In population genetics, frequencies describe how common alleles and genotypes are in a gene pool. Hardy–Weinberg calculations convert between allele and genotype frequencies, letting you predict expected genotype distributions in a nonevolving population.

Core frequency concepts

Allele frequency (gene pool level)

For a diploid population, each individual contributes two alleles per gene to the gene pool, so allele counts are out of $2N$ total gene copies (where $N$ is the number of individuals sampled).

Key ideas:

• Allele frequencies are typically symbolized as $p$ and $q$ for a two-allele system.

• For two alleles (e.g., A and a), the frequencies must sum to 1 because they represent all alleles at that locus in the population.

Genotype frequency (individuals level)

Genotype frequencies are based on counting individuals (not alleles), so they sum to 1 across all possible genotypes at that locus.

Hardy–Weinberg equations (how to calculate expected frequencies)

Hardy–Weinberg provides the mathematical relationship between allele frequencies and genotype frequencies in a nonevolving population.

This figure shows the three Hardy–Weinberg genotype frequency functions ($p^2$, $2pq$, and $q^2$) plotted against allele frequency. Reading the curves makes it easy to see how changing $p$ automatically determines all three expected genotype proportions under a nonevolving model. Source

For a gene with two alleles, once you know $p$ and $q$, you can calculate expected genotype frequencies.

These expressions let you move between:

This graph plots the Hardy–Weinberg expected genotype frequencies $p^2$, $2pq$, and $q^2$ across the full range of allele frequencies ($p$ from 0 to 1, with $q=1-p$). It visually emphasizes that heterozygotes ($2pq$) peak when $p=q=0.5$, while homozygotes dominate when one allele is rare. Source

• Allele frequencies ($p$, $q$) and

• Expected genotype frequencies ($p^2$, $2pq$, $q^2$)

How to obtain allele frequencies from genotype frequencies (and vice versa)

From genotype counts to genotype frequencies

You first convert raw counts into frequencies:

• Divide each genotype count by the total number of individuals, $N$.

• Check that genotype frequencies sum to 1 (allowing for rounding).

From genotype frequencies to allele frequencies

To calculate allele frequencies, count allele copies contributed by each genotype:

• Each homozygote contributes two copies of its allele.

• Each heterozygote contributes one copy of each allele.

Conceptual relationships for alleles A and a:

• $p$ increases with AA and Aa individuals (because both carry A).

• $q$ increases with aa and Aa individuals (because both carry a).

From allele frequencies to expected genotype frequencies

Once $p$ and $q$ are known:

• Square $p$ to get expected AA frequency ($p^2$).

• Multiply $2 \times p \times q$ to get expected Aa frequency ($2pq$).

• Square $q$ to get expected aa frequency ($q^2$).

What “nonevolving population” implies for calculations

Hardy–Weinberg calculations are used to generate expected genotype frequencies under a stable allele-frequency model. In this context:

• $p$ and $q$ are treated as constant from one generation to the next.

• Deviations between observed genotype frequencies and $p^2:2pq:q^2$ expectations suggest the population may not match the nonevolving model.

Common constraints and checks (good calculation habits)

• Ensure allele frequencies remain within bounds: $0 \le p \le 1$, $0 \le q \le 1$.

• For two alleles, verify $p+q=1$ after computing allele frequencies.

• Verify expected genotype frequencies sum to 1: $p^2 + 2pq + q^2 = 1$ (rounding may cause slight differences).

• Distinguish clearly between:

  • Allele frequency (counts alleles; denominator $2N$), and

  • Genotype frequency (counts individuals; denominator $N$).