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Population Standard Deviation Calculator

Paste your numbers and get the sample or population standard deviation, variance, mean, SEM and CV% — with the squared deviations shown step by step.

Separate numbers with commas, spaces or new lines. A decimal comma like 3,14 is understood — separate such numbers with spaces or new lines.

Processing... 0%
Parsing the numbers
Computing the mean
Summing the squared deviations

Result

This page opens the standard deviation calculator preset to the population formula: paste your numbers and you get σ, the population standard deviation, along with the variance, mean, standard error and coefficient of variation, all computed with the divisor n.

The population formula is the plain root mean square of the deviations: σ = √(Σ(xᵢ − x̄)² ÷ n). First the mean is computed, then each value's squared distance from it, and the average of those squares is the variance σ². For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the squared deviations sum to 32, and σ = √(32 ÷ 8) = 2 exactly.

Divide by n only when your list is complete in itself — the grades of every student in the class, the ages of every employee in the company, all twelve monthly totals of one year. When the data is just a sample drawn from a larger whole, the n divisor underestimates the true spread; flip the select to sample mode and the calculator switches to n − 1.

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Accepted Formats & Rules

  • Standard deviation, variance, mean, standard error of the mean (SEM) and coefficient of variation (CV%).
  • Exact two-pass algorithm: the mean first, then the true squared deviations — no streaming shortcuts.
  • Step-by-step deviations table (first 50 rows for long lists, with sums always covering every value).

Key Benefits

✓ Sample (n − 1) and population (n) formulas in one tool — pick the right one with a single select.
✓ The full working is shown: every deviation from the mean, its square, and the exact sums used.
✓ Everything is computed locally in your browser — your data never leaves the page.

Use Cases

  • • Paste a batch of lab or QA measurements and get the sample standard deviation with Bessel's correction (n − 1), ready to quote in a report.
  • • Compare volatility between two data sets — response times, prices, test scores — using the coefficient of variation, which normalizes SD by the mean.
  • • Hand-check homework: the step table prints every xᵢ − x̄ and (xᵢ − x̄)² so each squared deviation can be verified line by line.