Standard Deviation and Variance: Formula, Example and Interpretation

What do variance and standard deviation measure?

Two classes can have the same average exam score and still look completely different: in one, almost everyone scores close to the average; in the other, some people ace the exam and others fail. Measures of variability (dispersion) capture this difference.

The variance takes each value’s deviation from the mean, squares it (so that negative and positive deviations do not cancel out) and averages the squares. Because squaring changes the unit (points become points²), we take the square root and get the standard deviation, which is back in the original unit. You can read it as “the typical distance of a value from the mean”.

• Sample variance: s² = Σ(x − M)² / (n − 1)
• Sample standard deviation: s = √(Σ(x − M)² / (n − 1))
• Population variance: σ² = Σ(x − μ)² / N, population SD σ = √σ²

Worked example step by step

Eight students take a 10-point quiz and score: 2, 4, 4, 4, 5, 5, 7, 9. The sum is 40, so the mean is M = 40 / 8 = 5.00.

Why divide by n − 1?

When you compute deviations from the sample mean, they are on average a little too small, because the sample mean is by definition the point that minimizes the squared deviations in that sample. Dividing by n would therefore underestimate the population variance systematically. Dividing by n − 1 corrects this bias (Bessel’s correction).

Another way to see it: once you know the mean and n − 1 deviations, the last deviation is fixed, because all deviations sum to zero. Only n − 1 deviations are free to vary. This number is called the degrees of freedom, a concept you will meet again in the t-test and in ANOVA.

In practice: in a thesis you almost always analyze a sample, so use n − 1. The difference matters for small samples and becomes negligible for large ones.

Interpreting the SD: the 68–95–99.7 rule

If a variable is approximately normally distributed, the standard deviation tells you where most values lie. This empirical rule is a quick plausibility check for your data.

• A z-score expresses a value in SD units: z = (x − M) / SD. In the quiz, a score of 9 has z = (9 − 5) / 2.14 = 1.87.
• Values with |z| > 3 are rare in normal data and worth checking as possible outliers.
• The rule does not hold for strongly skewed data; there, use the median and interquartile range.

Standard deviation vs. standard error

These two are mixed up constantly, even in published papers. The standard deviation describes how much individual values vary. The standard error of the mean describes how much the mean itself would vary from sample to sample:

SE = SD / √n

In the quiz example: SE = 2.14 / √8 = 0.76. The SE shrinks as the sample grows, the SD does not: a larger class does not make students more similar, but it does make the class average more precise. Use the SD to describe your sample and the SE (or better, a confidence interval) to describe the precision of an estimate. Error bars in charts must say which one they show.

Related measures of spread

The SD is the standard measure for metric, roughly symmetric data. For other situations there are alternatives.

How to report the standard deviation

APA style reports the SD in parentheses after the mean, both with the same number of decimals (usually two). In tables, use separate columns for M and SD. Never use the ± notation without saying whether it means SD or SE.