Learning Exercise

Material: Bun & Bolt Math, Central limit theorem, https://bunandbolt.com/math/central-limit-theorem

Instructor setup:
Open the page and press Present for a projector. Check whether answers are currently visible and use H to hide or reveal them as needed. The setting can persist between pages. Set each population, n and width explicitly as described below, ask the question, and give pairs a minute to write a prediction before revealing. Page Up and Page Down navigate the highlights in presentation mode, but some highlights advance through several n values automatically within seconds. Use manual controls for prediction pauses; do not rely on a highlight remaining at its initial n. Keep keyboard focus outside an editable field when using presentation shortcuts.

Steps:
1. Two dice make a triangle. Population Die, n = 2. Ask: what shape will the averages of two dice make? Reveal: a triangle, because a total of 7 comes up in 6 of 36 ways. Try n = 3: the totals 10 and 11 each come up in 27 of 216 ways, and the shape is already rounding.

2. Same width, new shape. Choose Exponential and Magnified by √n; set n = 1, 4 and 30. Ask: is the average exactly normal by n = 30? Reveal: skewness falls from 2 to 1 to about 0.365. At n = 30, the maximum difference between its cumulative distribution and the matched normal is about 0.0243: close, but not exact. In this magnified view, the population and averages have the same standard deviation, so the changing shape is easier to compare.

3. Narrowing is another story. Keep Exponential selected (its standard deviation is 1), choose Actual width, and set n = 4, 16 and 100. Ask: what happens to the spread? Reveal: the standard deviation of the average is 0.5, 0.25 and 0.1. Four times as many independent draws halves the width. Distinguish this variance identity from the convergence in shape in step 2.

4. Population Rare, an event with a 1% chance, n = 30. Ask: does the average of 30 trials look normal? Reveal: the average is exactly 0 with probability about 0.740. Switch to ±Rare: symmetric, and still far from normal at n = 30. So the n you need depends on the population, and symmetry alone is not enough.

5. Population Cauchy, n = 1, then n = 100. Ask: does averaging 100 draws narrow it? Reveal: the average of 100 draws has exactly the same distribution as one draw; about 6.35% of values lie outside [−10, 10] in the two tails combined. The Cauchy distribution has no mean or variance, so the theorem's hypothesis fails.

6. Switch back to Exponential, n = 30, Magnified by √n (or select the final highlight, "The draws agree"). Replay 10,000 independent sample means and compare the histogram with the theoretical curve. The standard deviation of the unscaled mean is 1/√30 ≈ 0.1826, and its skewness is 2/√30 ≈ 0.365. Finite-sample measurements fluctuate around these values. Do not make this comparison while Cauchy is selected: it has no population standard deviation or skewness.

Extension:
Choose Draw and sketch your own population, for example two humps far apart. Predict how the shape changes with n, then compare it with the normal overlay. Visual closeness depends on the purpose and scale; this is not a universal pass/fail test. Curves for a drawn population are computed numerically. The window How distributions form shows how the distribution of a sum is built step by step.