Standard deviation explained: what it measures, how it's calculated, and the 68–95–99.7 rule
Two classes both average 70% on a test. In one, everyone scored between 65 and 75; in the other, half scored 40 and half scored 100. The average hides the difference; the standard deviation is the number that shows it. This guide explains what it measures, the calculation the standard deviation calculator performs, and how to read it when it appears in results.
Why the average isn't enough
An average locates the centre of a set of numbers (Mean, median and mode: which average to use, and when the mean lies); it says nothing about how far the numbers stray from it. Spread matters for almost every decision: whether a delivery time of "3 days on average" means 2–4 or 1–10; whether a fund's 7% average return came with 5% or 30% swings; whether a manufacturing process makes parts that fit. Standard deviation summarises spread in the same units as the data — days, percent, millimetres — which is what makes it readable.
How it's calculated
- Find the mean.
- Subtract the mean from each value (the deviations).
- Square each deviation (so negatives don't cancel and big deviations count more).
- Average the squares — that's the variance.
- Take the square root to get back to the original units — the standard deviation.
For 2, 4, 4, 4, 5, 5, 7, 9: mean 5; deviations −3, −1, −1, −1, 0, 0, 2, 4; squares 9, 1, 1, 1, 0, 0, 4, 16; variance 32 ÷ 8 = 4; standard deviation 2. A small SD relative to the mean means values cluster; a large one means they scatter.
Population versus sample
If your numbers are the whole population (every student in the class), divide the squared deviations by n. If they are a sample standing in for a larger population (100 customers out of a million), divide by n − 1 instead — Bessel's correction, which compensates for the sample mean being closer to its own data than the true mean is. Spreadsheets have both (STDEV.P and STDEV.S); the difference matters for small samples and vanishes for large ones. When in doubt, and your data is a sample, use n − 1.
The 68–95–99.7 rule
For data that follows a bell curve (a normal distribution — heights, measurement errors, test scores in a large group), about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. Adult male height in the UK has a mean around 175 cm and SD around 7 cm, so two-thirds of men are 168–182 cm and only one in 370 is over 196. The rule is what makes a standard deviation interpretable at a glance — and it fails for skewed data (incomes, waiting times), where the same SD says something different on each side of the mean.
Reading it in practice
- Quality control: a process with SD well inside the tolerance rarely produces rejects; "six sigma" means the tolerance is six SDs from the mean.
- Finance: volatility is the SD of returns; a 7% average with 20% SD loses money in many years.
- Grades and scores: a z-score — how many SDs a value sits from the mean — compares results across different tests.
- Experiments: the standard error (SD ÷ √n) says how precisely the mean is known, which is what How A/B testing works — and the three ways it lies is built on.
- Outliers: anything beyond three SDs on bell-shaped data deserves a look; on skewed data, use the median and quartiles instead.
The average calculator and the grade calculator supply the means; the SD calculator adds the spread and the z-scores.
Sources and further reading
The claims in this guide rest on these references, which were checked when the guide was last updated. Spotted an error? The contact page says how to report it.