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Standard deviation is taught in the note for Random Variables. Read it below and look for the section on this topic.

Random Variables

14 min read

Random variables

A random variable X takes numerical values determined by chance, for example the score on a die. A discrete random variable takes separate values and is described by a probability distribution P(X = x) with ΣP = 1. A continuous variable takes values on an interval and uses a density curve.

Expectation

The expectation (mean) is the long-run average value.

Variance and standard deviation

The variance measures the spread; the standard deviation has the same units as X.
Worked exampleA fair die

Find E(X) and Var(X) for a fair six-sided die.

  1. Expectation.
  2. E(X²).
  3. Variance.

Answer: E(X) = 3.5, Var(X) = 2.92, σ ≈ 1.71.

All six bars are equal: a uniform distribution.
DistributionMeanVariance
Uniform on 1 to n(n + 1)/2(n² − 1)/12
Binomial (n, p)npnpq
Poisson (λ)λλ
Normal (μ, σ²)μσ²
Try itA fair n-sided die

Choose the number of faces. See the mean, variance and standard deviation.

Check yourself For a random variable, the sum of all P(X = x) is
Check yourself If E(X) = 4 then E(3X + 2) =
Check yourself If Var(X) = 5 then Var(2X) =

Key terms

Random variable
A variable whose value depends on chance.
Expectation
The mean value of a random variable.
Variance
The mean of the squared distances from the mean.
Standard deviation
The square root of the variance.

Summary

  • A discrete probability distribution has ΣP = 1.
  • E(X) = Σ x P(x).
  • Var(X) = E(X²) − [E(X)]².
  • E(aX + b) = aE(X) + b and Var(aX + b) = a² Var(X).
Variance