Definition · Plain-language
Standard deviation
Standard deviation measures how spread out a set of values is around its mean. It is the square root of the variance, so a larger standard deviation means more dispersion.
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What standard deviation measures
Standard deviation quantifies how far, on average, the values in a dataset sit from their mean. To compute it, you find each value’s deviation from the mean, square those deviations, average them to get the variance, then take the square root. Taking the square root returns the figure to the original units of the data — pounds, seconds or marks — which is why standard deviation is usually preferred to variance for reporting. A standard deviation of zero means every value is identical; the larger it grows, the more scattered the data.
Reading it alongside the mean
A mean alone hides how representative it is, and the standard deviation supplies the missing context. Two classes can both average 60% in an exam, but one with a small standard deviation has marks bunched near 60, while one with a large standard deviation has marks ranging from very low to very high. For roughly normal data, the empirical rule applies: about 68% of values fall within one standard deviation of the mean and about 95% within two, which lets you judge how unusual any single value is.
Population versus sample
There are two versions of the standard deviation. The population standard deviation (σ) divides the summed squared deviations by N, the total number of values, and is used when you have data for an entire population. The sample standard deviation (s) divides by N − 1 instead — known as Bessel’s correction — which gives an unbiased estimate of the population spread when you only have a sample. The N − 1 denominator slightly inflates the result to compensate for using the sample mean rather than the true population mean.
Key facts
At a glance
- Definition: average spread of values around the mean
- Relationship: the square root of the variance
- Units: same units as the original data
- Symbols: σ (population) and s (sample)
- Population: divides squared deviations by N
- Sample: divides by N − 1 (Bessel’s correction) for an unbiased estimate
Common misconceptions
What people often get wrong
Often heard: Standard deviation and variance are the same thing.
Actually: They are closely related but not identical. The variance is the average squared deviation from the mean; the standard deviation is its square root, expressed in the data’s original units, which makes it easier to interpret.
Often heard: A larger standard deviation means the mean is wrong.
Actually: No. A large standard deviation simply means the data are more spread out around the mean. The mean can be perfectly accurate; the standard deviation just tells you how representative of individual values it is.
Often heard: You always divide by N when calculating standard deviation.
Actually: Only for a full population. For a sample you divide by N − 1 (Bessel’s correction) to get an unbiased estimate of the population standard deviation, because the sample mean tends to underestimate spread.
Going deeper
