Standard Deviation Calculator

How to use the standard deviation calculator

1. Type or paste your data in the Data set box. Separate the values with commas, spaces, semicolons or new lines; a column copied from a spreadsheet works directly. Use a point for decimals (2.5).
2. The results update as you type, or press Calculate.
3. Read both versions side by side: Sample (divide by n − 1) gives the Standard deviation s and Variance s²; Population (divide by n) gives the Standard deviation σ and Variance σ².
4. Below, you also get the Mean x̄, the Count n, the Sum Σx, the Sum of squares Σ(x − x̄)² and the Standard error s/√n.
5. The Step by step panel writes out the formula with your numbers: the mean, each squared deviation, their sum, then the variances and square roots. Copy copies every result as text.

Sample or population?

Use the population formula when your data contains every member of the group you care about: the heights of all 24 pupils of one class, the scores of every game of a season. Use the sample formula when your data is a subset used to estimate a larger group: 50 customers surveyed out of thousands, ten measurements of a process that could be repeated forever.

In practice, if you are unsure, the sample standard deviation s is usually what a teacher, a lab report or a statistics package (for example STDEV.S in spreadsheets) expects. The population version corresponds to STDEV.P.

The formulas

With n values x₁ … xₙ and their mean x̄ = Σx ÷ n:

• Sum of squared deviations: SS = Σ(x − x̄)²
• Population variance: σ² = SS ÷ n, and population standard deviation σ = √σ²
• Sample variance: s² = SS ÷ (n − 1), and sample standard deviation s = √s²
• Standard error of the mean: s ÷ √n

Dividing by n − 1 instead of n (Bessel’s correction) compensates for the fact that deviations are measured from the sample’s own mean, which is always a little closer to the data than the true mean. Without it, the sample variance would underestimate the population variance on average.

Worked example. For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 40 ÷ 8 = 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4 and 16, whose sum is 32. The population variance is 32 ÷ 8 = 4, so σ = 2. The sample variance is 32 ÷ 7 ≈ 4.571, so s ≈ 2.138.

Accurate even with large numbers

The textbook shortcut Σx² − (Σx)² ÷ n subtracts two huge, nearly equal numbers and can lose all its precision, for instance with timestamps or values around one billion. This tool uses Welford’s algorithm instead: it updates the mean and the sum of squared deviations one value at a time, which keeps the result accurate even when the values share a large offset.