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Measure how spread out your numbers are — with the mean included.
Calculate the standard deviation of any list of numbers to see how spread out the values are around the mean. The calculator shows the mean alongside, so you get the full picture of your dataset in one step.
Standard deviation measures spread: how far a typical value strays from the average. Consider {4, 5, 5, 6} versus {0, 5, 5, 10} — both have a mean of 5, but the second dataset is far more scattered, and its larger standard deviation captures exactly that. In investing it quantifies volatility (a fund with high SD swings harder), in manufacturing it quantifies consistency (low SD means every part matches spec), in test scores it tells you whether the class clustered together or split into strong and weak groups, and in science it sets the error bars on every measurement.
The calculation has four steps: find the mean, square each value's distance from it (squaring penalizes large deviations and keeps everything positive), average those squared distances — that average is the variance — and take the square root to return to the original units. One subtlety: for a full population you divide by N, but for a sample you divide by N - 1 (Bessel's correction), because a sample's values cluster slightly tighter around the sample mean than around the true mean, so dividing by N would underestimate the spread. Use population when you have every value (all students in a class); use sample when your data is a subset (20 voters polled out of millions). With large datasets the N versus N-1 difference fades to nothing.
This calculator uses the population standard deviation: the square root of the average squared distance of each value from the mean. A small value means the numbers cluster tightly; a large one means they are spread out.
Mean = (2+4+4+4+5+5+7+9) = 40 / 8 = 5. Squared deviations from 5: 9, 1, 1, 1, 0, 0, 4, 16 — sum = 32. Population variance = 32 / 8 = 4, so SD = sqrt(4) = 2. Interpretation: a typical value sits about 2 units from the mean of 5. Note the 9 (deviation 4, squared 16) contributes half the total variance — one outlier dominates the spread, which is why SD is sensitive to extremes.
Mean = 36 / 3 = 12. Squared deviations: 4, 0, 4 — sum = 8. Sample variance = 8 / (3 - 1) = 4, so SD = 2. Dividing by N - 1 = 2 instead of 3 corrects for the small sample; had we wrongly used the population formula we would get sqrt(8/3) = 1.63, understating the spread. Rule of thumb from the empirical rule: for roughly bell-shaped data, about 68% of values fall within one SD of the mean — here that predicts most values between 10 and 14, which is exactly the dataset.
It measures spread. A low standard deviation means values cluster near the mean; a high one means they are widely scattered. The classic dataset 2, 4, 4, 4, 5, 5, 7, 9 has a mean of 5 and a standard deviation of 2.
This calculator uses the population formula (divides by N). If you need the sample formula (divides by N−1), multiply the result by √(N/(N−1)).
No — it's a square root of squared distances, so it's always zero or positive. Zero means every value is identical.