The central limit theorem

Averaging steadies the wobble: A coffee machine pours cups whose volume varies: mean μ = 250 mL, standard deviation σ = 12 mL. One cup might be 231 mL or 268 mL — quite a spread.

Now pour n = 9 cups and average them. That average X̄ also varies from one batch of nine to the next — but it wobbles less than a single cup, because a few low cups get cancelled by a few high ones.

How much less? Its centre is still μ = 250, but its standard deviation shrinks to the standard error:

Why √n, not n: Averaging n values divides the spread by √n, not n. So to halve the standard error you need four times as much data (because √4 = 2). That is why bigger samples give steadier averages — but with diminishing returns.

The shape becomes a bell — even when the population isn't: Single delivery times at a depot are skewed — most are quick, a few are very long. That population is not Normal.

But the Central Limit Theorem says: average enough of them and the average behaves Normally anyway. For a large sample size n, the distribution of the sample mean X̄ is approximately Normal:

When does 'large enough' kick in?: If the population is already Normal, X̄ is exactly Normal for any n — even n = 2.

If the population is not Normal, you need a reasonably large n (a common rule of thumb is n ≥ 30) before the bell shape is a good approximation. Smaller-but-still-skewed → the approximation is rough.

Single value vs sample mean — don't mix them up: P(one delivery > 44) would use σ = 15. P(mean of 36 deliveries > 44) uses the standard error σ/√n = 2.5. The mean is far less likely to be extreme — that is why the second probability is so much smaller.