Descriptive statistics
Standard Deviation Calculator
The typical distance from the mean, in the same units as your data.
Also called stdev calculator, SD calculator, sample standard deviation calculator.
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How to read this result
The standard deviation is a typical distance from the mean, measured in the data's own units. If mean daily sales are 400 units with an SD of 60, a typical day lands somewhere around 60 units either side of 400.
The 68-95-99.7 rule turns it into probabilities — but only for normal data. About 68% of a normal distribution lies within one SD of the mean, 95% within two and 99.7% within three. On skewed data those percentages are wrong, sometimes badly, so check the histogram before using them.
It is not robust. Because deviations are squared, one extreme value can dominate the SD entirely. In the delivery-time example on the range and MAD page, a single 45-minute delivery among nines multiplies the standard deviation by more than six while barely moving the interquartile range.
Sample or population? Divide by n − 1 for a sample — the usual case — and by n only when the data are the whole population. Spreadsheets make this an easy mistake: STDEV.S and STDEV.P are different functions, and the older STDEV is the sample version.
The formula
- the sample standard deviation
- the population standard deviation
- the sample and population means
Worked example
Standard deviation of twelve exam scores
Scores: 72, 85, 78, 90, 68, 95, 88, 76, 82, 91, 79, 84. The mean is 82.33.
- Subtract the mean from each value and square the result.(72 − 82.33)² = 106.78, (85 − 82.33)² = 7.11, …
- Add the squares.Σ(x − x̄)² = 718.67
- Divide by n − 1 to get the variance.718.67 / 11 = 65.33
- Take the square root.s = √65.33 = 8.083
- Read it back onto the scale: one SD either side of the mean.82.33 ± 8.08 → 74.25 to 90.42
Eight of the twelve scores fall inside one standard deviation of the mean — 67%, almost exactly what the normal rule predicts, which is a sign the scores are reasonably bell-shaped.
Checked against R's sd.
The calculator above is loaded with these numbers by the Load the worked example button.
Assumptions, and when to use something else
The standard deviation assumes numeric data on an interval scale. It describes any distribution's spread, but the familiar interpretations — 68% within one SD, outliers beyond three — assume normality. On skewed data the SD is still computable and the rules of thumb are not.
It is also sensitive to sample size in a way people underestimate: with fewer than about ten observations, the SD itself is estimated poorly, and two samples from the same population can easily differ by 50% in their SDs.
- You want the squared versionVarianceWhat the mathematics uses, because variances add and standard deviations do not.
- Outliers are inflating itRange and MADThe median absolute deviation is a robust replacement, and MAD × 1.4826 estimates σ for normal data.
- You want to know how precise the mean isStandard errorSD ÷ √n — how much the mean itself would vary between samples.
- You want to compare spread across different unitsCoefficient of variationSD as a percentage of the mean.
Questions people ask
What is a good standard deviation?
There is no such thing in the abstract — it depends entirely on the scale and the context. An SD of 8 on exam scores out of 100 is moderate; an SD of 8 on a manufacturing tolerance measured in millimetres might be a crisis. The coefficient of variation is the version you can compare.
Is standard deviation the same as standard error?
No, and it is the most common confusion in reporting. The standard deviation describes how spread out the individual values are. The standard error describes how much the mean would vary if you repeated the study. SE = SD/√n, so the SE is always smaller and gets smaller with more data while the SD does not.
Which Excel function should I use?
STDEV.S for a sample — the usual case — and STDEV.P when the data are the entire population. The legacy STDEV is the sample version and STDEVP the population one.
Can the standard deviation be larger than the mean?
Easily, and it is common for counts and waiting times. It means the coefficient of variation is above 1, which for data that cannot go below zero implies substantial right skew.