Standard Deviation Calculator

How standard deviation is calculated

First find the mean (average) of your data. Then, for each value, subtract the mean and square the result — this makes every difference positive and weights larger deviations more heavily. Average those squared differences to get the variance, and take its square root to get the standard deviation, back in the same units as your original data.

In symbols, the sample standard deviation is s = √(Σ(x − x̄)² ÷ (n − 1)) and the population standard deviation is σ = √(Σ(x − μ)² ÷ N). The only difference is the divisor. If you only need the average, median or mode of a list, the average calculator covers those; this page is about how spread out the values are.

Worked example, step by step

Take the calculator's default data: 10, 12, 23, 23, 16, 23, 21, 16. There are 8 values and they add up to 144, so the mean is 144 ÷ 8 = 18. Next, subtract the mean from each value and square the difference:

The deviations always add up to 0, which is a handy check on your arithmetic. The squared deviations add up to 192. Treated as a sample, divide by n − 1 = 7: the variance is 27.4286 and the standard deviation is √27.4286 = 5.2372. Treated as a whole population, divide by n = 8: the variance is 24 and the standard deviation is √24 = 4.899. The "Show the math" panel under the results repeats these steps with whatever numbers you enter.

Sample vs. population standard deviation

Population standard deviation divides the summed squared differences by n (the full count). Sample standard deviation divides by n−1 instead — known as Bessel's correction — which slightly increases the result to correct for the fact that a sample tends to underestimate the true variability of the whole population it was drawn from. Use "Population" only when your data genuinely represents every member of the group you care about; "Sample" is the more common, more conservative choice.

A sample of one value has no sample standard deviation: n − 1 is 0, and one measurement tells you nothing about spread. The calculator shows "Undefined" in that case rather than a misleading 0. The population figure for a single value is 0, because a group of one has no spread at all.

The 68-95-99.7 rule

When data follows a roughly bell-shaped (normal) distribution, the standard deviation tells you where most values fall. About 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. If exam scores have a mean of 70 and a standard deviation of 10, roughly two-thirds of students scored between 60 and 80, and only about 1 in 20 scored below 50 or above 90.

The rule is a guide for symmetric, single-peaked data. It does not hold for skewed data such as incomes or wait times. To turn a single value into a percentile, compute its z-score, (value − mean) ÷ standard deviation, and look it up in the z-score mode of the probability calculator.

STDEV.S vs. STDEV.P in Excel and Sheets

Excel and Google Sheets both have a function for each version. =STDEV.S(A1:A8) returns the sample standard deviation (divides by n − 1) and matches this calculator's Sample mode. =STDEV.P(A1:A8) returns the population standard deviation (divides by n) and matches Population mode. The older names still work: STDEV is the same as STDEV.S, and STDEVP is the same as STDEV.P. For variance, use VAR.S and VAR.P in the same way.

A common spreadsheet mistake is reaching for STDEV.P because the data feels complete, when it is really a sample of something larger, such as last month's orders standing in for all future orders. If you are unsure, STDEV.S is the safer default. Text and blank cells in the range are ignored, so check the count matches the number of values you expect.

Mean and standard deviation together

The mean tells you the typical value; the standard deviation tells you how far values typically stray from it. Two data sets can have the same mean but very different spreads — this calculator always reports both side by side, along with the min, max, and range, so you get the full shape of your data at a glance, not just one number.

Relative standard deviation

Relative standard deviation (RSD) expresses standard deviation as a percentage of the mean: RSD = (standard deviation ÷ mean) × 100. Because it's unitless, RSD lets you compare the variability of data sets measured on completely different scales — for example, comparing the consistency of lab measurements in milligrams against measurements in liters, where the raw standard deviations alone wouldn't be comparable.