Standard deviation measures spread. Two datasets can share a mean and be nothing alike: scores of 50, 50, 50 and scores of 0, 50, 100 both average 50, but only one of them describes a consistent performance. Standard deviation is the number that tells them apart, and it is expressed in the same units as the original data, which makes it directly interpretable in a way variance is not.
A small standard deviation means values cluster near the mean. A large one means they scatter. That is the entire intuition; everything else is detail about how it is computed.
Population versus sample
If your data is the entire group you care about, divide by n — that is the population standard deviation, written σ. If it is a sample being used to estimate a wider population, divide by n − 1 instead, giving the sample standard deviation, s. The n − 1 adjustment, called Bessel's correction, exists because a sample tends to underestimate the spread of the population it came from. On large datasets the difference is negligible; on small ones it is substantial, which is exactly when it is most often ignored.
Calculating it step by step
- Find the mean of all values.
- Subtract the mean from each value to get its deviation.
- Square each deviation, which removes the signs and weights larger deviations more heavily.
- Average the squared deviations — dividing by n or n − 1 as appropriate. This is the variance.
- Take the square root, returning the result to the original units.
The 68-95-99.7 rule
| Range | Share of data | Reading |
|---|---|---|
| Within 1σ of the mean | ≈ 68% | Ordinary |
| Within 2σ | ≈ 95% | Common threshold for "unusual" |
| Within 3σ | ≈ 99.7% | Rare; used as a process-control limit |
| Beyond 3σ | ≈ 0.3% | Investigate the measurement first |
Reading it honestly
The 68-95-99.7 rule assumes a roughly normal distribution. Applied to skewed data — incomes, response times, anything with a long tail — it will mislead badly, and a median with an interquartile range is the more honest summary. Standard deviation is also sensitive to outliers, since deviations are squared: one extreme value can dominate the result. Always look at the distribution before quoting the number.
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