---
title: "Standard Deviation Calculator — Population (σ) and Sample (s)"
description: "Free standard deviation calculator. Get both population (σ, ÷n) and sample (s, ÷n−1) standard deviation from any data set, with mean, variance, and a step-by-step deviation table. Explains Bessel's correction and curriculum differences."
url: "https://worldcalculators.org/calculators/standard-deviation/"
canonical: "https://worldcalculators.org/calculators/standard-deviation/"
---

# Standard Deviation Calculator

Free standard deviation calculator. Get both population (σ, ÷n) and sample (s, ÷n−1) standard deviation from any data set, with mean, variance, and a step-by-step deviation table. Explains Bessel's correction and curriculum differences.

## Quick answer

Sample s = √[ Σ(x−x̄)² ÷ (n−1) ] — use for samples (nearly always).  ·  Population σ = √[ Σ(x−μ)² ÷ n ] — only if you have ALL the data.  ·  Excel STDEV() = sample · STDEVP() = population.

## Population vs Sample: Which One Do I Use?

Use population standard deviation (σ, ÷n) only when your data covers the entire group you care about — every employee in a company, every product made in one shift. Use sample standard deviation (s, ÷n−1) when your data is a sample used to estimate a larger population — a survey of 200 voters, test scores from one class representing all students. In nearly all real analysis you have a sample, so ÷n−1 is the safe default .

## Why ÷(n−1)? Bessel's Correction

A sample tends to cluster slightly closer to its own mean than the full population does, so dividing by n would underestimate the true spread. Dividing by n−1 — Bessel's correction — inflates the result just enough to make the sample standard deviation an unbiased estimator of σ. This is why Excel, R, and most scientific calculators default to ÷n−1 for the "s" key.

## Frequently Asked Questions

Population standard deviation (σ) divides the sum of squared deviations by n and is used when your data covers the ENTIRE group. Sample standard deviation (s) divides by n−1 (Bessel's correction) and is used when your data is a sample drawn from a larger population. In practice the sample formula (÷n−1) is almost always correct, because you rarely have data on the complete population.
Dividing by n−1 instead of n is called Bessel's correction. It compensates for the fact that a sample's spread tends to underestimate the true population spread, making s an unbiased estimator of σ. This is the default in Excel STDEV(), Python statistics.stdev(), R sd(), and the s/Sx key on Casio and TI calculators.

## Sources & Methodology

Population (÷n) and sample (÷n−1) standard deviation and variance are computed client-side from the squared deviations about the mean. Curriculum and software defaults reflect published exam-board specifications and documented function behaviour.
Standards and figures reviewed June 2026.

Source: https://worldcalculators.org/calculators/standard-deviation/
Markdown mirror of the HTML page. Prefer this URL for RAG; the interactive calculator still lives on the HTML page.