Sample Variance
Unbiased estimate of population variance.
Formula
Variables
Example
Uses n-1 (Bessel correction)
How it works
Sample variance measures how spread out a set of observations is around their average, using the squared distances from the mean. The division by n-1 instead of n corrects for the fact that the sample mean is itself estimated from the same data, which would otherwise make the spread look slightly too small on average. This makes s^2 an unbiased estimator of the population variance when the sample is randomly drawn.
When to use it
Use this when you want to summarize variability from a sample of data, such as test scores, experimental measurements, or survey responses. It is especially useful when the sample is meant to represent a larger population and you want a variance estimate that is not systematically too low.
Worked example, step by step
Data: 4, 7, 9, 10 Step 1: Find the mean: x̄ = (4 + 7 + 9 + 10) / 4 = 30 / 4 = 7.5 Step 2: Subtract the mean and square each result: (4 - 7.5)^2 = (-3.5)^2 = 12.25 (7 - 7.5)^2 = (-0.5)^2 = 0.25 (9 - 7.5)^2 = (1.5)^2 = 2.25 (10 - 7.5)^2 = (2.5)^2 = 6.25 Step 3: Add the squared deviations: 12.25 + 0.25 + 2.25 + 6.25 = 21 Step 4: Divide by n - 1 = 4 - 1 = 3 s^2 = 21 / 3 = 7 Final answer: the sample variance is 7
Common mistakes to avoid
- Dividing by n instead of n - 1 when the data are a sample
- Using x̄ from a different dataset or rounding it too early
- Forgetting to square the deviations before adding them
Did You Know?
Sample variance divides by n−1, not n — Bessel's correction that stops it from systematically underestimating spread.
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IntermediateAverage squared deviation from the mean.
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