How Do You Use the Hardy-Weinberg Equation?

Use the Hardy-Weinberg equations to estimate allele and genotype frequencies in a population. If you know the proportion of individuals showing a recessive phenotype, start with $q^2$, then work back to $q$ and $p$.

What the letters mean

For a gene with two alleles, $p$ is the frequency of the dominant allele and $q$ is the frequency of the recessive allele. An allele frequency is the proportion of copies of that gene that are a particular allele. Because these are the only two alleles:

$$p+q=1$$

The expected genotype frequencies are:

$$p^2+2pq+q^2=1$$

Here, $p^2$ is the proportion homozygous dominant, $2pq$ is the proportion heterozygous, and $q^2$ is the proportion homozygous recessive. Homozygous means having two copies of the same allele; heterozygous means having one of each.

A recessive phenotype identifies the homozygous recessive genotype. A dominant phenotype does not identify a single genotype: an individual showing it could be homozygous dominant or heterozygous. That is why the recessive group is often the easiest starting point.

Worked example

Suppose $16\%$ of a population shows a recessive trait. Assuming the trait is controlled by one gene with two alleles and complete dominance, those individuals are homozygous recessive. Convert the percentage to a decimal first: $16\%=0.16$. Frequencies are proportions between $0$ and $1$, so do not substitute $16$ into the equation. Therefore:

$$q^2=0.16,q=\sqrt{0.16}=0.4$$

The dominant allele frequency is $p=1-0.4=0.6$. The expected heterozygote frequency is:

$$2pq=2(0.6)(0.4)=0.48$$

So $48\%$ of the population is expected to be heterozygous. As a check, the expected homozygous dominant frequency is $p^2=0.36$, and $0.36+0.48+0.16=1$.

Exam mistake to avoid: the proportion showing the recessive trait is $q^2$, not $q$. Take its square root before finding $p$. These are Hardy-Weinberg predictions: they assume conditions such as random mating, a large population, and no selection, mutation or migration affecting allele frequencies. Random mating means mates are not chosen by genotype. If these conditions do not hold, observed genotype frequencies may differ from the predictions.