Editing Variance (section)

===Fair die<!--Singular: die; plural: dice. Don't change-->===
A fair [[dice|six-sided die]] can be modeled as a discrete random variable, {{mvar|X}}, with outcomes 1 through 6, each with equal probability 1/6.  The expected value of {{mvar|X}} is <math>(1 + 2 + 3 + 4 + 5 + 6)/6 = 7/2.</math> Therefore, the variance of {{mvar|X}} is
<math display="block">\begin{align}
  \operatorname{Var}(X) &= \sum_{i=1}^6 \frac{1}{6}\left(i - \frac{7}{2}\right)^2 \\[5pt]
    &= \frac{1}{6}\left((-5/2)^2 + (-3/2)^2 + (-1/2)^2 + (1/2)^2 + (3/2)^2 + (5/2)^2\right) \\[5pt]
    &= \frac{35}{12} \approx 2.92.
\end{align}</math>

The general formula for the variance of the outcome, {{mvar|X}}, of an {{nowrap|{{mvar|n}}-sided}} die is
<math display="block">\begin{align}
  \operatorname{Var}(X) &= \operatorname{E}\left(X^2\right) - (\operatorname{E}(X))^2 \\[5pt]
    &= \frac{1}{n}\sum_{i=1}^n i^2 - \left(\frac{1}{n}\sum_{i=1}^n i\right)^2 \\[5pt]
    &= \frac{(n + 1)(2n + 1)}{6} - \left(\frac{n + 1}{2}\right)^2 \\[4pt]
    &= \frac{n^2 - 1}{12}.
\end{align}</math>