Hardy-Weinberg Principle Formula | Derivation & Calculator

Core idea

Overview

Under ideal conditions including random mating and no mutation, migration, or natural selection, allele frequencies remain constant across generations. The equation p^2 + 2pq + q^2 = 1 represents the distribution of genotypes in a population, where homozygous dominant, heterozygous, and homozygous recessive frequencies are summed to unity. This model serves as a null hypothesis to detect evolutionary change in biological systems.

When to use: Use this when you are given data on phenotypic frequencies (usually the recessive phenotype) in a population and need to calculate the allele frequency or the frequency of carriers.

Why it matters: It allows biologists to quantify the rate of evolution and identify which evolutionary pressures, such as natural selection or genetic drift, are acting upon a population.

Symbols

Variables

p = Frequency of dominant allele, q = Frequency of recessive allele, $p^{2}$ = Frequency of homozygous dominant genotype, 2pq = Frequency of heterozygous genotype, $q^{2}$ = Frequency of homozygous recessive genotype

p

Frequency of dominant allele

Variable

q

Frequency of recessive allele

Variable

p^{2}

Frequency of homozygous dominant genotype

Variable

2pq

Frequency of heterozygous genotype

Variable

q^{2}

Frequency of homozygous recessive genotype

Variable

Walkthrough

Derivation

Derivation of Hardy-Weinberg Principle

The Hardy-Weinberg principle is derived by expanding the binomial expression of allele frequencies to predict genotype frequencies in a stable population.

The population is infinitely large.
Mating is random.
No mutations occur.
There is no natural selection.
There is no gene flow (no migration into or out of the population).

1

Define allele frequencies

Let p represent the frequency of the dominant allele (A) and q represent the frequency of the recessive allele (a). Since these are the only two alleles for the trait, their sum must equal 1 (100%).

p + q = 1

Note: Remember that p and q are allele frequencies, not genotype frequencies.

2

Expand the binomial expression

To find the genotype frequencies for the next generation, we square the allele frequency sum, reflecting the combination of gametes during random fertilization.

(p + q)^{2} = 1^{2}

Note: The expansion follows the algebraic rule (a+b)^2 = $a^{2}$ + 2ab + $b^{2}$ .

3

Distribute the squares

Expanding the equation gives the three possible genotype frequencies: $p^{2}$ (homozygous dominant AA), 2pq (heterozygous Aa), and $q^{2}$ (homozygous recessive aa).

p^{2} + 2 pq + q^{2} = 1

Note: The 2pq term accounts for the two different ways to inherit a heterozygote (Aa or aA).

Result

p^{2} + 2 pq + q^{2} = 1

Source: AQA/OCR/Edexcel A-Level Biology Specification

Why it behaves this way

Intuition

Imagine a square area representing all individuals in a population. The sides of the square are divided into two segments, p and q (where p + q = 1, the total length). By squaring the side (p + q)^2, the square is partitioned into four regions: a large square of area $p^{2}$ , two rectangles of area pq, and a smaller square of area $q^{2}$ . These represent the probability of picking two alleles from the gene pool and the resulting genotypes.

p

Frequency of the dominant allele

The 'market share' of the dominant gene variant within the entire population's total gene pool.

q

Frequency of the recessive allele

The 'market share' of the recessive gene variant within the entire population's total gene pool.

p^{2}

Frequency of homozygous dominant genotype

The probability of an individual inheriting the dominant allele from both parents (p ×p).

2pq

Frequency of heterozygous genotype

The two ways to get this genotype: inheriting dominant from mother and recessive from father (pq), or vice versa (qp), resulting in 2pq.

q^{2}

Frequency of homozygous recessive genotype

The probability of an individual inheriting the recessive allele from both parents (q ×q).

Signs and relationships

+: The plus signs represent the summation of distinct, mutually exclusive genotype categories that together must account for the entire population (100% or 1).
=: Equality to 1 signifies that the total probability space of all possible allele combinations in the population is equal to 100%.

One free problem

Practice Problem

In a population where 16% of individuals exhibit the recessive phenotype, what is the frequency of the recessive allele (q)?

qSquared0.16

Solve for: $q$

Hint: The frequency of the recessive phenotype represents $q^{2}$ . You must take the square root to find q.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In the prevalence of cystic fibrosis, an autosomal recessive disorder, to determine the frequency of carriers in a human population, Hardy-Weinberg Principle is used to calculate p^2 + 2pq + q^2 from Frequency of dominant allele, Frequency of recessive allele, and Frequency of homozygous dominant genotype. The result matters because it helps compare populations or ecosystems and decide whether the system is growing, stable, or under stress.

Study smarter

Tips

Always calculate q first using the recessive phenotype frequency, as this is the only genotype that directly reveals its allele frequency.
Remember that $q^{2}$ is the frequency of the homozygous recessive genotype (aa), while q is the frequency of the recessive allele (a).
Ensure the sum of your calculated p and q equals 1 before proceeding to calculate genotype frequencies.

Avoid these traps

Common Mistakes

Confusing the frequency of the homozygous recessive phenotype ( $q^{2}$ ) with the frequency of the recessive allele (q).
Assuming a population is in Hardy-Weinberg equilibrium when it is undergoing natural selection or significant mutation.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The Hardy-Weinberg principle is derived by expanding the binomial expression of allele frequencies to predict genotype frequencies in a stable population.

Use this when you are given data on phenotypic frequencies (usually the recessive phenotype) in a population and need to calculate the allele frequency or the frequency of carriers.

It allows biologists to quantify the rate of evolution and identify which evolutionary pressures, such as natural selection or genetic drift, are acting upon a population.

Confusing the frequency of the homozygous recessive phenotype (q^2) with the frequency of the recessive allele (q). Assuming a population is in Hardy-Weinberg equilibrium when it is undergoing natural selection or significant mutation.

In the prevalence of cystic fibrosis, an autosomal recessive disorder, to determine the frequency of carriers in a human population, Hardy-Weinberg Principle is used to calculate p^2 + 2pq + q^2 from Frequency of dominant allele, Frequency of recessive allele, and Frequency of homozygous dominant genotype. The result matters because it helps compare populations or ecosystems and decide whether the system is growing, stable, or under stress.

Always calculate q first using the recessive phenotype frequency, as this is the only genotype that directly reveals its allele frequency. Remember that q^2 is the frequency of the homozygous recessive genotype (aa), while q is the frequency of the recessive allele (a). Ensure the sum of your calculated p and q equals 1 before proceeding to calculate genotype frequencies.

References

Sources

Hardy, G. H. (1908). Mendelian proportions in a mixed population.
Campbell Biology, 12th Edition.
AQA/OCR/Edexcel A-Level Biology Specification

Hardy-Weinberg Principle

Overview

Variables

Derivation

Define allele frequencies

Expand the binomial expression

Distribute the squares

Intuition

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Effective Population Size

Hardy–Weinberg (Tri-allelic) — Heterozygote

Linkage Disequilibrium (D)

Frequently Asked Questions

Sources