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 are one standard deviation to each side.

What to try

• t vs. normal: pick Student's t, turn on the normal overlay, and drag df from 1 upward. The heavy tails shrink as df grows, and by df ≈ 100 the t is so close to the normal that a t-table adds little. (lesson)
• The normal approximation: pick Binomial with the overlay on. At n = 5 the bell fits badly, and by n = 50 it fits well. That is the Central Limit Theorem at work. (lesson)
• Chi-square skew: 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", and Beta(20, 50) says the same thing with more confidence. Bayesian updating sharpens a prior in the same way. (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 a uniform distribution: six faces, each with probability 1/6, and a flat plot. Roll two and add them, and the plot is no longer flat. 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. This shape is the triangular distribution. The sum of two independent uniforms always has it, and it is why a 7 turns up six times as often as a 12 in any board game played with two dice. Keep adding dice and the straight sides begin to curve.

The other settings count the same rolls to answer a different question. The difference between two dice is also triangular, centered on zero. A null distribution for a difference has this shape when the two things you subtract do not differ. 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. That is why rolling two dice and keeping the higher one counts as a reward in some tabletop games. Lowest-of-two is its mirror image, at 2.53.

Two dice can also surprise you. 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. These are the Sicherman dice, the only other way to do this with positive whole numbers. Flattening the triangle, on the other hand, is impossible. 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). If all eleven totals were equally likely, that would make a 7 at least twice as likely as itself.

The same idea applies well beyond dice. Adding independent random quantities combines their distributions, and after a few additions the result is close to a normal curve, whatever the starting distributions looked like. That is the Central Limit Theorem on the smallest possible example. It also explains why so many named distributions are sums: the binomial counts coin flips, chi-square adds up squared z-scores, and the sampling distribution of a mean is a sum divided by n. Switch the playground above to Binomial with the overlay on and you see the same process with coin flips in place of dice.