ThisCalculator

Statistics

Calculate the Standard Deviation

Updated · Standard formula, shown below

Sample standard deviation s 13.491
Population standard deviation σ
12.315
Result
Mean
18.000
Result
Count
6 values
Result

Numbers separated by commas, spaces or new lines.

This calculator finds the standard deviation of a list of numbers. Type or paste your values separated by commas and you get the count, the mean, the sample standard deviation and the population standard deviation.

Formula

Mean x̄ = Σx ÷ n. Sample standard deviation s = √( Σ(x − x̄)² ÷ (n − 1) ). Population standard deviation σ = √( Σ(x − x̄)² ÷ n ).

  • x: each data value (any unit; the results share it)
  • n: the number of values
  • Σ(x − x̄)²: the sum of squared deviations from the mean

Worked Example

Data: 4, 8, 15, 16, 23, 42.

  1. n = 6, Σx = 108, mean = 108 ÷ 6 = 18.000
  2. Deviations: −14, −10, −3, −2, 5, 24; squares: 196, 100, 9, 4, 25, 576; sum = 910
  3. Sample: 910 ÷ 5 = 182, s = √182 = 13.491
  4. Population: 910 ÷ 6 = 151.667, σ = √151.667 = 12.315

Limits and common mistakes

  • Use the sample value s when the numbers are a sample from a bigger group (the usual case); use σ only when you have every member of the population.
  • You need at least two values for s.
  • One extreme value (42 here) pulls both the mean and the standard deviation up; check for typing errors first.
Formula used
list x | Data values | 4, 8, 15, 16, 23, 42
out s | Sample standard deviation s | - | sd(x) | 3
out sp | Population standard deviation σ | - | psd(x) | 3
out avg | Mean | - | mean(x) | 3
out n | Count | values | count(x) | 0

Each line is one input (in, sel, list) or one result (out: label, unit, formula, decimals). How the calculators work

Method: the standard formula for this calculation, rounded as each result states. See how the calculators work or report an error.