What standard deviation measures
The mean tells you the center of a data set; the standard deviation tells you how spread out it is. Two classes can both average 75 on a test, but if one class's scores range from 70 to 80 and the other's from 40 to 100, their standard deviations — about 3 versus about 18 — capture that difference in a single number. It is the square root of the variance, the average squared distance from the mean, so it is in the same units as the data.
The formulas
Mean x̄ = Σx / n Population variance σ² = Σ(x − x̄)² / n σ = √σ² Sample variance s² = Σ(x − x̄)² / (n − 1) s = √s² Standard error SE = s / √n
Worked example
Data: 2, 4, 4, 4, 5, 5, 7, 9 (n = 8). Mean = 40 / 8 = 5.
Deviations from 5: −3, −1, −1, −1, 0, 0, 2, 4. Squared: 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32.
Population: σ² = 32 / 8 = 4, so σ = 2. Sample: s² = 32 / 7 = 4.571, so s ≈ 2.138.
Notes
- Standard deviation is sensitive to outliers: one extreme value can inflate it a lot. The interquartile range on the mean, median and mode calculator is more robust.
- The empirical rule only applies to roughly normal (bell-shaped) data.
- Results are shown to six significant decimals; the calculation uses full precision.
Sources: NIST/SEMATECH, e-Handbook of Statistical Methods, §1.3.5.6 Measures of Scale; OpenStax, Introductory Statistics 2e, §2.7 Measures of the Spread of the Data.
Frequently asked questions
Should I use sample or population standard deviation?
Use the population formula (divide by n) when your data includes every member of the group you care about — all 30 students in a class. Use the sample formula (divide by n − 1) when the data is a sample used to estimate a larger population — 30 students chosen from a whole school. Most statistics courses and software default to the sample version.
Why divide by n − 1 for a sample?
A sample's values sit closer to their own mean than to the true population mean, so dividing by n underestimates spread. Dividing by n − 1 (Bessel's correction) makes the sample variance an unbiased estimate of the population variance.
What does the standard deviation tell me?
How far values typically are from the mean, in the same units as the data. For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three — the empirical rule.
What is the standard error of the mean?
The standard deviation divided by √n. It describes how much the sample mean itself would vary from sample to sample, and shrinks as the sample grows. It is used to build confidence intervals for the mean.
How do I enter my numbers?
Paste or type them separated by commas, spaces, semicolons or line breaks — straight from a spreadsheet column works. Anything that is not a number is reported and ignored.