Central Limit Theorem Simulation

About this experiment

Take a handful of values from almost any population, work out their mean, and do it again and again. The means pile up into a bell-shaped curve, even when the population itself is flat, lopsided or has two peaks. This is the central limit theorem, and it is the reason the normal distribution appears all over statistics. Pick a population, change the sample size and watch the histogram of sample means grow narrower and more symmetric.

Learning goals

Distinguish the distribution of a population from the sampling distribution of the mean
State the central limit theorem and the conditions under which it applies
Show that the mean of the sample means equals the population mean
Show that the standard deviation of the sample means (standard error) is σ/√n
Explain why a larger sample size is needed for very skewed populations

How to use it

1With the skewed population and n = 1, draw 1000 samples. What does the histogram look like, and why?
2Set n = 5 and draw 1000 again. Describe the shape now. Is it still skewed?
3Try n = 30. Compare the SD of the sample means with the predicted σ/√n.
4Quadruple n from 5 to 20. By what factor does the spread of the means shrink?
5Switch to two peaks with n = 2. Where do the means pile up? Increase n and watch the dip fill in.
6Use dice with n = 2. Which totals are most common? Link this to the sums of two dice.
7For each population, find the smallest n where the histogram looks normal. Which needs the largest n?

The model

X ˉ = \frac{X _{1} + X _{2} + \dots + X _{n}}{n}

Mean of one sample

E (X ˉ) = μ, SD (X ˉ) = \frac{σ}{n}

Centre and spread of the sampling distribution

X ˉ \approx N (μ, \frac{σ ^{2}}{n}) for large n

Central limit theorem

Z = \frac{X ˉ - μ}{σ / n}

Standardised sample mean, used for confidence intervals and tests

Assumptions

Each value in a sample is drawn independently from the same population (sampling with replacement).
The skewed population is an exponential distribution with mean 2, cut off at 10. The two-peak population mixes two normal curves at 2.5 and 7.5.
Random draws come from a seeded generator, so the same sequence of samples appears each time after Clear.
The normal curve is the theoretical N(μ, σ²/n), scaled to the number of samples and the bar width.

Try this in class

Q1A population has mean 50 and SD 12. Find the mean and standard deviation of the mean of samples of size 36.
Q2How large must a sample be to halve the standard error obtained with n = 25?
Q3Why is the central limit theorem useful even when we do not know the shape of the population?
Q4Does the central limit theorem say that a large sample looks normal? Explain the difference.
Q5Why do polls of about 1000 people give useful results for a whole country of millions?

Frequently asked questions

Is n = 30 always enough?

It is a common rule of thumb, but not a law. For symmetric populations far fewer values are enough. For very skewed populations, such as incomes, much larger samples are needed before the means look normal.

When does the theorem fail?

It needs a finite variance. For some heavy-tailed distributions, such as the Cauchy distribution, the mean of a sample is no more stable than a single value, however large the sample.

Why does the spread shrink with √n and not with n?

Errors in individual values partly cancel out when averaged. The variance of a sum of n independent values grows like n, so the SD of the sum grows like √n, and dividing by n leaves σ/√n for the mean.