Variance & standard deviation
Variance measures spread: Var(X) = E[(X − μ)²], which expands to the computationally friendlier E[X²] − (E[X])². The standard deviation is its square root, restoring the original units. The scaling rule Var(aX + b) = a²Var(X) has two features worth memorising — the constant b disappears, and the multiplier is SQUARED.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Compute E[X] firstThe mean is needed for both formulas.
- Compute E[X²]Weight the squared values by their probabilities: Σ x²P(X = x).
- Apply the shortcutVar(X) = E[X²] − (E[X])². Far less arithmetic than the definitional form.
- Take the square root for the standard deviationAnd remember Var(aX + b) = a²Var(X), so SD(aX + b) = |a|·SD(X).
Worked example
Find the variance of a single fair die roll.
- E[X] = (1+2+3+4+5+6)/6 = 3.5.
- E[X²] = (1+4+9+16+25+36)/6 = 91/6.
- Var = 91/6 − 3.5² = 91/6 − 49/4.
- Common denominator 12: 182/12 − 147/12 = 35/12.
Answer. Var(X) = 35/12 ≈ 2.917, so SD ≈ 1.708.
Where marks get dropped
These are the specific errors that cost credit on variance & standard deviation questions — QED's rubric penalises each of them separately.
- Forgetting to square the constant in Var(aX). Doubling every value quadruples the variance.
- Subtracting a constant and expecting the variance to change. Var(X + b) = Var(X) — shifting does not affect spread.
- Computing E[X²] as (E[X])². These are different sums, and their difference is precisely the variance.
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Variance & standard deviation — frequently asked questions
Why square the deviations?
To stop positive and negative deviations cancelling — E[X − μ] is always 0. Squaring also makes the algebra tractable and connects to least squares.
Is Var(X + Y) = Var(X) + Var(Y)?
Only for independent (or uncorrelated) X and Y. In general you must add 2Cov(X,Y).
Why prefer the standard deviation?
It has the same units as X, so it can be compared with the mean directly. Variance is in squared units, which is unintuitive.
The rest of Probability
Events, conditional probability, random variables. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Sample spaces & events
- 2Conditional probability & independence
- 3Bayes’ theorem
- 4Random variables & expected value
- 5Variance & standard deviation
- 6Binomial & uniform distributions
- 7Law of total probability & tree diagrams
- 8Geometric & Poisson distributions
- 9Joint, marginal & conditional distributions
- 10Markov & Chebyshev bounds
- 11Equally likely outcomes & counting
- 12Linearity of expectation
- 13Indicator random variables
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