How spread out are your numbers?
The mean tells you where a set of numbers is centred, but not how tightly they cluster around that centre. Two classes can both average 70% on a test while one scores between 65 and 75 and the other swings from 30 to 100. Standard deviation measures that spread in the same units as the data itself: a small standard deviation means the values huddle near the mean; a large one means they are scattered widely. This calculator works through the full table of deviations for your numbers, so every figure can be checked by hand.
The standard deviation formula
For each value x, subtract the mean to get its deviation, square it (which removes minus signs and emphasises large gaps), average the squared deviations to get the variance, then take the square root to return to the original units. Squaring and then square-rooting is what makes standard deviation different from a simple average distance — it penalises outliers more, which is usually what you want when a few extreme values matter most.
Population vs sample: the N or N − 1 question
This page reports both versions because textbooks, spreadsheets and exam questions expect different ones. The population standard deviation divides by N and is correct when your numbers are the entire group of interest — all the students in one class, all the days in one year. The sample standard deviation divides by N − 1 instead, which makes the estimate slightly larger. That correction (called Bessel's correction) exists because a sample's own mean sits closer to its values than the true population mean would, so dividing by N would systematically understate the spread of the wider population you are trying to estimate. Rule of thumb: describing the data you have — population; generalising beyond it — sample.
Reading the number
For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three — the empirical rule. So a mean of 5 with a standard deviation of 2, the default example here, means most values sit roughly between 1 and 9. A standard deviation of zero means every value is identical. If your data has a heavy skew or extreme outliers, the mean and standard deviation can mislead, and the median-based view in the Mean Median Mode Calculator is worth checking alongside.
Worked example
For the numbers 2, 4, 6: the mean is 4, the deviations are −2, 0 and 2, and their squares are 4, 0 and 4, totalling 8. Population variance = 8 ÷ 3 = about 2.6667, so the population SD is about 1.6330. Sample variance = 8 ÷ 2 = 4, so the sample SD is exactly 2.
How to use this calculator
- Paste or type your numbers into the box, separated by commas or spaces.
- Check the count matches how many numbers you meant to enter.
- Read the population standard deviation in the green panel.
- Use the sample figures instead if your numbers are a sample of a larger group.
- Work through the deviation table below to see each squared deviation that builds the result.
Variance vs standard deviation
Variance is the average squared deviation; standard deviation is its square root. They carry the same information, but variance is in squared units (squared centimetres, squared dollars) which nobody can picture, while standard deviation is in the original units and can be drawn on a chart or compared with the mean directly. Statisticians keep variance for theory because variances of independent quantities add together neatly; everyone else should report the standard deviation. This page shows both, so whichever one a formula, spreadsheet function (VAR.P, VAR.S, STDEV.P, STDEV.S) or exam question asks for, you can match it exactly.