Free Standard Deviation Calculator with Steps

The formula, piece by piece

The sample standard deviation formula is s = √[ Σ(xᵢ − x̄)² / (n − 1) ]. It looks dense, but it is four small operations chained together, and each one has a job.

x̄ is the mean — add every value and divide by n, the number of values. Each (xᵢ − x̄) is a deviation: how far one value sits from the mean, negative if below it, positive if above. Squaring each deviation does two things: it makes everything positive so distances below the mean cannot cancel distances above it, and it weights big deviations more heavily than small ones. Σ just means "add them all up."

Dividing that sum by n − 1 gives the sample variance s²; dividing by n instead gives the population variance σ². The square root at the end undoes the squaring, so the final answer is in the same units as your data — if you measured heights in centimeters, the standard deviation is in centimeters too.

A worked example you can follow by hand

Take the data set 2, 4, 4, 4, 5, 5, 7, 9 — the same one pre-loaded in the calculator above, so you can check every line.

Step 1, the mean: the sum is 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40, and there are n = 8 values, so x̄ = 40 / 8 = 5.

Step 2, deviations and squares: subtracting 5 from each value gives −3, −1, −1, −1, 0, 0, 2, 4. Squaring each gives 9, 1, 1, 1, 0, 0, 4, 16. Their sum is 32.

Step 3, variance: as a sample, s² = 32 / (8 − 1) = 32 / 7 ≈ 4.5714. As a population, σ² = 32 / 8 = 4 exactly.

Step 4, square root: s = √4.5714 ≈ 2.1381 and σ = √4 = 2. Notice the sample answer is always a bit larger — that is the n − 1 divisor at work.

Sample or population: which divisor do you need?

Use the population formula (divide by n) only when your data is the entire group you care about — every student in one specific class, every game a team played this season. Nothing was left out, so the mean you computed is the true mean, and dividing by n is exact.

Use the sample formula (divide by n − 1) when your data is a subset standing in for a larger group — 50 surveyed customers out of thousands, 20 measured plants out of a whole field. A sample's values tend to sit closer to their own sample mean than to the true population mean, which makes the raw average of squared deviations come out slightly too small. Dividing by n − 1 instead of n corrects that bias; the correction is called Bessel's correction.

In practice, statistics courses and real research almost always work with samples, so when a problem does not say which to use, sample standard deviation is usually the intended answer. But read the problem: words like "the entire population," "all," or "every" signal the population formula.

Common mistakes that produce wrong answers

• Forgetting to square the deviations. The raw deviations always sum to zero — that is a property of the mean, not a coincidence — so skipping the squaring step gives a variance of 0 every time.
• Using n when the problem wants n − 1, or the reverse. This is the single most common lost point on homework. State which divisor you used and why.
• Taking the square root too early. The square root is the last step, applied to the variance — not to each squared deviation and not to the sum.
• Rounding the mean before computing deviations. If x̄ = 4.6667 and you round it to 4.7, every deviation inherits the error and the squares amplify it. Keep full precision until the final answer.
• Reporting the variance as the standard deviation. They differ by exactly one square root; if your answer's units are squared (cm², points²), you stopped one step early.
• Dropping a data value or typing one twice. Recount n before dividing — the calculator above shows n explicitly so you can compare it against your list.

How to check your answer

Three quick sanity checks catch most errors. First, the deviations in your table must sum to zero (allowing for tiny rounding). If they do not, your mean is wrong, and everything after it is too.

Second, compare your two answers: the sample standard deviation must be larger than the population standard deviation for the same data, because you divided by a smaller number. If yours came out equal or reversed, the divisors got swapped.

Third, check the scale. Standard deviation can never be negative, is zero only when every value is identical, and can never exceed the range of your data (the max minus the min). For most bell-shaped data, roughly two-thirds of the values fall within one standard deviation of the mean — if your data spans 2 to 9 and you got a standard deviation of 40, something went wrong. Re-enter the data in the calculator above and compare its step table against yours line by line to find where the numbers diverge.