</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo>=</mo>
        <mn>2</mn>
        <msub>
          <mi>n</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>11</mn>
          </mrow>
        </msub>
        <mo>+</mo>
        <msub>
          <mi>n</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>12</mn>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle n_{1}=2n_{11}+n_{12}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd3ca5e89bf410e93dc12aac27dc394adbb612d7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:16.192ex; height:2.509ex;" alt="n_{1}=2n_{{11}}+n_{{12}}" /></SPAN>.</P>
<P><B>An example</B> Using one of the examples from Emigh (1980),<SUP id="cite_ref-emigh_1980_4-0" class="reference"><A href="#cite_note-emigh_1980-4">[4]</A></SUP> we can consider the case where <I>n</I> = 100, and <I>p</I> = 0.34. The possible observed heterozygotes and their exact significance level is given in Table 4.</P>
<TABLE class="wikitable">
<CAPTION>Table 4: Example of Fisher's exact test for <I>n</I> = 100, <I>p</I> = 0.34.<SUP id="cite_ref-emigh_1980_4-1" class="reference"><A href="#cite_note-emigh_1980-4">[4]</A></SUP></CAPTION>
<TBODY><TR>
<TH>Number of heterozygotes</TH>
<TH>Significance level</TH>
</TR>
<TR>
<TD>0</TD>
<TD>0.000</TD>
</TR>
<TR>
<TD>2</TD>
<TD>0.000</TD>
</TR>
<TR>
<TD>4</TD>
<TD>0.000</TD>
</TR>
<TR>
<TD>6</TD>
<TD>0.000</TD>
</TR>
<TR>
<TD>8</TD>
<TD>0.000</TD>
</TR>
<TR>
<TD>10</TD>
<TD>0.000</TD>
</TR>
<TR>
<TD>12</TD>
<TD>0.000</TD>
</TR>
<TR>
<TD>14</TD>
<TD>0.000</TD>
</TR>
<TR>
<TD>16</TD>
<TD>0.000</TD>
</TR>
<TR>
<TD>18</TD>
<TD>0.001</TD>
</TR>
<TR>
<TD>20</TD>
<TD>0.007</TD>
</TR>
<TR>
<TD>22</TD>
<TD>0.034</TD>
</TR>
<TR>
<TD>34</TD>
<TD>0.067</TD>
</TR>
<TR>
<TD>24</TD>
<TD>0.151</TD>
</TR>
<TR>
<TD>32</TD>
<TD>0.291</TD>
</TR>
<TR>
<TD>26</TD>
<TD>0.474</TD>
</TR>
<TR>
<TD>30</TD>
<TD>0.730</TD>
</TR>
<TR>
<TD>28</TD>
<TD>1.000</TD>
</TR>
</TBODY></TABLE>
<P>Using this table, one must look up the significance level of the test based on the observed number of heterozygotes. For example, if one observed 20 heterozygotes, the significance level for the test is 0.007. As is typical for Fisher's exact test for small samples, the gradation of significance levels is quite coarse.</P>
<P>However, a table like this has to be created for every experiment, since the tables are dependent on both <I>n</I> and <I>p</I>.</P>
<H2><SPAN class="mw-headline" id="Inbreeding_coefficient">Inbreeding coefficient</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=13" title="Edit section: Inbreeding coefficient">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P>The inbreeding coefficient, <I>F</I> (see also <A href="/wiki/F-statistics" title="F-statistics"><I>F</I>-statistics</A>), is one minus the observed frequency of heterozygotes over that expected from Hardy–Weinberg equilibrium.</P>
<DL>
<DD><SPAN class="nowrap"><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>F</mi>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <mi mathvariant="normal">E</mi>
              <mo><!--  --></mo>
              <mrow class="MJX-TeXAtom-ORD">
                <mo stretchy="false">(</mo>
                <mi>f</mi>
                <mo stretchy="false">(</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mtext>Aa</mtext>
                </mrow>
                <mo stretchy="false">)</mo>
                <mo stretchy="false">)</mo>
              </mrow>
              <mo>−<!-- − --></mo>
              <mi mathvariant="normal">O</mi>
              <mo><!--  --></mo>
              <mo stretchy="false">(</mo>
              <mi>f</mi>
              <mo stretchy="false">(</mo>
              <mrow class="MJX-TeXAtom-ORD">
                <mtext>Aa</mtext>
              </mrow>
              <mo stretchy="false">)</mo>
              <mo stretchy="false">)</mo>
            </mrow>
            <mrow>
              <mi mathvariant="normal">E</mi>
              <mo><!--  --></mo>
              <mo stretchy="false">(</mo>
              <mi>f</mi>
              <mo stretchy="false">(</mo>
              <mrow class="MJX-TeXAtom-ORD">
                <mtext>Aa</mtext>
              </mrow>
              <mo stretchy="false">)</mo>
              <mo stretchy="false">)</mo>
            </mrow>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mn>1</mn>
        <mo>−<!-- − --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <mi mathvariant="normal">O</mi>
              <mo><!--  --></mo>
              <mo stretchy="false">(</mo>
              <mi>f</mi>
              <mo stretchy="false">(</mo>
              <mrow class="MJX-TeXAtom-ORD">
                <mtext>Aa</mtext>
              </mrow>
              <mo stretchy="false">)</mo>
              <mo stretchy="false">)</mo>
            </mrow>
            <mrow>
              <mi mathvariant="normal">E</mi>
              <mo><!--  --></mo>
              <mo stretchy="false">(</mo>
              <mi>f</mi>
              <mo stretchy="false">(</mo>
              <mrow class="MJX-TeXAtom-ORD">
                <mtext>Aa</mtext>
              </mrow>
              <mo stretchy="false">)</mo>
              <mo stretchy="false">)</mo>
            </mrow>
          </mfrac>
        </mrow>
        <mo>,</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle F={\frac {\operatorname {E} {(f({\text{Aa}}))}-\operatorname {O} (f({\text{Aa}}))}{\operatorname {E} (f({\text{Aa}}))}}=1-{\frac {\operatorname {O} (f({\text{Aa}}))}{\operatorname {E} (f({\text{Aa}}))}},}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea4f45d89b4813f711f93eed98621d668bf4d8e4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:46.418ex; height:6.509ex;" alt="F={\frac  {\operatorname {E}{(f({\text{Aa}}))}-\operatorname {O}(f({\text{Aa}}))}{\operatorname {E}(f({\text{Aa}}))}}=1-{\frac  {\operatorname {O}(f({\text{Aa}}))}{\operatorname {E}(f({\text{Aa}}))}}," /></SPAN></SPAN></DD>
</DL>
<P>where the expected value from Hardy–Weinberg equilibrium is given by</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi mathvariant="normal">E</mi>
        <mo><!--  --></mo>
        <mo stretchy="false">(</mo>
        <mi>f</mi>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>Aa</mtext>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo stretchy="false">)</mo>
        <mo>=</mo>
        <mn>2</mn>
        <mi>p</mi>
        <mi>q</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (f({\text{Aa}}))=2pq}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a319e6f6a1b69f3469bb84bf8bc9966c9252460c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:16.011ex; height:2.843ex;" alt="\operatorname {E}(f({\text{Aa}}))=2pq" /></SPAN></DD>
</DL>
<P>For example, for Ford's data above;</P>
<DL>
<DD><SPAN class="nowrap"><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>F</mi>
        <mo>=</mo>
        <mn>1</mn>
        <mo>−<!-- − --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>138</mn>
            <mn>141.2</mn>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mn>0.023.</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle F=1-{138 \over 141.2}=0.023.}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ead5791bfb0c65b2ad0182c41982504a7dd1d9c8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:24.184ex; height:5.176ex;" alt="F=1-{138 \over 141.2}=0.023." /></SPAN></SPAN></DD>
</DL>
<P>For two alleles, the chi-squared goodness of fit test for Hardy–Weinberg proportions is equivalent to the test for inbreeding, <I>F</I> = 0.</P>
<P>The inbreeding coefficient is unstable as the expected value approaches zero, and thus not useful for rare and very common alleles. For: <I>E</I> = 0, O > 0, <I>F</I> = −∞ and <I>E</I> = 0, O = 0, <I>F</I> is undefined.</P>
<H2><SPAN class="mw-headline" id="History">History</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=14" title="Edit section: History">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P><A href="/wiki/Mendelian_genetics" class="mw-redirect" title="Mendelian genetics">Mendelian genetics</A> were rediscovered in 1900. However, it remained somewhat controversial for several years as it was not then known how it could cause continuous characteristics. <A href="/wiki/Udny_Yule" title="Udny Yule">Udny Yule</A> (1902) argued against Mendelism because he thought that dominant alleles would increase in the population.<SUP id="cite_ref-5" class="reference"><A href="#cite_note-5">[5]</A></SUP> The <A href="/wiki/United_States" title="United States">American</A> <A href="/wiki/William_E._Castle" title="William E. Castle">William E. Castle</A> (1903) showed that without <A href="/wiki/Selection_(biology)" class="mw-redirect" title="Selection (biology)">selection</A>, the genotype frequencies would remain stable.<SUP id="cite_ref-6" class="reference"><A href="#cite_note-6">[6]</A></SUP> <A href="/wiki/Karl_Pearson" title="Karl Pearson">Karl Pearson</A> (1903) found one equilibrium position with values of <I>p</I> = <I>q</I> = 0.5.<SUP id="cite_ref-7" class="reference"><A href="#cite_note-7">[7]</A></SUP> <A href="/wiki/Reginald_Punnett" title="Reginald Punnett">Reginald Punnett</A>, unable to counter Yule's point, introduced the problem to <A href="/wiki/Godfrey_Harold_Hardy" class="mw-redirect" title="Godfrey Harold Hardy">G. H. Hardy</A>, a <A href="/wiki/United_Kingdom" title="United Kingdom">British</A> <A href="/wiki/Mathematician" title="Mathematician">mathematician</A>, with whom he played <A href="/wiki/Cricket" title="Cricket">cricket</A>. Hardy was a <A href="/wiki/Pure_mathematics" title="Pure mathematics">pure mathematician</A> and held <A href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</A> in some contempt; his view of biologists' use of mathematics comes across in his 1908 paper where he describes this as "very simple":<SUP id="cite_ref-8" class="reference"><A href="#cite_note-8">[8]</A></SUP></P>
<DL>
<DD><I>To the Editor of Science: I am reluctant to intrude in a discussion concerning matters of which I have no expert knowledge, and I should have expected the very simple point which I wish to make to have been familiar to biologists. However, some remarks of Mr. Udny Yule, to which Mr. R. C. Punnett has called my attention, suggest that it may still be worth making...</I></DD>
</DL>
<DL>
<DD><I>Suppose that Aa is a pair of Mendelian characters, A being dominant, and that in any given generation the number of pure dominants (AA), heterozygotes (Aa), and pure recessives (aa) are as</I> <I>p</I>:2<I>q</I>:<I>r</I>. <I>Finally, suppose that the numbers are fairly large, so that mating may be regarded as random, that the sexes are evenly distributed among the three varieties, and that all are equally fertile. A little mathematics of the multiplication-table type is enough to show that in the next generation the numbers will be as</I> (<I>p</I> + <I>q</I>)<SUP>2</SUP>:2(<I>p</I> + <I>q</I>)(<I>q</I> + <I>r</I>):(<I>q</I> + <I>r</I>)<SUP>2</SUP>, or as <I>p</I><SUB>1</SUB>:2<I>q</I><SUB>1</SUB>:<I>r</I><SUB>1</SUB>, <I>say.</I></DD>
</DL>
<DL>
<DD><I>The interesting question is: in what circumstances will this distribution be the same as that in the generation before? It is easy to see that the condition for this is</I> <I>q</I><SUP>2</SUP> = <I>pr</I>. And since <I>q</I><SUB>1</SUB><SUP>2</SUP> = <I>p</I><SUB>1</SUB><I>r</I><SUB>1</SUB>, <I>whatever the values of</I> <I>p</I>, <I>q</I>, <I>and</I> <I>r</I> <I>may be, the distribution will in any case continue unchanged after the second generation</I></DD>
</DL>
<P>The principle was thus known as <I>Hardy's law</I> in the <A href="/wiki/English-speaking_world" title="English-speaking world">English-speaking world</A> until 1943, when <A href="/wiki/Curt_Stern_(geneticist)" class="mw-redirect" title="Curt Stern (geneticist)">Curt Stern</A> pointed out that it had first been formulated independently in 1908 by the German physician <A href="/wiki/Wilhelm_Weinberg" title="Wilhelm Weinberg">Wilhelm Weinberg</A>.<SUP id="cite_ref-crow_9-0" class="reference"><A href="#cite_note-crow-9">[9]</A></SUP><SUP id="cite_ref-stern_10-0" class="reference"><A href="#cite_note-stern-10">[10]</A></SUP> <A href="/wiki/William_E._Castle" title="William E. Castle">William Castle</A> in 1903 also derived the ratios for the special case of equal allele frequencies, and it is sometimes (but rarely) called the Hardy–Weinberg–Castle Law.</P>
<H3><SPAN id="Derivation_of_Hardy's_equations"></SPAN><SPAN class="mw-headline" id="Derivation_of_Hardy.27s_equations">Derivation of Hardy's equations</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=15" title="Edit section: Derivation of Hardy's equations">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>Hardy's statement begins with a <A href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</A> for the frequencies <I>p</I>, <I>2q</I>, and <I>r</I>. These recurrence relations follow from fundamental concepts in probability, specifically <A href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">independence</A>, and <A href="/wiki/Conditional_probability" title="Conditional probability">conditional probability</A>. For example, consider the probability of an offspring from the generation <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mstyle displaystyle="false" scriptlevel="0">
          <mi>t</mi>
        </mstyle>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \textstyle t}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8212227d9434ab8b72032e0cdb0c6a1a93479b8d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:0.85ex; height:2.009ex;" alt="\textstyle t" /></SPAN> being homozygous dominant. Alleles are inherited independently from each parent. A dominant allele can be inherited from a homozygous dominant parent with probability 1, or from a heterozygous parent with probability 0.5. To represent this reasoning in an equation, let <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mstyle displaystyle="false" scriptlevel="0">
          <msub>
            <mi>A</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
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          </msub>
        </mstyle>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \textstyle A_{t}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/054ed004476d4bbeb669f3dfc9053e237bb173ff" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.587ex; height:2.676ex;" alt="{\displaystyle \textstyle A_{t}}" /></SPAN> represent inheritance of a dominant allele from a parent. Furthermore, let <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mstyle displaystyle="false" scriptlevel="0">
          <mi>A</mi>
          <msub>
            <mi>A</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
              <mo>−<!-- − --></mo>
              <mn>1</mn>
            </mrow>
          </msub>
        </mstyle>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \textstyle AA_{t-1}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b3970a0225a0d385a4ea2b56b30f961ad06ca4b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:6.456ex; height:2.676ex;" alt="{\displaystyle \textstyle AA_{t-1}}" /></SPAN> and <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mstyle displaystyle="false" scriptlevel="0">
          <mi>A</mi>
          <msub>
            <mi>a</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
              <mo>−<!-- − --></mo>
              <mn>1</mn>
            </mrow>
          </msub>
        </mstyle>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \textstyle Aa_{t-1}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d7e23add8e38fb0113e06bc2d345908546b60b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:5.942ex; height:2.676ex;" alt="{\displaystyle \textstyle Aa_{t-1}}" /></SPAN> represent potential parental genotypes in the preceding generation.</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
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        <mrow class="MJX-TeXAtom-ORD">
          <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
            <mtr>
              <mtd>
                <msub>
                  <mi>p</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mi>t</mi>
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              </mtd>
              <mtd>
                <mi></mi>
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                <mi>P</mi>
                <mo stretchy="false">(</mo>
                <msub>
                  <mi>A</mi>
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                <mo>,</mo>
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                    <mi>t</mi>
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                <mo stretchy="false">)</mo>
                <mo>=</mo>
                <mi>P</mi>
                <mo stretchy="false">(</mo>
                <msub>
                  <mi>A</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mi>t</mi>
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    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p_{t}&=P(A_{t},A_{t})=P(A_{t})^{2}\\&=\left(P(A_{t}|AA_{t-1})P(AA_{t-1})+P(A_{t}|Aa_{t-1})P(Aa_{t-1})\right)^{2}\\&=\left((1)p_{t-1}+(0.5)2q_{t-1}\right)^{2}\\&=\left(p_{t-1}+q_{t-1}\right)^{2}\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62604da4b8106b141b97a0ca7ea4e5f3e74881ed" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -6.069ex; margin-bottom: -0.269ex; width:57.245ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}p_{t}&=P(A_{t},A_{t})=P(A_{t})^{2}\\&=\left(P(A_{t}|AA_{t-1})P(AA_{t-1})+P(A_{t}|Aa_{t-1})P(Aa_{t-1})\right)^{2}\\&=\left((1)p_{t-1}+(0.5)2q_{t-1}\right)^{2}\\&=\left(p_{t-1}+q_{t-1}\right)^{2}\end{aligned}}}" /></SPAN></DD>
</DL>
<P>The same reasoning, applied to the other genotypes yields the two remaining recurrence relations. Equilibrium occurs when each proportion is constant between subsequent generations. More formally, a population is at equilibrium at generation <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
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    <annotation encoding="application/x-tex">{\displaystyle \textstyle t}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8212227d9434ab8b72032e0cdb0c6a1a93479b8d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:0.85ex; height:2.009ex;" alt="\textstyle t" /></SPAN> when</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
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          <mn>0</mn>
          <mo>=</mo>
          <msub>
            <mi>p</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
            </mrow>
          </msub>
          <mo>−<!-- − --></mo>
          <msub>
            <mi>p</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
              <mo>−<!-- − --></mo>
              <mn>1</mn>
            </mrow>
          </msub>
        </mstyle>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \textstyle 0=p_{t}-p_{t-1}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7640e046aeb7597b5c7209fd8f9ae3fdcf2478b2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:13.274ex; height:2.509ex;" alt="{\displaystyle \textstyle 0=p_{t}-p_{t-1}}" /></SPAN>, <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mstyle displaystyle="false" scriptlevel="0">
          <mn>0</mn>
          <mo>=</mo>
          <msub>
            <mi>q</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
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          <mo>−<!-- − --></mo>
          <msub>
            <mi>q</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
              <mo>−<!-- − --></mo>
              <mn>1</mn>
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \textstyle 0=q_{t}-q_{t-1}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e756bb9dad95cdff5801043d1ababf19d6a18f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:13.01ex; height:2.509ex;" alt="{\displaystyle \textstyle 0=q_{t}-q_{t-1}}" /></SPAN>, and <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mstyle displaystyle="false" scriptlevel="0">
          <mn>0</mn>
          <mo>=</mo>
          <msub>
            <mi>r</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
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          <mo>−<!-- − --></mo>
          <msub>
            <mi>r</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
              <mo>−<!-- − --></mo>
              <mn>1</mn>
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \textstyle 0=r_{t}-r_{t-1}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ab8b84da062bf12f39f26cab6d0df44bcbe544" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:13.033ex; height:2.509ex;" alt="{\displaystyle \textstyle 0=r_{t}-r_{t-1}}" /></SPAN></DD>
</DL>
<P>By solving these equations necessary and sufficient conditions for equilibrium to occur can be determined. Again, consider the frequency of homozygous dominant animals. Equilibrium implies</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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            <mtr>
              <mtd>
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                  <mrow class="MJX-TeXAtom-ORD">
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    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}0&=p_{t}-p_{t-1}\\&=p_{t-1}^{2}+2p_{t-1}q_{t-1}+q_{t-1}^{2}-p_{t-1}\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f68a3441d8928bb4b8df86e9476e50eb31bbb125" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:35.137ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}0&=p_{t}-p_{t-1}\\&=p_{t-1}^{2}+2p_{t-1}q_{t-1}+q_{t-1}^{2}-p_{t-1}\end{aligned}}}" /></SPAN></DD>
</DL>
<P>First consider the case, where <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
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        <mstyle displaystyle="false" scriptlevel="0">
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            <mi>p</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
              <mo>−<!-- − --></mo>
              <mn>1</mn>
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          </msub>
          <mo>=</mo>
          <mn>0</mn>
        </mstyle>
      </mstyle>
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    <annotation encoding="application/x-tex">{\displaystyle \textstyle p_{t-1}=0}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e293b4481c02ae5909f6c04da81f41023264d8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.079ex; width:8.489ex; height:2.509ex;" alt="{\displaystyle \textstyle p_{t-1}=0}" /></SPAN>, and note that it implies that <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mstyle displaystyle="false" scriptlevel="0">
          <msub>
            <mi>q</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
              <mo>−<!-- − --></mo>
              <mn>1</mn>
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          </msub>
          <mo>=</mo>
          <mn>0</mn>
        </mstyle>
      </mstyle>
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    <annotation encoding="application/x-tex">{\displaystyle \textstyle q_{t-1}=0}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b7ebd89d9e622af4939b5d103c5d7bec4bba77d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:8.278ex; height:2.509ex;" alt="{\displaystyle \textstyle q_{t-1}=0}" /></SPAN> and <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mstyle displaystyle="false" scriptlevel="0">
          <msub>
            <mi>r</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
              <mo>−<!-- − --></mo>
              <mn>1</mn>
            </mrow>
          </msub>
          <mo>=</mo>
          <mn>1</mn>
        </mstyle>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \textstyle r_{t-1}=1}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d317f7d503b5e54b81e26831a68651329006f4d2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:8.29ex; height:2.509ex;" alt="{\displaystyle \textstyle r_{t-1}=1}" /></SPAN>. Now consider the remaining case, where <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mstyle displaystyle="false" scriptlevel="0">
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            <mi>p</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
              <mo>−<!-- − --></mo>
              <mn>1</mn>
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          </msub>
        </mstyle>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \textstyle p_{t-1}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/640921e3c023a8ee233a974821c3db525f3eb4fb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.079ex; width:4.207ex; height:2.009ex;" alt="{\displaystyle \textstyle p_{t-1}}" /></SPAN> ≠ <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mstyle displaystyle="false" scriptlevel="0">
          <mn>0</mn>
        </mstyle>
      </mstyle>
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    <annotation encoding="application/x-tex">{\displaystyle \textstyle 0}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/917913183a2d19526fd19377fc666a6958facb00" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.173ex; height:2.176ex;" alt="\textstyle 0" /></SPAN></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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            <mtr>
              <mtd>
                <mn>0</mn>
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                <mo>=</mo>
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                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>2</mn>
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                <mrow class="MJX-TeXAtom-ORD">
                  <mo>/</mo>
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                  <mi>p</mi>
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                    <mo>−<!-- − --></mo>
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                <mo>−<!-- − --></mo>
                <mn>1</mn>
                <mo stretchy="false">)</mo>
              </mtd>
            </mtr>
            <mtr>
              <mtd></mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <msubsup>
                  <mi>q</mi>
                  <mrow class="MJX-TeXAtom-ORD">
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                <mrow class="MJX-TeXAtom-ORD">
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                  <mrow class="MJX-TeXAtom-ORD">
                    <mi>t</mi>
                    <mo>−<!-- − --></mo>
                    <mn>1</mn>
                  </mrow>
                </msub>
                <mo>−<!-- − --></mo>
                <msub>
                  <mi>r</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mi>t</mi>
                    <mo>−<!-- − --></mo>
                    <mn>1</mn>
                  </mrow>
                </msub>
              </mtd>
            </mtr>
          </mtable>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}0&=p_{t-1}(p_{t-1}+2q_{t-1}+q_{t-1}^{2}/p_{t-1}-1)\\&=q_{t-1}^{2}/p_{t-1}-r_{t-1}\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d492fe268fa12f8e04c248b5e30c8df1a5590a72" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:39.313ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}0&=p_{t-1}(p_{t-1}+2q_{t-1}+q_{t-1}^{2}/p_{t-1}-1)\\&=q_{t-1}^{2}/p_{t-1}-r_{t-1}\end{aligned}}}" /></SPAN></DD>
</DL>
<P>Where the final equality holds because the allele proportions must sum to one. In both cases, <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mstyle displaystyle="false" scriptlevel="0">
          <msubsup>
            <mi>q</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
              <mo>−<!-- − --></mo>
              <mn>1</mn>
            </mrow>
            <mrow class="MJX-TeXAtom-ORD">
              <mn>2</mn>
            </mrow>
          </msubsup>
          <mo>=</mo>
          <msub>
            <mi>p</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
              <mo>−<!-- − --></mo>
              <mn>1</mn>
            </mrow>
          </msub>
          <msub>
            <mi>r</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>t</mi>
              <mo>−<!-- − --></mo>
              <mn>1</mn>
            </mrow>
          </msub>
        </mstyle>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \textstyle q_{t-1}^{2}=p_{t-1}r_{t-1}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/825f6f9fb9d86a58a629ef4fc97cf22acb026035" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:15.241ex; height:3.176ex;" alt="{\displaystyle \textstyle q_{t-1}^{2}=p_{t-1}r_{t-1}}" /></SPAN>. It can be shown that the other two equilibrium conditions imply the same equation. Together, the solutions of the three equilibrium equations imply sufficiency of Hardy's condition for equilibrium. Since the condition always holds for the second generation, all succeeding generations have the same proportions.</P>
<H3><SPAN class="mw-headline" id="Numerical_example">Numerical example</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=16" title="Edit section: Numerical example">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>An example computation of the genotype distribution given by Hardy's original equations is instructive. The phenotype distribution from Table 3 above will be used to compute Hardy's initial genotype distribution. Note that the <I>p</I> and <I>q</I> values used by Hardy are not the same as those used above.</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
            <mtr>
              <mtd>
                <mrow class="MJX-TeXAtom-ORD">
                  <mtext>sum</mtext>
                </mrow>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mrow class="MJX-TeXAtom-ORD">
                    <mi mathvariant="normal">o</mi>
                    <mi mathvariant="normal">b</mi>
                    <mi mathvariant="normal">s</mi>
                  </mrow>
                  <mo stretchy="false">(</mo>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mtext>AA</mtext>
                  </mrow>
                  <mo stretchy="false">)</mo>
                  <mo>+</mo>
                  <mn>2</mn>
                  <mo>×<!-- × --></mo>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mi mathvariant="normal">o</mi>
                    <mi mathvariant="normal">b</mi>
                    <mi mathvariant="normal">s</mi>
                  </mrow>
                  <mo stretchy="false">(</mo>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mtext>Aa</mtext>
                  </mrow>
                  <mo stretchy="false">)</mo>
                  <mo>+</mo>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mi mathvariant="normal">o</mi>
                    <mi mathvariant="normal">b</mi>
                    <mi mathvariant="normal">s</mi>
                  </mrow>
                  <mo stretchy="false">(</mo>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mtext>aa</mtext>
                  </mrow>
                  <mo stretchy="false">)</mo>
                </mrow>
                <mo>=</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>1469</mn>
                  <mo>+</mo>
                  <mn>2</mn>
                  <mo>×<!-- × --></mo>
                  <mn>138</mn>
                  <mo>+</mo>
                  <mn>5</mn>
                </mrow>
              </mtd>
            </mtr>
            <mtr>
              <mtd></mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mn>1750</mn>
              </mtd>
            </mtr>
          </mtable>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\text{sum}}&={\mathrm {obs} ({\text{AA}})+2\times \mathrm {obs} ({\text{Aa}})+\mathrm {obs} ({\text{aa}})}={1469+2\times 138+5}\\&=1750\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821f276bafdf2c030d8c072a89496539a51cb56c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:64.457ex; height:5.843ex;" alt="{\begin{aligned}{\text{sum}}&={{\mathrm  {obs}}({\text{AA}})+2\times {\mathrm  {obs}}({\text{Aa}})+{\mathrm  {obs}}({\text{aa}})}={1469+2\times 138+5}\\&=1750\end{aligned}}" /></SPAN></DD>
</DL>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
            <mtr>
              <mtd>
                <mi>p</mi>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mfrac>
                    <mn>1469</mn>
                    <mn>1750</mn>
                  </mfrac>
                </mrow>
                <mo>=</mo>
                <mn>0.83943</mn>
              </mtd>
            </mtr>
            <mtr>
              <mtd>
                <mn>2</mn>
                <mi>q</mi>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mfrac>
                    <mrow>
                      <mn>2</mn>
                      <mo>×<!-- × --></mo>
                      <mn>138</mn>
                    </mrow>
                    <mn>1750</mn>
                  </mfrac>
                </mrow>
                <mo>=</mo>
                <mn>0.15771</mn>
              </mtd>
            </mtr>
            <mtr>
              <mtd>
                <mi>r</mi>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mfrac>
                    <mn>5</mn>
                    <mn>1750</mn>
                  </mfrac>
                </mrow>
                <mo>=</mo>
                <mn>0.00286</mn>
              </mtd>
            </mtr>
          </mtable>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p&={1469 \over 1750}=0.83943\\2q&={2\times 138 \over 1750}=0.15771\\r&={5 \over 1750}=0.00286\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b351ef2429e49cf0e9ced3fcab155d0e1cf00e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -7.41ex; margin-bottom: -0.261ex; width:25.296ex; height:16.509ex;" alt="{\begin{aligned}p&={1469 \over 1750}=0.83943\\2q&={2\times 138 \over 1750}=0.15771\\r&={5 \over 1750}=0.00286\end{aligned}}" /></SPAN></DD>
</DL>
<P>As checks on the distribution, compute</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>p</mi>
        <mo>+</mo>
        <mn>2</mn>
        <mi>q</mi>
        <mo>+</mo>
        <mi>r</mi>
        <mo>=</mo>
        <mn>0.83943</mn>
        <mo>+</mo>
        <mn>0.15771</mn>
        <mo>+</mo>
        <mn>0.00286</mn>
        <mo>=</mo>
        <mn>1.00000</mn>
        <mspace width="thinmathspace"></mspace>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle p+2q+r=0.83943+0.15771+0.00286=1.00000\,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2953abcd2d6686b433cc01726eb3ceae24619e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.079ex; width:53.358ex; height:2.509ex;" alt="p+2q+r=0.83943+0.15771+0.00286=1.00000\," /></SPAN></DD>
</DL>
<P>and</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>E</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>0</mn>
          </mrow>
        </msub>
        <mo>=</mo>
        <msup>
          <mi>q</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>−<!-- − --></mo>
        <mi>p</mi>
        <mi>r</mi>
        <mo>=</mo>
        <mn>0.00382.</mn>
        <mspace width="thinmathspace"></mspace>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle E_{0}=q^{2}-pr=0.00382.\,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42f67aaf923fa8cd6935d395f8cef54b96b7b5f6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:24.986ex; height:3.009ex;" alt="E_{0}=q^{2}-pr=0.00382.\," /></SPAN></DD>
</DL>
<P>For the next generation, Hardy's equations give</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
            <mtr>
              <mtd>
                <mi>q</mi>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mfrac>
                    <mn>0.15771</mn>
                    <mn>2</mn>
                  </mfrac>
                </mrow>
                <mo>=</mo>
                <mn>0.07886</mn>
              </mtd>
            </mtr>
            <mtr>
              <mtd></mtd>
            </mtr>
            <mtr>
              <mtd>
                <msub>
                  <mi>p</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>1</mn>
                  </mrow>
                </msub>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mo stretchy="false">(</mo>
                <mi>p</mi>
                <mo>+</mo>
                <mi>q</mi>
                <msup>
                  <mo stretchy="false">)</mo>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>2</mn>
                  </mrow>
                </msup>
                <mo>=</mo>
                <mn>0.84325</mn>
              </mtd>
            </mtr>
            <mtr>
              <mtd>
                <mn>2</mn>
                <msub>
                  <mi>q</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>1</mn>
                  </mrow>
                </msub>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mn>2</mn>
                <mo stretchy="false">(</mo>
                <mi>p</mi>
                <mo>+</mo>
                <mi>q</mi>
                <mo stretchy="false">)</mo>
                <mo stretchy="false">(</mo>
                <mi>q</mi>
                <mo>+</mo>
                <mi>r</mi>
                <mo stretchy="false">)</mo>
                <mo>=</mo>
                <mn>0.15007</mn>
              </mtd>
            </mtr>
            <mtr>
              <mtd>
                <msub>
                  <mi>r</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>1</mn>
                  </mrow>
                </msub>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mo stretchy="false">(</mo>
                <mi>q</mi>
                <mo>+</mo>
                <mi>r</mi>
                <msup>
                  <mo stretchy="false">)</mo>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>2</mn>
                  </mrow>
                </msup>
                <mo>=</mo>
                <mn>0.00668.</mn>
              </mtd>
            </mtr>
          </mtable>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}q&={0.15771 \over 2}=0.07886\\\\p_{1}&=(p+q)^{2}=0.84325\\2q_{1}&=2(p+q)(q+r)=0.15007\\r_{1}&=(q+r)^{2}=0.00668.\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7be59af63e75c30476a16448a56240f2b6f818b1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -8.505ex; width:32.88ex; height:18.176ex;" alt="{\begin{aligned}q&={0.15771 \over 2}=0.07886\\\\p_{1}&=(p+q)^{2}=0.84325\\2q_{1}&=2(p+q)(q+r)=0.15007\\r_{1}&=(q+r)^{2}=0.00668.\end{aligned}}" /></SPAN></DD>
</DL>
<P>Again as checks on the distribution, compute</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo>+</mo>
        <mn>2</mn>
        <msub>
          <mi>q</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo>+</mo>
        <msub>
          <mi>r</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo>=</mo>
        <mn>0.84325</mn>
        <mo>+</mo>
        <mn>0.15007</mn>
        <mo>+</mo>
        <mn>0.00668</mn>
        <mo>=</mo>
        <mn>1.00000</mn>
        <mspace width="thinmathspace"></mspace>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle p_{1}+2q_{1}+r_{1}=0.84325+0.15007+0.00668=1.00000\,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424e5589880c9780206729960bfa5cd88cb07193" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.079ex; width:56.511ex; height:2.509ex;" alt="p_{1}+2q_{1}+r_{1}=0.84325+0.15007+0.00668=1.00000\," /></SPAN></DD>
</DL>
<P>and</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>E</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo>=</mo>
        <msubsup>
          <mi>q</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>1</mn>
          </mrow>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msubsup>
        <mo>−<!-- − --></mo>
        <msub>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>1</mn>
          </mrow>
        </msub>
        <msub>
          <mi>r</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo>=</mo>
        <mn>0.00000</mn>
        <mspace width="thinmathspace"></mspace>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle E_{1}=q_{1}^{2}-p_{1}r_{1}=0.00000\,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9976770b7060d59b728d7b9fc559595dbc242e99" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:26.452ex; height:3.176ex;" alt="E_{1}=q_{1}^{2}-p_{1}r_{1}=0.00000\," /></SPAN></DD>
</DL>
<P>which are the expected values. The reader may demonstrate that subsequent use of the second-generation values for a third generation will yield identical results.</P>
<H2><SPAN class="mw-headline" id="Graphical_representation">Graphical representation</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=17" title="Edit section: Graphical representation">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:262px;"><A href="/wiki/File:De_finetti_diagram.png" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/De_finetti_diagram.png/260px-De_finetti_diagram.png" width="260" height="260" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/De_finetti_diagram.png/390px-De_finetti_diagram.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/52/De_finetti_diagram.png 2x" data-file-width="500" data-file-height="500" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:De_finetti_diagram.png" class="internal" title="Enlarge"></A></DIV>
A <A href="/wiki/De_Finetti_diagram" title="De Finetti diagram">de Finetti diagram</A> representing a distribution of genotype frequencies</DIV>
</DIV>
</DIV>
<P>It is possible to represent the distribution of genotype frequencies for a bi-allelic locus within a population graphically using a <A href="/wiki/De_Finetti_diagram" title="De Finetti diagram">de Finetti diagram</A>. This uses a triangular plot (also known as trilinear, triaxial or <A href="/wiki/Ternary_plot" title="Ternary plot">ternary plot</A>) to represent the distribution of the three genotype frequencies in relation to each other. It differs from many other such plots in that the direction of one of the axes has been reversed.<SUP id="cite_ref-11" class="reference"><A href="#cite_note-11">[11]</A></SUP> The curved line in the diagram is the Hardy–Weinberg <A href="/wiki/Parabola" title="Parabola">parabola</A> and represents the state where <A href="/wiki/Allele" title="Allele">alleles</A> are in Hardy–Weinberg equilibrium. It is possible to represent the effects of <A href="/wiki/Natural_selection" title="Natural selection">natural selection</A> and its effect on allele frequency on such graphs.<SUP id="cite_ref-12" class="reference"><A href="#cite_note-12">[12]</A></SUP> The <A href="/wiki/De_Finetti_diagram" title="De Finetti diagram">de Finetti diagram</A> has been developed and used extensively by <A href="/wiki/A._W._F._Edwards" title="A. W. F. Edwards">A. W. F. Edwards</A> in his book <I>Foundations of Mathematical Genetics</I>.<SUP id="cite_ref-13" class="reference"><A href="#cite_note-13">[13]</A></SUP></P>
<H2><SPAN class="mw-headline" id="See_also">See also</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=18" title="Edit section: See also">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<UL>
<LI><A href="/wiki/Regression_toward_the_mean" title="Regression toward the mean">Regression toward the mean</A></LI>
<LI><A href="/wiki/Multinomial_distribution" title="Multinomial distribution">Multinomial distribution</A> (Hardy–Weinberg is a trinomial distribution with probabilities <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mo stretchy="false">(</mo>
        <msup>
          <mi>θ<!-- θ --></mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>,</mo>
        <mn>2</mn>
        <mi>θ<!-- θ --></mi>
        <mo stretchy="false">(</mo>
        <mn>1</mn>
        <mo>−<!-- − --></mo>
        <mi>θ<!-- θ --></mi>
        <mo stretchy="false">)</mo>
        <mo>,</mo>
        <mo stretchy="false">(</mo>
        <mn>1</mn>
        <mo>−<!-- − --></mo>
        <mi>θ<!-- θ --></mi>
        <msup>
          <mo stretchy="false">)</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mo stretchy="false">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle (\theta ^{2},2\theta (1-\theta ),(1-\theta )^{2})}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/040deed1d4b68e39c12df9a6cf7585c39a2b4e01" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:23.327ex; height:3.176ex;" alt="(\theta ^{2},2\theta (1-\theta ),(1-\theta )^{2})" /></SPAN>)</LI>
</UL>
<H2><SPAN class="mw-headline" id="References">References</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=19" title="Edit section: References">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DIV class="reflist columns references-column-width" style="-moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;">
<OL class="references">
<LI id="cite_note-1"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-1"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">The term <I>frequency</I> usually refers to a number or count, but in this context, it is synonymous with <I>probability</I>.</SPAN></LI>
<LI id="cite_note-2"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-2"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation web">Carr, Dr. Steven M. <A rel="nofollow" class="external text" href="https://www.mun.ca/biology/scarr/2900_HW_for_dioecious.html">"Hardy-Weinberg in dioecious organisms"</A>. <I>www.mun.ca</I>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mun.ca&rft.atitle=Hardy-Weinberg+in+dioecious+organisms&rft.aulast=Carr&rft.aufirst=Dr.+Steven+M.&rft_id=https%3A%2F%2Fwww.mun.ca%2Fbiology%2Fscarr%2F2900_HW_for_dioecious.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></SPAN></LI>
<LI id="cite_note-Hartl2007-3"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-Hartl2007_3-0"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Hartl DL, Clarke AG (2007) Principles of population genetics. Sunderland, MA: Sinauer</SPAN></LI>
<LI id="cite_note-emigh_1980-4"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-emigh_1980_4-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-emigh_1980_4-1"><SUP><I><B>b</B></I></SUP></A></SPAN> <SPAN class="reference-text"><CITE class="citation journal">Emigh, Ted H. (1980). "A Comparison of Tests for Hardy–Weinberg Equilibrium". <I>Biometrics</I>. <B>36</B> (4): 627–642. <A href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</A>:<A rel="nofollow" class="external text" href="//doi.org/10.2307%2F2556115">10.2307/2556115</A>. <A href="/wiki/JSTOR" title="JSTOR">JSTOR</A> <A rel="nofollow" class="external text" href="//www.jstor.org/stable/2556115">2556115</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrics&rft.atitle=A+Comparison+of+Tests+for+Hardy%E2%80%93Weinberg+Equilibrium&rft.volume=36&rft.issue=4&rft.pages=627-642&rft.date=1980&rft_id=info%3Adoi%2F10.2307%2F2556115&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2556115&rft.aulast=Emigh&rft.aufirst=Ted+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></SPAN></LI>
<LI id="cite_note-5"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-5"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Yule, 1902</SPAN></LI>
<LI id="cite_note-6"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-6"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Castle, 1903</SPAN></LI>
<LI id="cite_note-7"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-7"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Pearson, 1903</SPAN></LI>
<LI id="cite_note-8"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-8"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Hardy, 1908</SPAN></LI>
<LI id="cite_note-crow-9"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-crow_9-0"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation journal"><A href="/wiki/James_F._Crow" title="James F. Crow">Crow, James F.</A> (1999). <A rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pmc/articles/PMC1460671">"Hardy, Weinberg and language impediments"</A>. <I>Genetics</I>. <B>152</B> (3): 821–825. <A href="/wiki/PubMed_Central" title="PubMed Central">PMC</A> <SPAN class="plainlinks"><A rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pmc/articles/PMC1460671">1460671</A> <IMG alt="Freely accessible" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" title="Freely accessible" width="9" height="14" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></SPAN>. <A href="/wiki/PubMed_Identifier" class="mw-redirect" title="PubMed Identifier">PMID</A> <A rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pubmed/10388804">10388804</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Genetics&rft.atitle=Hardy%2C+Weinberg+and+language+impediments&rft.volume=152&rft.issue=3&rft.pages=821-825&rft.date=1999&rft_id=%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1460671&rft_id=info%3Apmid%2F10388804&rft.aulast=Crow&rft.aufirst=James+F.&rft_id=%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1460671&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></SPAN></LI>
<LI id="cite_note-stern-10"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-stern_10-0"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation journal"><A href="/wiki/Curt_Stern_(geneticist)" class="mw-redirect" title="Curt Stern (geneticist)">Stern, Curt</A> (1962). "Wilhelm Weinberg". <I>Genetics</I>. <B>47</B>: 1–5.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Genetics&rft.atitle=Wilhelm+Weinberg&rft.volume=47&rft.pages=1-5&rft.date=1962&rft.aulast=Stern&rft.aufirst=Curt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></SPAN></LI>
<LI id="cite_note-11"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-11"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation journal">Cannings, C.; Edwards, A.W.F. (1968). "Natural selection and the de Finetti diagram". <I>Annals of Human Genetics</I>. <B>31</B>: 421–428. <A href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</A>:<A rel="nofollow" class="external text" href="//doi.org/10.1111%2Fj.1469-1809.1968.tb00575.x">10.1111/j.1469-1809.1968.tb00575.x</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Human+Genetics&rft.atitle=Natural+selection+and+the+de+Finetti+diagram&rft.volume=31&rft.pages=421-428&rft.date=1968&rft_id=info%3Adoi%2F10.1111%2Fj.1469-1809.1968.tb00575.x&rft.au=Cannings%2C+C.&rft.au=Edwards%2C+A.W.F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></SPAN></LI>
<LI id="cite_note-12"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-12"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">See e.g. Ineichen & Batschelet 1975</SPAN></LI>
<LI id="cite_note-13"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-13"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Edwards, 1977</SPAN></LI>
</OL>
</DIV>
<H2><SPAN class="mw-headline" id="Bibliography">Bibliography</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=20" title="Edit section: Bibliography">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<UL>
<LI><CITE class="citation journal">Castle, W. E. (1903). "The laws of Galton and Mendel and some laws governing race improvement by selection". <I>Proceedings of the American Academy of Arts and Sciences</I>. <B>35</B>: 233–242.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Academy+of+Arts+and+Sciences&rft.atitle=The+laws+of+Galton+and+Mendel+and+some+laws+governing+race+improvement+by+selection&rft.volume=35&rft.pages=233-242&rft.date=1903&rft.aulast=Castle&rft.aufirst=W.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></LI>
<LI><CITE class="citation journal">Crow, Jf (Jul 1999). <A rel="nofollow" class="external text" href="http://www.genetics.org/cgi/pmidlookup?view=long&pmid=10388804">"Hardy, Weinberg and language impediments"</A>. <I>Genetics</I>. <B>152</B> (3): 821–5. <A href="/wiki/International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</A> <A rel="nofollow" class="external text" href="//www.worldcat.org/issn/0016-6731">0016-6731</A>. <A href="/wiki/PubMed_Central" title="PubMed Central">PMC</A> <SPAN class="plainlinks"><A rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pmc/articles/PMC1460671">1460671</A> <IMG alt="Freely accessible" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" title="Freely accessible" width="9" height="14" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></SPAN>. <A href="/wiki/PubMed_Identifier" class="mw-redirect" title="PubMed Identifier">PMID</A> <A rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pubmed/10388804">10388804</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Genetics&rft.atitle=Hardy%2C+Weinberg+and+language+impediments&rft.volume=152&rft.issue=3&rft.pages=821-5&rft.date=1999-07&rft_id=%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1460671&rft.issn=0016-6731&rft_id=info%3Apmid%2F10388804&rft.au=Crow%2C+Jf&rft_id=http%3A%2F%2Fwww.genetics.org%2Fcgi%2Fpmidlookup%3Fview%3Dlong%26pmid%3D10388804&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></LI>
<LI>Edwards, A.W.F. 1977. <I>Foundations of Mathematical Genetics.</I> Cambridge University Press, Cambridge (2nd ed., 2000). <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-521-77544-2" title="Special:BookSources/0-521-77544-2">0-521-77544-2</A></LI>
<LI><CITE class="citation journal">Emigh, T.H. (1980). "A comparison of tests for Hardy–Weinberg equilibrium". <I>Biometrics</I>. <B>36</B> (4): 627–642. <A href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</A>:<A rel="nofollow" class="external text" href="//doi.org/10.2307%2F2556115">10.2307/2556115</A>. <A href="/wiki/JSTOR" title="JSTOR">JSTOR</A> <A rel="nofollow" class="external text" href="//www.jstor.org/stable/2556115">2556115</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrics&rft.atitle=A+comparison+of+tests+for+Hardy%E2%80%93Weinberg+equilibrium&rft.volume=36&rft.issue=4&rft.pages=627-642&rft.date=1980&rft_id=info%3Adoi%2F10.2307%2F2556115&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2556115&rft.aulast=Emigh&rft.aufirst=T.H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></LI>
<LI>Ford, E.B. (1971). <I>Ecological Genetics</I>, London.</LI>
<LI><CITE class="citation journal">Guo, Sw; <A href="/wiki/Elizabeth_A._Thompson" title="Elizabeth A. Thompson">Thompson, Elizabeth A.</A> (Jun 1992). "Performing the exact test of Hardy–Weinberg proportion for multiple alleles". <I>Biometrics</I>. Biometrics, Vol. 48, No. 2. <B>48</B> (2): 361–72. <A href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</A>:<A rel="nofollow" class="external text" href="//doi.org/10.2307%2F2532296">10.2307/2532296</A>. <A href="/wiki/International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</A> <A rel="nofollow" class="external text" href="//www.worldcat.org/issn/0006-341X">0006-341X</A>. <A href="/wiki/JSTOR" title="JSTOR">JSTOR</A> <A rel="nofollow" class="external text" href="//www.jstor.org/stable/2532296">2532296</A>. <A href="/wiki/PubMed_Identifier" class="mw-redirect" title="PubMed Identifier">PMID</A> <A rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pubmed/1637966">1637966</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrics&rft.atitle=Performing+the+exact+test+of+Hardy%E2%80%93Weinberg+proportion+for+multiple+alleles&rft.volume=48&rft.issue=2&rft.pages=361-72&rft.date=1992-06&rft_id=info%3Apmid%2F1637966&rft.issn=0006-341X&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2532296&rft_id=info%3Adoi%2F10.2307%2F2532296&rft.au=Guo%2C+Sw&rft.au=Thompson%2C+Elizabeth+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></LI>
<LI><CITE class="citation journal">Hardy, G. H. (Jul 1908). <A rel="nofollow" class="external text" href="http://www.esp.org/foundations/genetics/classical/hardy.pdf">"Mendelian Proportions in a Mixed Population"</A> <SPAN style="font-size:85%;">(PDF)</SPAN>. <I>Science</I>. <B>28</B> (706): 49–50. <A href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</A>:<A rel="nofollow" class="external text" href="//doi.org/10.1126%2Fscience.28.706.49">10.1126/science.28.706.49</A>. <A href="/wiki/International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</A> <A rel="nofollow" class="external text" href="//www.worldcat.org/issn/0036-8075">0036-8075</A>. <A href="/wiki/PubMed_Identifier" class="mw-redirect" title="PubMed Identifier">PMID</A> <A rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pubmed/17779291">17779291</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Mendelian+Proportions+in+a+Mixed+Population&rft.volume=28&rft.issue=706&rft.pages=49-50&rft.date=1908-07&rft.issn=0036-8075&rft_id=info%3Apmid%2F17779291&rft_id=info%3Adoi%2F10.1126%2Fscience.28.706.49&rft.aulast=Hardy&rft.aufirst=G.+H.&rft_id=http%3A%2F%2Fwww.esp.org%2Ffoundations%2Fgenetics%2Fclassical%2Fhardy.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></LI>
<LI><CITE class="citation journal">Ineichen, Robert; Batschelet, Eduard (1975). "Genetic selection and de Finetti diagrams". <I>Journal of Mathematical Biology</I>. <B>2</B>: 33. <A href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</A>:<A rel="nofollow" class="external text" href="//doi.org/10.1007%2FBF00276014">10.1007/BF00276014</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Biology&rft.atitle=Genetic+selection+and+de+Finetti+diagrams&rft.volume=2&rft.pages=33&rft.date=1975&rft_id=info%3Adoi%2F10.1007%2FBF00276014&rft.au=Ineichen%2C+Robert&rft.au=Batschelet%2C+Eduard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></LI>
<LI><CITE class="citation journal"><A href="/wiki/Joanna_Masel" title="Joanna Masel">Masel, Joanna</A> (2012). "Rethinking Hardy–Weinberg and genetic drift in undergraduate biology". <I>BioEssays</I>. <B>34</B> (8): 701–10. <A href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</A>:<A rel="nofollow" class="external text" href="//doi.org/10.1002%2Fbies.201100178">10.1002/bies.201100178</A>. <A href="/wiki/PubMed_Identifier" class="mw-redirect" title="PubMed Identifier">PMID</A> <A rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pubmed/22576789">22576789</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=BioEssays&rft.atitle=Rethinking+Hardy%E2%80%93Weinberg+and+genetic+drift+in+undergraduate+biology&rft.volume=34&rft.issue=8&rft.pages=701-10&rft.date=2012&rft_id=info%3Adoi%2F10.1002%2Fbies.201100178&rft_id=info%3Apmid%2F22576789&rft.au=Masel%2C+Joanna&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></LI>
<LI><CITE class="citation journal">Pearson, K. (1903). "Mathematical contributions to the theory of evolution. XI. On the influence of natural selection on the variability and correlation of organs". <I><A href="/wiki/Philosophical_Transactions_of_the_Royal_Society_A" title="Philosophical Transactions of the Royal Society A">Philosophical Transactions of the Royal Society A</A></I>. <B>200</B> (321–330): 1–66. <A href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</A>:<A rel="nofollow" class="external text" href="//doi.org/10.1098%2Frsta.1903.0001">10.1098/rsta.1903.0001</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+A&rft.atitle=Mathematical+contributions+to+the+theory+of+evolution.+XI.+On+the+influence+of+natural+selection+on+the+variability+and+correlation+of+organs&rft.volume=200&rft.issue=321%E2%80%93330&rft.pages=1-66&rft.date=1903&rft_id=info%3Adoi%2F10.1098%2Frsta.1903.0001&rft.aulast=Pearson&rft.aufirst=K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></LI>
<LI><CITE class="citation journal">Stern, C. (1943). "The Hardy–Weinberg law". <I><A href="/wiki/Science_(journal)" title="Science (journal)">Science</A></I>. <B>97</B> (2510): 137–138. <A href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</A>:<A rel="nofollow" class="external text" href="//doi.org/10.1126%2Fscience.97.2510.137">10.1126/science.97.2510.137</A>. <A href="/wiki/JSTOR" title="JSTOR">JSTOR</A> <A rel="nofollow" class="external text" href="//www.jstor.org/stable/1670409">1670409</A>. <A href="/wiki/PubMed_Identifier" class="mw-redirect" title="PubMed Identifier">PMID</A> <A rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pubmed/17788516">17788516</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=The+Hardy%E2%80%93Weinberg+law&rft.volume=97&rft.issue=2510&rft.pages=137-138&rft.date=1943&rft_id=info%3Apmid%2F17788516&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F1670409&rft_id=info%3Adoi%2F10.1126%2Fscience.97.2510.137&rft.aulast=Stern&rft.aufirst=C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></LI>
<LI><CITE class="citation journal">Weinberg, W. (1908). "Über den Nachweis der Vererbung beim Menschen". <I>Jahreshefte des Vereins für vaterländische Naturkunde in Württemberg</I>. <B>64</B>: 368–382.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Jahreshefte+des+Vereins+f%C3%BCr+vaterl%C3%A4ndische+Naturkunde+in+W%C3%BCrttemberg&rft.atitle=%C3%9Cber+den+Nachweis+der+Vererbung+beim+Menschen&rft.volume=64&rft.pages=368-382&rft.date=1908&rft.aulast=Weinberg&rft.aufirst=W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></LI>
<LI><CITE class="citation journal">Wigginton, Je; Cutler, Dj; Abecasis, Gr (May 2005). <A rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pmc/articles/PMC1199378">"A Note on Exact Tests of Hardy–Weinberg Equilibrium"</A>. <I>American Journal of Human Genetics</I>. <B>76</B> (5): 887–93. <A href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</A>:<A rel="nofollow" class="external text" href="//doi.org/10.1086%2F429864">10.1086/429864</A>. <A href="/wiki/International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</A> <A rel="nofollow" class="external text" href="//www.worldcat.org/issn/0002-9297">0002-9297</A>. <A href="/wiki/PubMed_Central" title="PubMed Central">PMC</A> <SPAN class="plainlinks"><A rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pmc/articles/PMC1199378">1199378</A> <IMG alt="Freely accessible" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" title="Freely accessible" width="9" height="14" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></SPAN>. <A href="/wiki/PubMed_Identifier" class="mw-redirect" title="PubMed Identifier">PMID</A> <A rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pubmed/15789306">15789306</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Human+Genetics&rft.atitle=A+Note+on+Exact+Tests+of+Hardy%E2%80%93Weinberg+Equilibrium&rft.volume=76&rft.issue=5&rft.pages=887-93&rft.date=2005-05&rft_id=%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1199378&rft.issn=0002-9297&rft_id=info%3Apmid%2F15789306&rft_id=info%3Adoi%2F10.1086%2F429864&rft.au=Wigginton%2C+Je&rft.au=Cutler%2C+Dj&rft.au=Abecasis%2C+Gr&rft_id=%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1199378&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></LI>
<LI><CITE class="citation journal">Yule, G. U. (1902). "Mendel's laws and their probable relation to intra-racial heredity". <I>New Phytol</I>. <B>1</B> (193–207): 222–238. <A href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</A>:<A rel="nofollow" class="external text" href="//doi.org/10.1111%2Fj.1469-8137.1902.tb07336.x">10.1111/j.1469-8137.1902.tb07336.x</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=New+Phytol&rft.atitle=Mendel%27s+laws+and+their+probable+relation+to+intra-racial+heredity&rft.volume=1&rft.issue=193%E2%80%93207&rft.pages=222-238&rft.date=1902&rft_id=info%3Adoi%2F10.1111%2Fj.1469-8137.1902.tb07336.x&rft.aulast=Yule&rft.aufirst=G.+U.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHardy%E2%80%93Weinberg+principle" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></LI>
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<H2><SPAN class="mw-headline" id="External_links">External links</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Hardy%E2%80%93Weinberg_principle&action=edit&section=21" title="Edit section: External links">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
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<LI><A rel="nofollow" class="external text" href="http://bio.waikato.ac.nz/staff/gemmill/"><I>EvolutionSolution</I> (at bottom of page)</A></LI>
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<LI><A rel="nofollow" class="external text" href="http://www.sph.umich.edu/csg/abecasis/Exact/">Source code (C/C++/Fortran/R) for Wigginton <I>et al.</I> 2005</A></LI>
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<LI><A href="/wiki/Linkage_disequilibrium" title="Linkage disequilibrium">Linkage disequilibrium</A></LI>
<LI><A href="/wiki/Fisher%27s_fundamental_theorem_of_natural_selection" title="Fisher's fundamental theorem of natural selection">Fisher's fundamental theorem</A></LI>
<LI><A href="/wiki/Neutral_theory_of_molecular_evolution" title="Neutral theory of molecular evolution">Neutral theory</A></LI>
<LI><A href="/wiki/Shifting_balance_theory" title="Shifting balance theory">Shifting balance theory</A></LI>
<LI><A href="/wiki/Price_equation" title="Price equation">Price equation</A></LI>
<LI><A href="/wiki/Coefficient_of_relationship" title="Coefficient of relationship">Coefficient of relationship</A></LI>
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<LI><A href="/wiki/Natural_selection" title="Natural selection">Natural</A></LI>
<LI><A href="/wiki/Sexual_selection" title="Sexual selection">Sexual</A></LI>
<LI><A href="/wiki/Selective_breeding" title="Selective breeding">Artificial</A></LI>
<LI><A href="/wiki/Ecological_selection" title="Ecological selection">Ecological</A></LI>
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<DIV style="padding:0.1em 0;line-height:1.2em;line-height:1.1em;">Effects of selection<BR />
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<LI><A href="/wiki/Genetic_hitchhiking" title="Genetic hitchhiking">Genetic hitchhiking</A></LI>
<LI><A href="/wiki/Negative_selection_(natural_selection)" title="Negative selection (natural selection)">Background selection</A></LI>
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<LI><A href="/wiki/Small_population_size" title="Small population size">Small population size</A></LI>
<LI><A href="/wiki/Population_bottleneck" title="Population bottleneck">Population bottleneck</A></LI>
<LI><A href="/wiki/Founder_effect" title="Founder effect">Founder effect</A></LI>
<LI><A href="/wiki/Coalescent_theory" title="Coalescent theory">Coalescence</A></LI>
<LI><A href="/wiki/Balding%E2%80%93Nichols_model" title="Balding–Nichols model">Balding–Nichols model</A></LI>
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<LI><A href="/wiki/Ronald_Fisher" title="Ronald Fisher">R. A. Fisher</A></LI>
<LI><A href="/wiki/J._B._S._Haldane" title="J. B. S. Haldane">J. B. S. Haldane</A></LI>
<LI><A href="/wiki/Sewall_Wright" title="Sewall Wright">Sewall Wright</A></LI>
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<LI><A href="/wiki/Evolution" title="Evolution">Evolution</A></LI>
<LI><A href="/wiki/Microevolution" title="Microevolution">Microevolution</A></LI>
<LI><A href="/wiki/Evolutionary_game_theory" title="Evolutionary game theory">Evolutionary game theory</A></LI>
<LI><A href="/wiki/Fitness_landscape" title="Fitness landscape">Fitness landscape</A></LI>
<LI><A href="/wiki/Genetic_genealogy" title="Genetic genealogy">Genetic genealogy</A></LI>
<LI><A href="/wiki/Quantitative_genetics" title="Quantitative genetics">Quantitative genetics</A></LI>
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document_title: Hardy–Weinberg principle

token: Hardy

token: Weinberg

token: principle

token: wikipedia

token: Jump

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token: proportions

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token: alleles

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token: shows

token: allele

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token: expected

token: genotype

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token: also

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token: equilibrium

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token: other

token: evolutionary

token: influences

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token: include

token: mate

token: choice

token: mutation

token: selection

token: genetic

token: drift

token: gene

token: flow

token: meiotic

token: drive

token: In

token: српскохрватски

question_text: reasons why a population may not be in hardy-weinberg equilibrium