Free tool
Start from the homozygous recessive frequency, an allele frequency, or observed genotype counts, and get p, q and the expected genotype frequencies - plus a chi-square test when you enter counts. Every step is shown.
Built for General Biology II and AP BiologyOnly the aa individuals show the recessive phenotype, so their frequency is q^2.
Adds expected numbers of individuals for each genotype.
Enter a frequency or counts to see p, q and the genotype frequencies.
Hardy-Weinberg problems get fast once the setup is automatic. Encodr keeps the equations and the five conditions on a spaced-repetition schedule so they are still there on exam day.
Get started freeFor a gene with two alleles, A at frequency p and a at frequency q:
p + q = 1 and p^2 + 2pq + q^2 = 1
p^2 is the frequency of AA individuals, 2pq of Aa carriers, and q^2 of aa individuals. The first equation is about alleles, the second about genotypes. The prediction only holds if all five conditions are met: no mutation, random mating, no gene flow, a very large population, and no natural selection. No real population meets all five, so the model is used as a baseline - a population that does not match it is evolving.
16% of a population shows a recessive trait, so q^2 = 0.16. Then q = √0.16 = 0.4, p = 1 - 0.4 = 0.6, p^2 = 0.36 and 2pq = 2 × 0.6 × 0.4 = 0.48. So 36% are AA, 48% are carriers and 16% are aa. Start from q^2, never from p^2: the dominant phenotype lumps AA and Aa together, so its frequency is p^2 + 2pq, not p^2.
A sample of 100 has 40 AA, 40 Aa and 20 aa. Count alleles: p = (2 × 40 + 40) ÷ 200 = 0.6 and q = 0.4. Expected counts are 0.36 × 100 = 36, 0.48 × 100 = 48 and 0.16 × 100 = 16. Chi-square = (40 - 36)^2/36 + (40 - 48)^2/48 + (20 - 16)^2/16 = 0.444 + 1.333 + 1.000 = 2.78. With 1 degree of freedom (3 genotype classes, minus 1, minus 1 for estimating p from the same data) the critical value at p = 0.05 is 3.84. 2.78 is below it, so the data do not show a significant departure from Hardy-Weinberg equilibrium. The full walkthrough is in the Hardy-Weinberg guide, and the General Biology II course has the practice cards.
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