Central Limit Theorem

How the model works

The central limit theorem: if you average n independent draws from almost any population with mean μ and standard deviation σ, the distribution of that average approaches a normal distribution as n grows:

X̄ ≈ Normal(μ, σ/√n)

σ/√n is the standard error. Quadrupling n halves the spread of the sample means. This is why averages are more reliable than single measurements, and why so many measured quantities look normal.

How fast the bell appears depends on the population. Symmetric populations (uniform) look normal by n ≈ 5. Strongly skewed ones (exponential) need n ≈ 30 or more; the readout's skewness tracks this.

Requires a finite variance. Heavy-tailed populations such as the Cauchy distribution never settle down; that case is not included here.

Guided experiments

1. Dice averages. Uniform 1–6 population, n = 2, then 10. Prediction: n = 2 gives a triangle; n = 10 is already a clear bell with SD 1.71/√10 ≈ 0.54. Average dice
2. Skewed waiting times. Exponential population, n = 5 then 50. Prediction: n = 5 is still visibly skewed (skewness ≈ 2/√5 ≈ 0.9); n = 50 is almost symmetric (≈ 0.28). Exponential, n = 5
3. Two humps. Bimodal population, n = 30. Prediction: the sample means form one single bell between the humps, even though no individual is likely to be there. Bimodal, n = 30

Challenge

A machine fills bags with a mean of 500 g and SD 12 g (not normally distributed). A box holds 36 bags. Roughly what is the chance that the average bag weight in a box is below 497 g?

References

• OpenStax, Introductory Statistics, ch. 7 “The Central Limit Theorem”.