Standard deviation calculator

How to use it

Paste numbers separated by spaces, commas, semicolons, tabs or new lines — a column copied from a spreadsheet works. Use a point for decimals (1.5, not 1,5). Results update as you type and when you press the button; copy them or download the tables as CSV.

Choose the quartile method your course or software uses; the median and every other statistic are the same under each method.

Formulas

• Mean x̄ = Σx / n. Sample variance s² = Σ(x − x̄)² / (n − 1); population variance σ² = Σ(x − x̄)² / n; the standard deviation is the square root.
• Standard error of the mean = s / √n. Coefficient of variation = s / |x̄|.
• Quartiles — R type 7 (Excel QUARTILE.INC, NumPy): position h = (n − 1)p + 1 in the sorted data, interpolating between neighbours. Type 6 (Minitab, SPSS, Excel QUARTILE.EXC): h = (n + 1)p. Tukey’s hinges: medians of the lower and upper halves, the median counted in both halves when n is odd.
• Outliers use Tukey’s fences Q1 − 1.5 × IQR and Q3 + 1.5 × IQR (3 × IQR marks far outliers).
• Skewness G1 = g1 √(n(n − 1)) / (n − 2) with g1 = m3 / m2^3/2; excess kurtosis G2 = ((n + 1) g2 + 6)(n − 1) / ((n − 2)(n − 3)) with g2 = m4 / m2² − 3, where m_r = Σ(x − x̄)^r / n. G1 and G2 match Excel SKEW and KURT, SPSS and SciPy with bias=False; g1 and g2 are the population (biased) versions.

Which standard deviation?

Use the sample standard deviation (n − 1) when your data are a sample used to estimate a larger population — the usual case. Use the population standard deviation (n) only when the data are the entire population of interest.

Assumptions and limits

• Up to 100,000 values. Skewness needs at least 3 values and kurtosis at least 4.
• Quartiles of small samples differ noticeably between methods; state the method you used.
• Fences flag unusual values for checking; they are not a rule for deleting data.
• All arithmetic runs in your browser in double precision. Sums are compensated and variances use the corrected two-pass method; tail probabilities come from the regularized incomplete beta and gamma functions and keep about 12 significant digits even for p-values as small as 1e-300. The NIST StRD NumAcc4 dataset (values near 10,000,000.2) returns its certified standard deviation 0.1 to 8 significant digits.

Worked example

For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5 and Σ(x − x̄)² = 32, so the population standard deviation is √(32 / 8) = 2 and the sample standard deviation is √(32 / 7) = 2.138. With type 7 quartiles Q1 = 4, Q3 = 5.5 and IQR = 1.5; the upper fence is 7.75, so 9 is flagged as an outlier. The mode is 4.

References

• NIST/SEMATECH e-Handbook of Statistical Methods: measures of scale
• NIST/SEMATECH e-Handbook of Statistical Methods: measures of skewness and kurtosis
• NIST/SEMATECH e-Handbook of Statistical Methods: what are outliers in the data?
• Hyndman and Fan (1996), Sample quantiles in statistical packages, The American Statistician 50(4)
• Joanes and Gill (1998), Comparing measures of sample skewness and kurtosis, The Statistician 47(1)
• R documentation: quantile (types 1–9)
• NIST Statistical Reference Datasets (StRD)