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# Hardy–Weinberg principle

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 Title: Hardy–Weinberg principle Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Hardy–Weinberg principle

Hardy–Weinberg proportions for two alleles: the horizontal axis shows the two allele frequencies p and q and the vertical axis shows the expected genotype frequencies. Each line shows one of the three possible genotypes.
G. H. Hardy
Wilhelm Weinberg

The Hardy–Weinberg principle, also known as the Hardy–Weinberg equilibrium, model, theorem, or law, states that allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences. These influences include mate choice, mutation, selection, genetic drift, gene flow and meiotic drive. Because one or more of these influences are typically present in real populations, the Hardy–Weinberg principle describes an ideal condition against which the effects of these influences can be analyzed.

In the simplest case of a single locus with two alleles denoted A and a with frequencies f(A) = p and f(a) = q, respectively, the expected genotype frequencies are f(AA) = p2 for the AA homozygotes, f(aa) = q2 for the aa homozygotes, and f(Aa) = 2pq for the heterozygotes. The genotype proportions p2, 2pq, and q2 are called the Hardy–Weinberg proportions. Note that the sum of all genotype frequencies of this case is the binomial expansion of the square of the sum of p and q, and such a sum, as it represents the total of all possibilities, must be equal to 1. Therefore, (p + q)2 = p2 + 2pq + q2 = 1. A solution of this equation is q = 1 − p.

If union of gametes to produce the next generation is random, it can be shown that the new frequency f satisfies \textstyle f'(\text{A}) = f(\text{A}) and \textstyle f'(\text{a}) = f(\text{a}). That is, allele frequencies are constant between generations.

This principle was named after G. H. Hardy and Wilhelm Weinberg, who first demonstrated it mathematically.

## Contents

• Derivation 1
• Deviations from Hardy–Weinberg equilibrium 2
• Sex linkage 3
• Generalizations 4
• Generalization for more than two alleles 4.1
• Generalization for polyploidy 4.2
• Complete generalization 4.3
• Applications 5
• Application to cases of complete dominance 5.1
• Significance tests for deviation 6
• Example χ2 test for deviation 6.1
• Fisher's exact test (probability test) 6.2
• Inbreeding coefficient 7
• History 8
• Derivation of Hardy's equations 8.1
• Numerical example 8.2
• Graphical representation 9
• See also 10
• References 11
• Bibliography 12
• External links 13

## Derivation

Consider a population of

• (at bottom of page)EvolutionSolution
• Hardy–Weinberg Equilibrium Calculator
• genetics Population Genetics Simulator
• HARDY C implementation of Guo & Thompson 1992
• 2005et al.Source code (C/C++/Fortran/R) for Wigginton
• Online de Finetti Diagram Generator and Hardy–Weinberg equilibrium tests
• Online Hardy–Weinberg equilibrium tests and drawing of de Finetti diagrams
• Hardy–Weinberg Equilibrium Calculator

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