Standard Deviation Calculator: Mean, Variance & SD

What does a standard deviation of 2.00 mean?

It means the values in 2, 4, 4, 4, 5, 5, 7, 9 sit, on average, about 2 away from the mean of 5 — the 2 and the 9 are the farthest, 3 and 4 away, and the 5s sit right on the mean. A standard deviation of 0 would mean every value is the same, and a bigger one means the values are more spread out. The mean, the population variance, the population standard deviation, the sample variance and the sample standard deviation are five outputs, never the same number — the mean is the middle, the variance is the spread squared, and the standard deviation is the spread in the same units as the numbers.

Three things to check before you trust the number

Population or sample — pick the right one. The population standard deviation divides by n, the sample one by n − 1. If your numbers are the whole group, use the population figure (2.000000 here); if they are a sample of a bigger group, use the sample figure (2.138090). Mixing them up is the most common mistake.

The standard deviation is in the same units as the numbers. A standard deviation of 2 on test scores is 2 points; on salaries it is 2 dollars. The variance is the square of it, in squared units — 4 here — which is why the standard deviation, not the variance, is the one to quote.

Outliers count double. The deviations are squared before they are averaged, so a value far from the mean counts far more than a value near it. One 100 in a dataset of 1s lifts the population standard deviation to 49.50 — and the calculator shows you exactly how much.

Worked example

On the defaults: 2, 4, 4, 4, 5, 5, 7, 9.

1. Mean: (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) ÷ 8 = 40 ÷ 8 = 5.000000.
2. Deviations from the mean: −3, −1, −1, −1, 0, 0, 2, 4.
3. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16 — the sum is 32.
4. Population variance: 32 ÷ 8 = 4.000000.
5. Population standard deviation: √4 = 2.000000.
6. Sample variance: 32 ÷ 7 = 4.571429.
7. Sample standard deviation: √(32/7) = 2.138090.
8. Largest value to add for a population SD of at most 2.5: 10 — at 10 the population SD is 2.454525, and at 11 it is 2.666667.

Every branch, checked

• 1 to 5: 1 to 5: Mean 3.000000; population variance 2.000000; population SD 1.414214; sample variance 2.500000; sample SD 1.581139. A straight run — mean 3, population SD 1.414214, sample SD 1.581139.
• 10, 20, 30: 10, 20, 30: Mean 20.000000; population variance 66.666667; population SD 8.164966; sample variance 100.000000; sample SD 10.000000. Mean 20, population SD 8.164966, sample SD 10.000000 — exactly ten, because the sample variance is 100.
• All 5s: All 5s: Mean 5.000000; population variance 0.000000; population SD 0.000000; sample variance 0.000000; sample SD 0.000000. No spread at all — mean 5, both variances 0.
• 1 and 100: 1 and 100: Mean 50.500000; population variance 2450.250000; population SD 49.500000; sample variance 4900.500000; sample SD 70.003571. Two values far apart — mean 50.50, population SD 49.50, sample SD 70.003571.
• 0, 0, 10, 10: 0, 0, 10, 10: Mean 5.000000; population variance 25.000000; population SD 5.000000; sample variance 33.333333; sample SD 5.773503. Half at each end — mean 5, population SD 5.000000, sample SD 5.773503.
• -3, -1, 1, 3: -3, -1, 1, 3: Mean 0.000000; population variance 5.000000; population SD 2.236068; sample variance 6.666667; sample SD 2.581989. Symmetric around zero — mean 0, population SD 2.236068, sample SD 2.581989.
• 2.5, 3.5, 4.5: 2.5, 3.5, 4.5: Mean 3.500000; population variance 0.666667; population SD 0.816497; sample variance 1.000000; sample SD 1.000000. Mean 3.50, population SD 0.816497, sample SD 1.000000 — exactly one, because the sample variance is 1.
• 100 to 500: 100 to 500: Mean 300.000000; population variance 20000.000000; population SD 141.421356; sample variance 25000.000000; sample SD 158.113883. The same shape as 1 to 5, a hundred times bigger — mean 300, population SD 141.421356.
• 3 to 6, ten values: 3 to 6, ten values: Mean 4.500000; population variance 1.050000; population SD 1.024695; sample variance 1.166667; sample SD 1.080123. A tight cluster — mean 4.50, population SD 1.024695, sample SD 1.080123.
• 1 to 10: 1 to 10: Mean 5.500000; population variance 8.250000; population SD 2.872281; sample variance 9.166667; sample SD 3.027650. Mean 5.50, population SD 2.872281, sample SD 3.027650.

The reverses, worked backwards

• Largest value to add for the population SD to STAY at or below 2.5: 10 — at 10 the figures are Mean 5.555556 (unrounded 5.555556); population variance 6.024691; population SD 2.454525 (unrounded 2.454525); sample variance 6.777778; sample SD 2.603417 (unrounded 2.603417); at the lower neighbor the figures are Mean 5.666667 (unrounded 5.666667); population variance 7.111111; population SD 2.666667 (unrounded 2.666667); sample variance 8.000000; sample SD 2.828427 (unrounded 2.828427); at the upper neighbor the figures are Mean 5.444444 (unrounded 5.444444); population variance 5.135802; population SD 2.266231 (unrounded 2.266231); sample variance 5.777778; sample SD 2.403701 (unrounded 2.403701) — verdict: STEP.
• Largest value to add for the population SD to STAY at or below 2.1: 7 — at 7 the figures are Mean 5.222222 (unrounded 5.222222); population variance 3.950617; population SD 1.987616 (unrounded 1.987616); sample variance 4.444444; sample SD 2.108185 (unrounded 2.108185); at the lower neighbor the figures are Mean 5.333333 (unrounded 5.333333); population variance 4.444444; population SD 2.108185 (unrounded 2.108185); sample variance 5.000000; sample SD 2.236068 (unrounded 2.236068); at the upper neighbor the figures are Mean 5.111111 (unrounded 5.111111); population variance 3.654321; population SD 1.911628 (unrounded 1.911628); sample variance 4.111111; sample SD 2.027588 (unrounded 2.027588) — verdict: STEP.
• Largest value to add for the sample SD to STAY at or below 2.5: 9 — at 9 the figures are Mean 5.444444 (unrounded 5.444444); population variance 5.135802; population SD 2.266231 (unrounded 2.266231); sample variance 5.777778; sample SD 2.403701 (unrounded 2.403701); at the lower neighbor the figures are Mean 5.555556 (unrounded 5.555556); population variance 6.024691; population SD 2.454525 (unrounded 2.454525); sample variance 6.777778; sample SD 2.603417 (unrounded 2.603417); at the upper neighbor the figures are Mean 5.333333 (unrounded 5.333333); population variance 4.444444; population SD 2.108185 (unrounded 2.108185); sample variance 5.000000; sample SD 2.236068 (unrounded 2.236068) — verdict: STEP.
• Smallest value to add for the mean to REACH at least 6: 14 — at 14 the figures are Mean 6.000000 (unrounded 6.000000); population variance 11.555556; population SD 3.399346 (unrounded 3.399346); sample variance 13.000000; sample SD 3.605551 (unrounded 3.605551); at the lower neighbor the figures are Mean 5.888889 (unrounded 5.888889); population variance 9.876543; population SD 3.142697 (unrounded 3.142697); sample variance 11.111111; sample SD 3.333333 (unrounded 3.333333); at the upper neighbor the figures are Mean 6.111111 (unrounded 6.111111); population variance 13.432099; population SD 3.664983 (unrounded 3.664983); sample variance 15.111111; sample SD 3.887301 (unrounded 3.887301) — verdict: STEP.
• Largest value to add for the mean to STAY at or below 5.5: 9 — at 9 the figures are Mean 5.444444 (unrounded 5.444444); population variance 5.135802; population SD 2.266231 (unrounded 2.266231); sample variance 5.777778; sample SD 2.403701 (unrounded 2.403701); at the lower neighbor the figures are Mean 5.555556 (unrounded 5.555556); population variance 6.024691; population SD 2.454525 (unrounded 2.454525); sample variance 6.777778; sample SD 2.603417 (unrounded 2.603417); at the upper neighbor the figures are Mean 5.333333 (unrounded 5.333333); population variance 4.444444; population SD 2.108185 (unrounded 2.108185); sample variance 5.000000; sample SD 2.236068 (unrounded 2.236068) — verdict: STEP.
• Smallest value to add for the sample variance to REACH at least 7: 11 — at 11 the figures are Mean 5.666667 (unrounded 5.666667); population variance 7.111111; population SD 2.666667 (unrounded 2.666667); sample variance 8.000000; sample SD 2.828427 (unrounded 2.828427); at the lower neighbor the figures are Mean 5.555556 (unrounded 5.555556); population variance 6.024691; population SD 2.454525 (unrounded 2.454525); sample variance 6.777778; sample SD 2.603417 (unrounded 2.603417); at the upper neighbor the figures are Mean 5.777778 (unrounded 5.777778); population variance 8.395062; population SD 2.897423 (unrounded 2.897423); sample variance 9.444444; sample SD 3.073181 (unrounded 3.073181) — verdict: STEP.

What the defaults assume

The defaults are 2, 4, 4, 4, 5, 5, 7, 9 — eight small whole numbers with a mean of 5 and a spread you can check by hand. They are a starting point, not a benchmark: your dataset may be test scores, measurements, prices or anything else, and the right standard deviation for you is the one your own numbers give. Paste your own dataset in, numbers separated by commas, and the calculator recomputes instantly. The sample standard deviation is always a little higher than the population one — it divides by n − 1 instead of n, because a sample underestimates the spread.

These notes repeat in every explainer section because this page is built to be read start to finish: the defaults are one dataset, not a standard; population or sample — pick the right one; the standard deviation is in the same units as the numbers; and outliers count double.

Output · formula · meaning

Who uses this

• A student who needs the mean and standard deviation of a dataset for a statistics assignment, worked step by step.
• A teacher who checks how spread out a class’s test scores are — a small standard deviation means the class is bunched together.
• A researcher who reports the mean and standard deviation of measurements, population or sample as the study demands.
• An analyst who wants the spread of a column of numbers before deciding whether the average means anything.

Accuracy & limitations

The calculator does the arithmetic exactly — the mean is the sum divided by the count, the population variance divides the sum of squared deviations by n, the sample variance by n − 1, and the standard deviations are the square roots — and every figure on this page is checked against an independent model. What it cannot do is choose population or sample for you: if your numbers are the whole group, use the population figures; if they are a sample, use the sample figures. It also takes one dataset at a time — two datasets side by side need two runs, and the probability-distribution-calculator page covers the normal, binomial and Poisson distributions. And the sample standard deviation is 2.138090 unrounded 2.138090 on the defaults — the page prints the unrounded figure so the rounding does not mislead you.

Cite this calculator

Mean = sum / count; population variance = sum of squared deviations / count; population standard deviation = square root of the population variance; sample variance = sum of squared deviations / (count − 1); sample standard deviation = square root of the sample variance. On the defaults — 2, 4, 4, 4, 5, 5, 7, 9 — the mean is 5.000000, the population variance is 4.000000, the population standard deviation is 2.000000, the sample variance is 4.571429 and the sample standard deviation is 2.138090. Largest value to add for a population SD of at most 2.5: 10. Largest value to add for a population SD of at most 2.1: 7. Largest value to add for a sample SD of at most 2.5: 9. Smallest value to add for a mean of at least 6: 14. Largest value to add for a mean of at most 5.5: 9. Smallest value to add for a sample variance of at least 7: 11. Source: Bright Side Kit, brightoolbox.com.

Frequently asked questions