Quick answer: Standard deviation measures how spread out a data set is. For the data set 2, 4, 4, 4, 5, 5, 7, 9, the population standard deviation is 2.00 and the sample standard deviation is 2.14. Paste your own numbers into this standard deviation calculator below to get both, plus variance, mean, and range.
Last reviewed: September 2026
Key facts
- Standard deviation is the square root of variance — it measures spread in the same units as your data.
- Use population SD (÷N) when your data is the whole group; use sample SD (÷n−1) when it is a sample of a larger group.
- Identical values give SD = 0; standard deviation can never be negative.
- In a roughly bell-shaped data set, about 68% of values fall within ±1 SD of the mean.
This free standard deviation calculator works on any data set you paste in — test scores, measurements, prices, or workout logs. It shows population and sample standard deviation side by side, so the sample-vs-population choice is made for you, and also functions as a variance calculator with mean, sum, range, and per-value squared deviations. Results update live as you type.
Standard deviation answers a question the mean (average) cannot: not "what is typical?" but "how much does it vary?" Two classes can both average 75% on a test, yet one has every student near 75 while the other swings from 40 to 100 — the standard deviation is what tells them apart.
How to use
- Paste your numbers into the box, separated by commas, spaces, semicolons, or line breaks — any mix works.
- You need at least 2 values; decimals and negative numbers are fine.
- Read the population SD (σ) in the blue card and the sample SD (s) in the green card.
- Check the summary table for mean, sum, min, max, and range.
- Open "Show per-value squared deviations" to see exactly how each value contributes to the total.
The standard deviation formulas
Population variance: σ² = Σ(x − μ)² ÷ N
Sample variance: s² = Σ(x − x̄)² ÷ (n − 1)
SD = √(variance)
The recipe is the same for both: find the mean, measure how far each value sits from it, square those distances, average them, and take the square root. Squaring stops positive and negative deviations from cancelling out, and the final square root returns the answer to the original units (dollars, centimetres, seconds — not squared dollars).
Sample vs population standard deviation
This is the one decision that trips people up, so this calculator simply shows you both:
- Population (σ, divide by N): use it when your data set is the entire group you care about — all employees in a department, every measurement from today's experiment.
- Sample (s, divide by n−1): use it when your data is a sample drawn from a larger group — 30 survey respondents standing in for a whole market, 8 test runs standing in for a machine's lifetime behaviour.
The sample version divides by n−1 (Bessel's correction) because a sample's own mean always sits slightly closer to its values than the true population mean does; without the correction, variance would be systematically underestimated. With large data sets the two answers converge anyway.
Worked examples
Data set: 2, 4, 4, 4, 5, 5, 7, 9
n = 8, sum = 40, mean = 40 ÷ 8 = 5.00. Squared deviations from the mean: (2−5)² = 9, three × (4−5)² = 1 each, two × (5−5)² = 0, (7−5)² = 4, (9−5)² = 16. Total = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32.
Population: variance = 32 ÷ 8 = 4.00, SD = √4 = 2.00. Sample: variance = 32 ÷ 7 = 4.5714, SD = √4.5714 = 2.14 (2.1381).
Data set: 10, 12, 23, 23, 16, 23, 21, 16
n = 8, sum = 144, mean = 144 ÷ 8 = 18.00. Squared deviations: 64 + 36 + 25 + 25 + 4 + 25 + 9 + 4 = 192.
Population: variance = 192 ÷ 8 = 24.00, SD = √24 = 4.90 (4.8990). Sample: variance = 192 ÷ 7 = 27.43 (27.4286), SD = √27.4286 = 5.24 (5.2372).
Reference: what different spreads look like
| Data set pattern | Typical SD | What it means |
|---|---|---|
| All values identical (e.g. 5, 5, 5, 5) | 0 | No spread at all — every value equals the mean. |
| Two values only: a and b | |a−b| ÷ 2 (population) | Each value sits exactly half the gap from the mean. |
| Tightly clustered values | Small | Values hug the mean; the process is consistent. |
| Values spanning a wide range | Large | High variability — the mean alone is a weak summary. |
| Roughly bell-shaped data | — | About 68% of values within ±1 SD, 95% within ±2 SD of the mean. |
Standard deviation in real life
- Strength training: paste your last 8 weeks of lift numbers into the calculator. A steady upward mean with a shrinking SD means real progress; a flat mean with a big SD means your training is just noisy. Estimate your current one-rep max from any set with the one-rep max calculator, then track a series of those estimates here.
- Running: paste your per-kilometre splits or recent race paces. A falling SD across weeks means your pacing is getting steadier — usually worth more than a faster average. Compute each run's pace with the running pace calculator first.
- School and exams: two classes averaging 75% look identical until you check the SD — a small SD means the teaching reached everyone, a large SD means it reached some and lost others.
- Money: investment returns are quoted as mean return and standard deviation — the SD is literally the "risk" number in the brochure.
Frequently asked questions
Population vs sample: which standard deviation should I use?
Use population SD (÷N) when your data is the entire group you care about — every value that exists. Use sample SD (÷n−1) when your data is a sample standing in for a larger group you did not fully measure. When in doubt, most statistics textbooks default to the sample version. This calculator shows both so you never have to guess.
Why does the sample formula divide by n−1?
This is Bessel's correction. A sample's own mean is always pulled slightly toward its values, which makes the raw squared deviations a little too small on average — the sample would look less spread out than the population it came from. Dividing by n−1 instead of n exactly compensates for that bias.
Is this page also a variance calculator?
Yes. Every standard deviation result comes with its variance: population variance σ² and sample variance s², shown under each SD card. Variance is simply the standard deviation squared — it measures the same spread but in squared units, which is why SD (back in the original units) is usually the number people report.
What does the standard deviation actually tell me?
It tells you the typical distance of a value from the mean. An SD of 2 on a mean of 5 means values usually sit within a couple of units of 5. As a rule of thumb for bell-shaped data: about 68% of values fall within ±1 SD of the mean and about 95% within ±2 SD.
What is the standard deviation of a single value, or of identical values?
If every value is identical, the SD is exactly 0 — there is no spread. A single value has no defined standard deviation at all (you cannot measure spread with one point), which is why this calculator asks for at least 2 numbers.
Can standard deviation be negative?
No — never. It is the square root of a sum of squared numbers, and squares cannot be negative. The smallest possible standard deviation is 0, when all values are identical. If a spreadsheet ever shows you a negative SD, something is broken in the formula.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods — measures of spread and variability
- Khan Academy — Statistics and Probability — standard deviation and variance explained
Key takeaways
- Standard deviation = typical distance from the mean; variance = the same spread, squared.
- Population SD divides by N; sample SD divides by n−1 (Bessel's correction) — this page shows both.
- SD is in the original units, never negative, and 0 for identical values.
- Use it wherever the mean alone misleads: training progress, pacing consistency, grades, and investment risk.
Related calculators
- Average calculator — find the mean of any data set first, then measure its spread here.
- One-rep max calculator — estimate your 1RM from any set, then track a series of estimates for progress.
- Running pace calculator — compute per-run paces, then check their consistency with standard deviation.
- Percentage calculator — express spread as a percentage of the mean (coefficient of variation).
- Online calculator — quick single-number arithmetic for manual spot-checks.
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