Standard Deviation Calculator
Find standard deviation and variance for any dataset — population or sample — with every step shown.
Calculate
Population = your data is the whole group. Sample = your data is a subset of something bigger. Not sure? See the guide below.
| Standard deviation | 2.0000 |
| Variance | 4.0000 |
| Mean | 5.0000 |
| Count | 8 |
| Min / Max / Range | 2 / 9 / 7 |
Step by step
| x | x − mean | (x − mean)² |
|---|---|---|
| 2 | -3.0000 | 9.0000 |
| 4 | -1.0000 | 1.0000 |
| 4 | -1.0000 | 1.0000 |
| 4 | -1.0000 | 1.0000 |
| 5 | 0.0000 | 0.0000 |
| 5 | 0.0000 | 0.0000 |
| 7 | 2.0000 | 4.0000 |
| 9 | 4.0000 | 16.0000 |
Variance = 32 ÷ 8 = 4 · SD = √4 = 2
What Standard Deviation Means
Standard deviation (SD) measures spread: how far numbers typically stray from the average. A small SD means the data clusters tightly around the mean; a large SD means it's scattered. Two classes can both average 75% on a test — but if one has an SD of 5 and the other 20, the first class performed uniformly while the second had both stars and strugglers. The mean alone never tells you that.
Think of it as the "typical distance from average." An SD of 2 on quiz scores means most students landed within about 2 points of the mean — immediately more informative than the raw list of scores.
The Formula: Population vs Sample
The only difference is the divisor: N for a population, N−1 for a sample. The steps are identical otherwise: find the mean, subtract it from each value (deviations), square each deviation, average the squares (that's the variance), and take the square root. We square because raw deviations always sum to zero — positives cancel negatives — and the square root at the end brings the units back to the original (points, not squared points).
Worked Example With Steps
Dataset: 2, 4, 4, 4, 5, 5, 7, 9 (population).
Step 1 — mean: (2+4+4+4+5+5+7+9) ÷ 8 = 40 ÷ 8 = 5.
Step 2 — deviations: −3, −1, −1, −1, 0, 0, 2, 4.
Step 3 — square them: 9, 1, 1, 1, 0, 0, 4, 16.
Step 4 — variance: (9+1+1+1+0+0+4+16) ÷ 8 = 32 ÷ 8 = 4.
Step 5 — SD: √4 = 2.
So a "typical" value sits about 2 away from the mean of 5. The table above shows this exact working for your own data — the same steps a teacher expects to see.
When to Use Population vs Sample
- Population (σ, divide by N): your data is the entire group you care about — all students in your class, every sale this quarter.
- Sample (s, divide by N−1): your data is a subset standing in for something larger — 50 survey respondents representing a city, 30 patients in a trial.
The N−1 adjustment (Bessel's correction) fixes a subtle bias: a sample's own mean always fits the sample data a little too well, so dividing by N would understate the true spread. If you're unsure, sample is the safer default in most coursework — and the calculator labels which one you used right in the result.
How to Interpret the Result
Always read SD next to the mean. An SD of 2 with a mean of 100 is negligible; with a mean of 5, it's enormous. A handy rule of thumb (the empirical rule): for roughly bell-shaped data, about 68% of values fall within one SD of the mean, 95% within two, and 99.7% within three.
Compare SDs to compare consistency: the steadier process has the smaller SD. And remember what SD can't do — it doesn't tell you why data spreads, and a single extreme outlier inflates it dramatically. If one value looks wrong, check it before trusting the summary.
Standard Deviation in Real Life
SD quietly runs a lot of decisions. Investing: a fund's volatility is quoted as the standard deviation of its returns — higher SD, bumpier ride. Manufacturing: quality control tracks the SD of part dimensions; a shrinking SD means the process is getting more consistent, which is often worth more than a better average. Sports: two basketball players can average 20 points per game, but the one with the lower SD is the reliable scorer coaches trust in close games.
Test scores and grading curves are the classic classroom use: many curved grades are set as "mean ± SD" bands, so knowing the SD tells you exactly how far above the pack you need to be. And in polling, the margin of error you see in headlines is built directly from a standard-deviation-style calculation — it's why a "±3 point" poll can't really distinguish 47% from 49%.
The pattern is always the same: the average tells you where the center is, and the standard deviation tells you how much the world wobbles around it. Any time someone quotes you an average without a spread, ask for the SD — you're missing half the story without it.
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Frequently Asked Questions
How do you calculate standard deviation?
Find the mean, subtract it from each value, square each difference, average those squares (that is the variance), then take the square root. For the sample 2, 4, 4, 4, 5, 5, 7, 9: mean = 5, variance = 4, SD = 2.
What is the difference between population and sample standard deviation?
Population SD (σ) divides by N and describes the entire group. Sample SD (s) divides by N−1, which corrects for the bias of estimating from a subset — use it when your data is a sample of something larger.
What is a good standard deviation?
There is no universal "good" value — it depends on the data. A small SD means values cluster tightly around the mean; a large one means they spread out. Compare it to the mean: an SD of 2 on a mean of 100 is tiny, but on a mean of 5 it is huge.
What is the formula for variance?
Variance is the average of the squared differences from the mean: population variance σ² = Σ(x − μ)² / N, sample variance s² = Σ(x − x̄)² / (N − 1). Standard deviation is the square root of variance.
Why N−1 for sample standard deviation?
A sample mean is always closer to the sample data than the true population mean is, so dividing by N would systematically underestimate spread. Dividing by N−1 (Bessel's correction) removes that bias.