Watch the central limit theorem happen

What does the central limit theorem say?

Start with a population: every value you could draw, with mean μ and standard deviation σ. Draw a random sample of n values and take its mean, x̄. Do that again and again, and the means form a distribution of their own, the sampling distribution of the mean.

The theorem describes that distribution. Its centre is μ. Its spread, called the standard error, is σ/√n. And as n grows, its shape gets closer to a normal curve, whatever the population looks like.

What should I try in the simulator?

1. 01

   Start with a sample of one

   A sample of one is a single draw, so its mean is that draw. At n = 1 the means simply copy the population: lopsided, and nothing like the curve.

2. 02

   Raise n to 5, then 30

   The means crowd towards the middle and the long tail shrinks. Every value still comes from the same skewed population; only their averages change shape.

3. 03

   Switch to two peaks

   Almost no single value sits in the middle, yet that is where the means pile up. The theorem is about averages, not about the data.

4. 04

   Watch the spread

   Compare the spread of the means with σ/√n beside it. Move from n = 5 to n = 20 and it halves: four times the sample, half the standard error.

5. 05

   Try yes or no

   The mean of 0s and 1s is a proportion. At small n the means jump in steps; by n = 100 they form a bell around 0.3.

Where do students lose marks on it?

• Saying the data become normal. Only the distribution of the sample mean does. One large sample still looks like its population.
• Using it for a single value. A question about one observation needs the population's own distribution, not σ/√n.
• Dropping the square root. The standard error is σ/√n, not σ/n.
• Treating n ≥ 30 as part of the theorem. It is a classroom rule of thumb. A strongly skewed population needs larger samples, and a normal population gives normal means at any n.
• Skipping the check for proportions. The normal approximation needs enough of both outcomes: np and n(1 − p) each above five, and better at ten or more.

How is it different from the law of large numbers?

The law of large numbers says one sample's mean gets closer to μ as n grows. The central limit theorem says how the means of many samples are spread around μ: normally, with standard error σ/√n. The first tells you a bigger sample is better; the second tells you how much better.

Your course will set this up in its own notation, with its own worked examples and its own kind of question. A practice exam built from your lecture slides tests it the way your course does. AgoFact builds simulators for courses too: see the one built for UCT's STA1000F.