Standard Deviation
A measure of how spread out a set of values is around its mean, expressed in the same units as the data.
The standard deviation measures how spread out a set of values is around their mean. It is the square root of the variance: variance is the average of the squared distances from the mean, and taking the square root returns the measure to the original units of the data, which makes it directly interpretable. A small standard deviation means the values cluster tightly around the mean; a large one means they are widely dispersed.
The population standard deviation σ of N values x₁ … xₙ with mean μ is
σ = sqrt( (1/N) · Σ (xᵢ − μ)² )
and for a sample of n values the usual estimator divides by n − 1 instead of n:
s = sqrt( (1/(n−1)) · Σ (xᵢ − x̄)² )
The n − 1 correction (Bessel's correction) makes s an unbiased estimator of the population parameter when the mean itself is estimated from the same sample.
The standard deviation is the natural scale for many statistical statements. For an approximately normal distribution, about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three — the empirical rule. More generally, Chebyshev's inequality guarantees that for any distribution with finite variance, at least 1 − 1/k² of the values lie within k standard deviations of the mean, so the measure has meaning even without normality.
Standard deviation is sensitive to outliers because squaring distances gives extreme values heavy weight; the interquartile range is a more robust alternative in their presence. It underlies z-scores, effect sizes such as Cohen's d, and the standard error of the mean, and it is the σ that appears in the statement of the central limit theorem.
Tags
data analysis mathematics probability statistics
Related articles
- Central Limit Theorem
- Bayes' Theorem
- Fundamental Theorem of Arithmetic
- Non-Euclidean Geometry
- Pythagorean Theorem
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