Standard Deviation

The standard deviation σ is the square root of the variance: σ = √Var(X). It measures how spread out values are from the mean in the original units of measurement. A standard deviation of 0 means no spread (all values equal); a large standard deviation means values are widely dispersed.

For a normal distribution, the standard deviation determines the shape of the bell curve: about 68% of values fall within ±1σ of the mean, about 95% within ±2σ, and about 99.7% within ±3σ. This “68-95-99.7 rule” (or empirical rule) gives a quick way to interpret spread for approximately normal data.

The sample standard deviation s estimates the population standard deviation. It is computed as s = √[(1/(n−1)) Σ (xᵢ − x̄)²], using n − 1 (Bessel’s correction) rather than n to produce a less biased estimate. Standard deviation is preferred over variance for reporting because it has the same units as the data — if heights are measured in centimeters, the standard deviation is also in centimeters, while the variance is in centimeters squared.