Distribution Playground

Every distribution in one sandbox. Pick one, drag its parameters, and watch the shape respond. The dashed line is the mean, and the dotted lines sit one standard deviation to each side. Formulas can't show you this; sliders can.

What to try

• t vs. normal: pick Student's t, turn on the normal overlay, and drag df from 1 upward. The fat tails melt away. That's why t-tables stop mattering around df ≈ 100. (lesson)
• The normal approximation: pick Binomial with the overlay on. At n = 5 the bell is a bad costume; by n = 50 it fits beautifully — the Central Limit Theorem in action. (lesson)
• Skew that won't quit: chi-square with small df is heavily right-skewed; drag df up and watch it slowly normalize. (lesson)
• Beta as belief: Beta(2, 5) says "probably a low proportion"; Beta(20, 50) says the same thing with confidence. This is exactly how Bayesian updating sharpens a prior. (lesson)
• Poisson's variance equals its mean, so the spread grows with λ. When real count data spread more than that, you've met overdispersion. (lesson)

Add two dice and you get a triangle

One die is the uniform distribution in its purest form: six faces, six probabilities of 1/6, a plot as flat as a table. Roll two and add them, and the flatness is gone. There is one way to total 2 and there are six ways to total 7, so the 36 equally likely rolls pile up in the middle. That shape has a name. It is the triangular distribution, which is what you always get from the sum of two independent uniforms, and it is the reason a 7 turns up six times as often as a 12 in every board game you have ever lost. Keep adding dice and the straight sides begin to curve.

The other settings are the same counting exercise with a different question. The difference between two dice is triangular as well, centered on zero, which is the shape a null distribution takes when the two things you subtracted really are alike. Recording only the highest of two dice breaks the symmetry: the counts climb 1, 3, 5, 7, 9, 11 out of 36, and the average rises from 3.5 to 4.47, which is why a tabletop game hands you a second die as a reward. Lowest-of-two is its mirror image, at 2.53.

Two dice have room for a surprise, too. Label one of them 1, 2, 2, 3, 3, 4 and the other 1, 3, 4, 5, 6, 8, and the totals follow exactly the same triangle as an ordinary pair. Those are the Sicherman dice, and they are the only other way to do it with positive whole numbers. Flattening the triangle, on the other hand, cannot be done at all: no weighting of two six-sided dice makes the eleven totals equally likely, because the chance of a 7 is always at least twice the square root of P(2) × P(12), and equal totals would need it to be twice its own size.

None of this is really about dice. Adding independent random quantities blends their distributions, and a few rounds of blending pull the result toward the normal curve whatever the ingredients looked like. That is the Central Limit Theorem demonstrated on the smallest dataset there is, and it explains why so many named distributions are sums underneath: the binomial counts coin flips, chi-square adds up squared z-scores, and the sampling distribution of a mean is a sum with a division at the end. Switch the playground above to Binomial with the overlay on and you are watching the same machinery run on different parts.