Standard Deviation Calculator

Paste your dataset into the input box. The calculator accepts numbers separated by commas, spaces, or new lines (so you can drop a column straight from a spreadsheet without reformatting). It works out the count, mean, median, mode, range, variance and standard deviation, and shows where the data sits within one, two and three standard deviations of the mean.

Pick population or sample mode at the top. Population uses divisor n; sample uses divisor n-1 (Bessel's correction). The difference matters for small datasets and almost vanishes for large ones. Most real-world statistics, including pretty much every survey, every controlled experiment, and every test from a class of pupils, are samples drawn from a wider population, so sample standard deviation is the safer default unless you genuinely have every value in the population.

Standard deviation tells you how spread out a set of numbers is around the mean. A small standard deviation means most values are clustered tightly around the average; a large one means they're scattered widely. Two datasets can share the same mean and look completely different in shape. Class A scoring 70, 70, 70, 70, 70 has a mean of 70 and a standard deviation of 0. Class B scoring 40, 55, 70, 85, 100 also has a mean of 70 but a standard deviation of about 23. The mean alone hides this entirely.

The formula is sqrt(sum((x - mean)^2) / n) for population, or sqrt(sum((x - mean)^2) / (n-1)) for sample. You square the deviations to make negatives positive (otherwise they'd cancel), average them to get variance, then square-root to bring the units back to the original scale. The 68-95-99.7 rule says that for a normal distribution, roughly 68% of values fall within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs.

Comparing consistency. Two football pundits make scoring predictions; whichever has the lower SD between predicted and actual scores is more reliable, even if their averages are similar. Quality control. A factory wants its product weights to cluster tightly around the target; rising SD signals a problem on the line before any single weight goes badly wrong. Investment risk. Standard deviation of monthly returns is the textbook definition of volatility; a fund with low SD is steady, a fund with high SD swings.

It's also the basis of z-scores, where a value is reported as 'X standard deviations above or below the mean'. A z-score of 0 is exactly average; a score of +2 means well above average (about top 2.5% if the data is normally distributed); a score of -3 means very far below. For more on combinations, sequences and core statistics, see the [Probability Calculator](/probability-calculator).