d20 or 2d10? The Shape of a Dice Roll

One twenty-sided die and two ten-sided dice both average about 11, but they play differently. Every d20 face is equally likely, while 2d10 piles up in the middle, so swingy and steady rolls can share the same average.

How it works

Discrete distributions, expected value and standard deviation

In one sentence

A discrete random variable $X$ has a probability mass function $P(x) \ge 0$ with $\sum P(x) = 1$ . Its expected value is $\mu = \sum x\,P(x)$ and its standard deviation is $\sigma = \sqrt{\sum (x-\mu)^2 P(x)}$ ; for constants $a$ and $b$ , $E(aX + b) = aE(X) + b$ and $\sigma_{aX+b} = |a|\,\sigma_X$ .

General formula

{\htmlData{sym=0}{\mu}} = \sum {\htmlData{sym=1}{x}}\,{\htmlData{sym=2}{P(x)}}, \qquad {\htmlData{sym=3}{\sigma}} = \sqrt{\sum ({\htmlData{sym=1}{x}}-{\htmlData{sym=0}{\mu}})^2\,{\htmlData{sym=2}{P(x)}}}

The mean is where the bar chart balances; the standard deviation is how far a typical roll lands from it.

What the symbols mean

$\mu$
Expected value
The long-run average roll: the balance point of the bars.
$x$
A possible total
One value the roll can come up.
$P(x)$
Its probability
The chance of rolling exactly that total.
$\sigma$
Standard deviation
How far a typical roll lands from the average.

A discrete distribution lists every outcome $x$ with its probability $P(x)$ ; the probabilities add up to 1.
One die with $2s$ sides is uniform: each face has probability $\frac{1}{2s}$ . Two $s$ -sided dice make a triangle, because there are more ways to roll a middle total (5+6, 6+5, 4+7, …) than an extreme one.
The expected value $\mu = \sum x P(x)$ is the long-run average roll, the balance point of the bars. Both rolls average about $s+1$ .
The standard deviation $\sigma$ measures the typical swing. Adding a flat bonus shifts $\mu$ by that bonus but leaves $\sigma$ unchanged.

Learn the math

Discrete Probability DistributionsStats · Lesson 13, Discrete Random Variables Expected Value and Standard DeviationStats · Lesson 14, Discrete Random Variables