Arithmetic Mean of Two Numbers

The arithmetic mean of two numbers is their sum halved: the balance point, equally far from each. Worked example: the mean of 12 and 20 is 32/2 = 16, and sure enough 16 − 12 = 20 − 16 = 4. The rearrangement answers the question students meet more often than the forward one — "my average across two tests is 78 and I scored 71 on the first, so what did I get on the second?" — where the answer is 2(78) − 71 = 85, not 78 + 7.

The Pythagoreans studied this alongside two rivals, the geometric and harmonic means, and named all three; Archytas of Tarentum in the fourth century BCE set out the definitions that survive. For any two positive numbers they are strictly ordered, A ≥ G ≥ H, with equality only when the numbers are identical, and that ordering is the reason the arithmetic mean overstates an average growth rate and an average speed — use the geometric mean for the first and the harmonic mean for the second. The commonest real-world trap is averaging averages: a class of 30 averaging 70% and a class of 10 averaging 90% do not combine to 80%, but to (30 × 70 + 10 × 90)/40 = 75%, because the plain arithmetic mean assumes equal weight.