Standard Deviation Calculator

What this calculator does

This calculator parses a list of numbers and returns count, sum, mean, variance, standard deviation, minimum, maximum, and range. It also shows a step-by-step deviation table so you can see how each data point contributes to the spread.

Formula

Sample: s = √( Σ(xᵢ − x̄)² ÷ (n − 1) )

Population: σ = √( Σ(xᵢ − μ)² ÷ n )

Variable definitions

• xᵢ — Each value in the dataset.
• x̄ / μ — Sample mean / population mean.
• n — Number of values in the dataset.

Step-by-step calculation

1. Calculate the mean: sum the values and divide by n.
2. For each value, subtract the mean and square the result.
3. Sum the squared deviations.
4. Divide by (n − 1) for sample or by n for population — this is the variance.
5. Take the square root of the variance to get the standard deviation.

Worked example

Dataset: 4, 8, 6, 5, 3, 7 (n = 6).

• Mean: 33 ÷ 6 = 5.5
• Squared deviations: 2.25, 6.25, 0.25, 0.25, 6.25, 2.25 → sum = 17.5
• Sample variance: 17.5 ÷ 5 = 3.5; sample SD ≈ 1.871
• Population variance: 17.5 ÷ 6 ≈ 2.917; population SD ≈ 1.708
• Min = 3, Max = 8, Range = 5

How to use this calculator

1. Paste or type your numbers separated by commas, spaces, or new lines.
2. Choose Sample or Population.
3. Read the summary and the per-value deviation table.

Common mistakes

• Sample vs population: using the wrong divisor changes the answer. Use sample (n − 1) unless you have literally every data point in the population.
• Confusing SD with variance: variance is SD squared and uses the same units squared (e.g. dollars²).

Frequently asked questions