Variance, standard deviation & spread
Variance averages the squared deviations from the mean. The catch is the divisor: the population variance divides by n, while the SAMPLE variance divides by n − 1 to correct the downward bias that arises from using the sample mean. Exams state which is intended, and choosing wrongly changes the answer — with small n, substantially.
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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 the meanEvery deviation is measured from it, so an error here contaminates everything.
- Square the deviations and addΣ(x − x̄)². Alternatively use Σx² − n x̄², which is quicker by hand.
- Divide by n or n − 1n for a population, n − 1 for a sample. When in doubt for real data, use n − 1.
- Take the square root for the SDWhich restores the original units and can be compared with the mean directly.
Worked example
Find the sample standard deviation of 2, 4, 4, 6, 9.
- Mean = 25/5 = 5.
- Deviations: −3, −1, −1, 1, 4. Squares: 9, 1, 1, 1, 16, summing to 28.
- Sample variance = 28/(5−1) = 7.
- SD = √7.
Answer. Sample variance 7 and standard deviation √7 ≈ 2.646. (The population version would divide by 5, giving 5.6 and SD ≈ 2.366.)
Where marks get dropped
These are the specific errors that cost credit on variance, standard deviation & spread questions — QED's rubric penalises each of them separately.
- Using n when the data are a sample. With n = 5 the two answers differ by 25%, so it matters.
- Forgetting to square-root the variance when the standard deviation is asked for.
- Comparing standard deviations across data with different units or scales. Use the coefficient of variation for that.
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Variance, standard deviation & spread — frequently asked questions
Why divide by n − 1?
Because the sample mean is itself estimated from the data, the deviations are systematically too small. Dividing by n − 1 (the degrees of freedom) makes the estimator unbiased — Bessel’s correction.
What is the range and why is it weak?
Max minus min. It uses only two values and is extremely sensitive to outliers, so the interquartile range is preferred for robust spread.
How do I interpret a standard deviation?
For roughly bell-shaped data, about 68% of values lie within one SD of the mean and 95% within two — the empirical rule.
The rest of Statistics
Describing data, distributions, estimation and hypothesis tests. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Mean, median & mode
- 2Variance, standard deviation & spread
- 3Shape, skew & outliers
- 4Boxplots, histograms & quartiles
- 5The normal distribution & z-scores
- 6Sampling, bias & the sampling distribution
- 7The central limit theorem
- 8Confidence intervals for a mean
- 9Hypothesis testing & p-values
- 10t-tests & comparing two means
- 11Chi-square tests for independence
- 12Correlation & least-squares regression
- 13Type I / Type II errors & power
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