Variance and standard deviation
Use the whole valid distribution and retain exact intermediate moments
- Definition Var(X)=Σpᵢ(xᵢ−μ)²≥0.
- Identity Var(X)=E(X²)−[E(X)]²
- Standard deviation The nonnegative square root of variance; round only as requested
- 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
\[E(X)=\frac32,\quad E(X^2)=\frac43+\frac{25}{6}=\frac{11}{2}\]
- \(\displaystyle \operatorname{Var}(X)=\frac{11}{2}-\frac94=\frac{13}{4}\)Subtract the squared mean, not the mean
- \(\displaystyle \sigma=\frac{\sqrt{13}}2\approx1.80\)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 varianceUsing 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 estimateDifficulty 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