Topic 10 of 14

Sampling distributions and the CLT

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Sample means have their own distribution. The Central Limit Theorem says it is near normal for big n, centered on μ, with spread σ / √n (the standard error).

One die roll is flat. The average of 30 rolls is a bell. Averaging cancels noise.

It is why 1,000 people can speak for millions, and why one day of data should never convince you.

AnalyticsWeekly averages are steadier than daily numbers; √n says by how much.
PollingA poll of 1,000 has about 1.6 points of noise on each candidate's share.
FinanceMany small bets average out far more predictably than one big bet.

Tennis example

One match's first-serve percentage bounces around. Average 36 matches and the noise shrinks 6-fold (√36). At 62% with a match SD of 6 points, the standard error is 1 point: the average almost always lands within 2 of 62.

Every possible sample: is the estimator on target?

A tiny population of four values: 1, 2, 3 and 6, so μ = 3 and σ² = 3.5. Draw n of them with replacement, list every sample there could be, and see where each statistic lands on average.

Sample size n = 2

Statistic

16 possible samples of n = 2 (4 choices per draw, so 4²). The sample means average 3 against a target of μ = 3: unbiased. Spread across samples (SD): 1.323, exactly σ / √n = √3.5 / √2.

(1, 1) → 1
(1, 2) → 1.5
(1, 3) → 2
(1, 6) → 3.5
(2, 1) → 1.5
(2, 2) → 2
(2, 3) → 2.5
(2, 6) → 4
(3, 1) → 2
(3, 2) → 2.5
(3, 3) → 3
(3, 6) → 4.5
(6, 1) → 3.5
(6, 2) → 4
(6, 3) → 4.5
(6, 6) → 6

CLT: sample means turn into a bell

Population shape

Sample size n = 5

Pick a population, set n, and draw. The dashed line is the true mean.

Check yourself

μ = 50, σ = 12, n = 36. P(49 < x bar < 51)?
Why can we use the normal there?
Repair bills: mean 640, SD 480, n = 64. SE?

The trap

The mistake people make most on this one.

Dividing σ by n instead of √n. The standard error shrinks with the square root: four times the sample only halves the noise.

Formulas

6 formulas and a way to remember them.

Sample mean: Centered on μ with standard error σ / √n
z for a sample mean: z = (x bar − μ) / (σ / √n)
Sum of n values: Mean nμ, SD σ√n
Sample proportion: Mean p, SE = √[p(1 − p) / n], normal if np ≥ 10 and n(1 − p) ≥ 10
Central Limit Theorem: Close to normal for large n (rule of thumb 30), whatever the population looks like
Good estimators: Unbiased: E(estimator) = parameter. Efficient: the smallest variance among the unbiased ones. x bar, p hat and s² (÷ n − 1) are unbiased.
Remember it: Averaging is noise cancelling. The √n is the price: four times the data buys half the noise.

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