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Variance, standard deviation and linear variance of a discrete random variable

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Mean alone does not describe spread

Two distributions can share a mean and have different variability. Values 0 and 4 with equal probabilities have mean 2, as do values 1 and 3 with equal probabilities. The first distribution lies farther from the common mean

Year 11, VCE Units 1 & 2, Chapter 6, 3 difficulty levels, Calculator allowedBuild a worksheet

Variance and standard deviation

Use the whole valid distribution and retain exact intermediate moments

  1. Definition Var(X)=Σpᵢ(xᵢ−μ)²≥0.
  2. Identity Var(X)=E(X²)−[E(X)]²
  3. Standard deviation The nonnegative square root of variance; round only as requested
  4. Zero spread Variance zero means the variable is constant with probability one

Worked example

For X taking 0, 2 and 5 with respective probabilities one half, one third and one sixth, calculate variance and SD

\[E(X)=\frac32,\quad E(X^2)=\frac43+\frac{25}{6}=\frac{11}{2}\]
  1. \(\displaystyle \operatorname{Var}(X)=\frac{11}{2}-\frac94=\frac{13}{4}\)Subtract the squared mean, not the mean
  2. \(\displaystyle \sigma=\frac{\sqrt{13}}2\approx1.80\)The exact standard deviation is shown above; to two decimal places it is 1.80

Common mistakes

Squaring after averaging
✕E(X²)=[E(X)]²✓E(X²)=Σxᵢ²pᵢ
The two quantities generally differ; their difference is variance
Using a sample divisor
✕divide the probability-weighted sum by n−1✓the probabilities already sum to one
This is the variance of a supplied distribution, not an unbiased sample estimate

Difficulty levels

introExact variance from supplied data
standardVariance or exact standard deviation
examVariance and SD, including explicit rounding when required

Questions about this topic

Why is challenge absent from this placement?+

The frozen bank uses challenge for linear variance transformations, which are taught in the next section

Can a negative exact variance be correct?+

No. Recheck the probabilities, the squared mean and arithmetic; the defining weighted squared deviations are nonnegative

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 6: Random variables and sampling

In this chapter

Expected value and linear expectation of a discrete random variableVariance, standard deviation and linear variance of a discrete random variableDistribution of a linear transformation of a discrete random variableConstruct finite independent sums and compare repeated copies with scalingDesign and interpret a probability simulationPopulations, samples, parameters, statistics and sampling methodsExact sampling distribution of a sample proportionConstruct sample means and interpret finite sampling distributions