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Research Methods

Central Limit Theorem Simulator

Three things that get confused with one another: the population, one sample from it, and the distribution of the means of many such samples. Only the third one is what the theorem is about.

Simulated — generated populations, drawn from a documented seed

Learning objective

By the end you should be able to say what becomes normal as the sample grows, and what does not.

About 25 minutes. Nothing you do here is saved or sent anywhere.

The sentence to strike out

"With a large enough sample the data become normal" is wrong, and it is wrong in a way that leads people to run the wrong tests. The data are whatever the population made them. What becomes closer to normal is the distribution of the sample mean across repeated samples — a distribution nobody ever observes, which is precisely why it has to be simulated to be seen.

  1. Answer the three prediction questions to open the simulator.
  2. Draw one sample at a time first, and watch its mean land.
  3. Then draw a thousand and compare the picture with the population above it.
  4. Change n and check the observed spread against the standard error.

First, three predictions

A fictional population is strongly right-skewed: most values are small, a long tail runs to the right, the mean is 40.0 and the standard deviation is 20.0. Samples of 25 are drawn from it over and over, and each sample's mean is written down.

The shape: how will the distribution of those sample means compare with the population?
The width: how spread out will the sample means be?
The centre: where will the sample means pile up?

Challenge — what the theorem says

Which statements are correct? Select all that apply — three of the seven are.

What this demonstrates

The sample mean is a random variable

Draw one sample and you get one mean; draw another and you get a different one. That variation is not error in any everyday sense — nothing was measured badly — it is the consequence of having sampled rather than counted everybody. The lower panel is the distribution of that variation, and it is a distribution no real study ever observes, because a real study contributes exactly one dot to it.

Centre, spread and shape, in that order

The means are centred on the population mean whatever shape the population has: sampling is unbiased, and the skew does not drag them. Their spread is the population standard deviation divided by the square root of n, which is what the standard error is and why quadrupling the sample only halves it. Their shape is the last thing to arrive, and the only part that depends on the population looking anything like normal.

How fast the shape arrives depends on the population

The skewness of the sample mean is the population's skewness divided by the square root of n, and its excess kurtosis is the population's divided by n. Both are printed in the table beside their simulated counterparts. That makes "n = 30 is enough" checkable rather than a rule to memorise. For the strongly skewed population here, thirty is not enough for the skewness to disappear. For the bimodal population, ten is already plenty.

The data do not change

Nothing about the upper panel improves as n grows. The population is what it is, and a larger sample simply describes it better. If the analysis you plan requires the data to be roughly normal — and many do not — a big sample does not supply that. What a big sample supplies is a well-behaved sampling distribution for the mean, which is a different claim and a narrower one.

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