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S02.1 · Populations, samples and sampling distributions

A free, illustrated module of 4 lessons and one interactive in S02 Statistical Inference: A statistic has a distribution; The standard error; The shape of a sampling distribution; Normal samples: the chi-square and t distributions.

S02.1 L2 · Intermediate 4 lessons · about 100 min Live · in review

Part of S02 Statistical Inference in the Statistics and Data Analysis track.

A number computed from a random sample is itself random, and everything inference says about uncertainty is a statement about its distribution over samples. These four lessons separate the population, the sample and the sampling distribution, measure the width of a sampling distribution as the standard error (with the square-root law and the finite population correction), follow its shape as the central limit theorem works on skewed rents, and find the exact chi-square and t distributions that appear when a normal sample’s spread must be estimated.

What you need first. Populations, samples and sampling bias (S01.2), the mean, variance and skewness of a table (S01.3), and expectation, variance, the law of large numbers and the central limit theorem (M06.7, M06.11). The code uses NumPy and scipy.stats.

Lessons

1. A statistic has a distribution S02.1.1 25 min

   A number computed from a random sample is itself random. Tell apart the population, one sample and the sampling distribution of a statistic, list one exactly for five flats, simulate one for a city of 500, and see why a test-set score has one too.

2. The standard error S02.1.2 25 min

   The standard error is the standard deviation of a statistic’s sampling distribution. Derive σ/√n and the square-root law, the finite population correction for sampling without replacement, the standard error of a proportion, and estimate them all from one sample.

3. The shape of a sampling distribution S02.1.3 25 min

   Watch the central limit theorem work on skewed rents: the skewness of a mean falls like 1/√n, derived from the third moment of a sum. Judge the normal approximation by its tails, treat a proportion as a mean, and see the median settle into a bell while the maximum never does.

4. Normal samples: the chi-square and t distributions S02.1.4 25 min

   For a normal population the sampling distributions are exact. Meet the chi-square distribution as a sum of squared normals, the distribution of the sample variance, and the t distribution that appears when s replaces σ, with heavier tails that need 2.776 instead of 1.96 at n = 5.

The interactive

S02.1-X1 · Sampling-distribution simulator. Draw samples from 40 skewed rents or from a coin with a chosen p, one at a time or a thousand at once. Each sample’s mean, median, maximum or proportion drops onto a growing histogram, a switch lays the normal curve over it, and the read-out compares the spread with sigma over root n and shows the mean settling into a bell while the maximum does not. It lives in The shape of a sampling distribution.