Standard Deviation

Standard deviation summarizes how spread out a set of numbers is. A small standard deviation means values cluster near the mean; a large one means they are scattered. It is the square root of the variance, and the NIST/SEMATECH e-Handbook notes that taking the root restores the units of the spread to the original data units, since variance squares them. Two versions exist: the sample standard deviation (s), which divides by n − 1 and estimates the spread of a larger population, and the population standard deviation (σ), which divides by n and describes a complete group.

Formula

s = √[ Σ (xᵢ − x̄)² ÷ (n − 1) ]     (sample)
σ = √[ Σ (xᵢ − μ)² ÷ n ]           (population)

• xᵢ: each value.
• x̄ or μ: the mean of the values.
• n: the number of values.
• Σ: the sum over all values.

In Excel, STDEV.S returns the sample version and STDEV.P the population version.

Example

Data: 2, 4, 4, 4, 5, 5, 7, 9 (n = 8).

1. Mean = 40 ÷ 8 = 5.
2. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32.
3. Population: σ = √(32 ÷ 8) = √4 = 2.
4. Sample: s = √(32 ÷ 7) ≈ √4.571 ≈ 2.138.

The standard deviation calculator returns both versions, and the step-by-step guide walks through each stage. The mean, median and mode calculator or average calculator handles step 1.

Common mistakes

• Using the population formula on a sample, which understates the spread, most noticeably in small samples.
• Reporting variance when standard deviation was asked for, or the reverse. Variance is in squared units.
• Treating standard deviation as robust. NIST notes it is less reliable for long-tailed data, where measures such as the interquartile range may suit better.