A population is considered to be in Hardy-Weinberg equilibrium if the allele and genotype frequencies are as follows.

A goodness-of-fit test can be used to determine if a population is significantly different from the expections of Hardy-Weinberg equilibrium. If we have a series of genotype counts from a population, then we can compare these counts to the ones predicted by the Hardy-Weinberg model. We conclude that the population is not in Hardy-Weinberg equilibrium if the probability that the counts were drawn under the Hardy-Weinberg model is too small for the deviations to be considered due to random chance. The significance level that is typically used is , i.e. the genotype counts have less than a one in twenty chance of being caused by a population in Hardy-Weinberg equilibrium.

In order to calculate this probability, we will use a test statistic, , which was devised in 1900 by Karl Pearson and has a well characterized distribution. If  are the set of observed counts, and  are the set of expected counts, then

This test statistic has a “chi-square” distribution with  degrees of freedom. Since we are testing Hardy-Weinberg equilibrium with two alleles,  (rationale not shown). Furthermore, it can be shown that if  then . Therefore, if  we will reject the null model and conclude that there is significant statistical support that the population is not in Hardy-Weinberg equilibrium.

Consider the following samples from a population.

Calculate the  value.

Since , we conclude that the genotype frequencies in this population are not significantly different than what would be expected if the population is in Hardy-Weinberg equilibrium.

Race and Sanger (1975) determined the blood groups of 1000 Britons as follows (from Hartl and Clarke 1997).

This results in . As in the previous example, the measured genotype frequencies are not significantly different from the expectations of Hardy-Weinberg equilibrium.

Matthijis et al. (1998) surveyed a group of 54 people suffering from Jaeken syndrome (from Freeman and Herron 2004).

This results in . Unlike the previous two examples, the measured genotype frequencies are significantly different from the expectations of Hardy-Weinberg equilibrium. This indicates that one or more of the Hardy-Weinberg conditions are being violated; although, it does not tell us which ones.

Although to derive the Hardy-Weinberg principle, we assumed that the size of the population was infinite, these statistical tests demonstrate that finite populations can approximately exist in Hardy-Weinberg equilibrium.