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Central Limit Theorem · predict, then test

What shape will the dice make?

The textbook statement

Draw independent random values from any population with mean μ and a finite standard deviation σ. As the sample size n grows, the distribution of their sum approaches a normal distribution with mean nμ and standard deviation σ√n — no matter what shape the original population had.

Correct, and almost impossible to picture from the words alone. So don’t take it on faith. Answer the two questions below from instinct first, then roll the dice and find out whether you were right.

Question 1

Roll a single die a few hundred times and plot how often each face comes up. What shape do you expect those bars to form?

Commit to an answer before you roll.

Question 2

Now roll ten dice at once and plot their total. Same dice, same fairness. Do you expect the same shape as before?

Try question 1 first.

Rolls recorded 0 Last total —
Observed frequency Bell curve (normal fit) True probability
Shape check
—

True shape vs a bell curve —

Dice per roll
1die

Changing the count starts a fresh tally, because the range of possible totals changes.

Roll
Overlays

Leave these off until you have made your prediction. Revealing the answer switches them on.

Numbers
Observed mean—
Expected mean3.50
Observed SD—
Expected SD1.708
Excess kurtosis−1.20

A perfect bell curve has excess kurtosis 0. For the total of n dice it equals −1.2 ÷ n, so it shrinks towards 0 as dice are added.

Sum of n fair six-sided dice · mean 3.5n · standard deviation √(35n/12)