Population (σ) divides the sum of squared differences by N. Sample (s) divides by N−1 (Bessel's correction), giving an unbiased estimate of population variance when working with a sample.
Carregando ferramenta… / Loading tool…
Calculate standard deviation, variance, mean and dispersion from a list of numbers.
Standard deviation measures how far the data lies, on average, from its mean, the larger it is, the more spread out the values are. The computation follows four steps: find the mean (μ = sum ÷ N), measure each value's deviation from the mean, square each deviation and add them up, then divide that sum and take the square root. The divisor is what distinguishes the two versions: the population one (σ) divides by N and applies when you have every data point in the universe; the sample one (s) divides by N−1, Bessel's correction, used when the data is only a sample of the full population. The variance is simply the standard deviation squared (σ² or s²). The tool also shows the mean, minimum, maximum and range, accepts lists separated by comma, semicolon, space or newline (with dot or comma decimals) and uses a two-pass algorithm to reduce floating-point error. Everything runs locally in your browser.
It all starts with the mean: μ = sum of the values ÷ N. Then, for each value, you compute the deviation (xᵢ − μ) and square it, squaring keeps deviations above and below from canceling and gives more weight to large departures. The sum of those squares, Σ(xᵢ − μ)², concentrates all the spread.
The final step is to divide and take the root. For the population variance, divide by N: σ² = Σ(xᵢ − μ)² ÷ N. For the sample, divide by N−1: s² = Σ(xᵢ − μ)² ÷ (N − 1). The standard deviation is the square root of the variance, which brings it back to the same unit as the data (the variance is in unit²).
Example 1, population: 2, 4, 4, 4, 5, 5, 7, 9. The mean is 40 ÷ 8 = 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4 and 16, which sum to 32. The population variance is 32 ÷ 8 = 4, so σ = √4 = 2.
Example 2, sample vs population: 2, 4, 6, 8. The mean is 5 and the sum of squared deviations is 9 + 1 + 1 + 9 = 20. As a sample, s² = 20 ÷ (4 − 1) ≈ 6.667 and s ≈ 2.582; as a population, σ² = 20 ÷ 4 = 5 and σ ≈ 2.236. Dividing by N−1 always gives a slightly larger deviation.
For data that follows a normal (bell-shaped) distribution, the empirical 68–95–99.7 rule holds: about 68% of values fall within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. This turns the standard deviation into a ruler for judging whether a value is typical or unusual. Because deviations are squared, the result is quite sensitive to outliers, a single distant value inflates it a lot.
The choice between population and sample depends on what you have: use σ (÷N) when the data is the entire universe (the grades of every student in a class), and s (÷N−1) when it is a sample estimating something larger (a survey, a production batch). Standard deviation is everywhere in finance (volatility/risk), quality control, grades and lab results.
Paste the code into your HTML and the tool shows up on your page, without J-Kit's navigation and ads. It still runs in the browser of whoever visits your site.
<iframe
src="https://jkit.tools/embed/en-US/standard-deviation-calculator"
width="100%"
height="600"
style="border:0"
loading="lazy"
title="Standard Deviation Calculator"
></iframe>These references help contextualize formulas, standards, APIs and limitations used on this page. They do not replace professional validation when a result has legal, financial, medical or operational impact.
Population (σ) divides the sum of squared differences by N. Sample (s) divides by N−1 (Bessel's correction), giving an unbiased estimate of population variance when working with a sample.
All calculations stay in your browser. No data is sent to any server.