From sample to world: the central limit theorem

In the previous lesson we learned: an estimate (say, a mean) is a random variable with its own sampling distribution. Now we learn the SHAPE of that distribution for the mean — a surprisingly simple fact on which all further inference rests.

Samples collected: 0n: 10Standard error σ/√n ≈ 3.40x̄ deviation from truth: 0.2

Sample size n10

Before you are three panels. Top — the population: skewed, e.g. service response time (many fast requests, rare very slow ones). Middle — one random sample from it: its points and the green line, the sample mean x̄. Bottom — the distribution of those means, accumulating. Press "take a sample" and notice: x̄ almost never equals the true mean — it misses slightly (the deviation is shown on the right).

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What it means

It is thanks to the CLT that polling 1,500 people yields a reliable estimate of a whole country's mood — no need to ask everyone. A random sample's mean is predictably close to the truth.

And the larger the sample, the narrower the error bell, i.e. the more precise the estimate — which is why sample size is always center stage.

Where it shows up

Quality control checks the mean of 30 parts, not the whole batch. Ratings and indexes average samples. All of it works because means behave predictably.

A/B tests compare the means of two user groups — and lean on exactly this fact about the behavior of means.

Definitions

Population

all the objects we want to conclude about (e.g. all users).

Sample

a randomly selected part of the population that we actually observe.

Central limit theorem (CLT)

the distribution of the means of random samples tends to normal (a bell) — even if the data itself is not normal.

Standard errorσ / √n

the width of the distribution of sample means: how far a sample mean typically strays from the truth. Falls like √n.

When the method lies (assumptions)

The CLT needs independent observations and finite variance. With ultra-heavy tails (a power law with α ≤ 2) the variance is infinite — no bell forms, and the mean does not stabilize even for large n.

With strong skew and small n the mean's distribution has not yet turned normal — you need a bigger sample. Dependent or clustered data (one user contributing many observations) effectively shrinks the real n.

Deep dive: the math and the mechanism (optional)

Почему именно колокол? Среднее — это сумма n независимых вкладов, делённая на n. Каждое наблюдение тянет среднее то вверх, то вниз; при сложении эти случайные отклонения частично гасят друг друга, и крайние комбинации (все наблюдения разом большие) астрономически редки — поэтому результат кучкуется у центра симметрично. Удивительно то, что форма исходных данных при этом почти не важна: складывая много независимых вкладов, мы всегда приходим к нормальной форме.

Откуда √n: дисперсии независимых складываются (из прошлого урока), поэтому Var(суммы) = n·σ², а деление на n уменьшает дисперсию в n² раз → Var(среднего) = σ²/n, и σ(среднего) = σ/√n. Корень — прямое следствие того, что складываются именно дисперсии (квадраты), а не сами разбросы.