To see whether a population is evolving, we need to know what "not evolving" looks like. In 1908 the mathematician G. H. Hardy and the physician Wilhelm Weinberg each worked it out, using the product rule and square roots.
Key idea
For a gene with alleles A and a, let p be the frequency of A and q the frequency of a, so $p + q = 1$. In a population that is not evolving, the genotype frequencies are $$p^2 + 2pq + q^2 = 1$$ (AA, Aa, aa), and p and q stay the same each generation.
Why it works
Imagine all the gametes of the population in one pool. A gamete carries A with chance p and a with chance q. If parents pair up at random, an offspring gets two gametes independently: AA with chance $p \times p = p^2$, aa with chance $q^2$, and Aa two ways (A then a, or a then A), $2pq$. It is a Punnett square for a whole population.
Worked example: Starting from the recessive share
In a population, 16% of individuals show a recessive trait. Find q, p and the share of carriers.
- Recessive individuals are aa, so $q^2 = 0.16$.
- $q = \sqrt{0.16} = 0.4$.
- $p = 1 - 0.4 = 0.6$.
- Carriers: $2pq = 2 \times 0.6 \times 0.4 = 0.48$.
Answer: q = 0.4, p = 0.6, and 48% are carriers (Aa).
Worked example: From counts
A sample has 36 AA, 48 Aa and 16 aa individuals. What is p?
- There are 100 individuals, so 200 alleles.
- A alleles: 2 × 36 + 48 = 120.
- p = 120 ÷ 200.
Answer: p = 0.6.
- The equilibrium holds only when no evolution is going on: no mutation, no migration (gene flow), a very large population (no drift), parents pair up at random, and no natural selection.
- Real populations rarely meet all five, so Hardy-Weinberg is a baseline: if measured frequencies differ from it, something is changing the population (see the mechanisms of evolution).
Common mistake
Two classic slips. One is taking q to be the share of recessive-looking individuals. That share is $q^2$; take its square root to get q. The other is thinking dominant alleles must become more common over time. Dominance only decides which allele shows. Without selection or another force, p and q do not change at all.