Population spread
For 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5 and the sum of squared deviations is 32. Population variance is 32/8 = 4 and population standard deviation is 2.
Calculations run locally in your browser. This tool does not upload the numbers you enter.
How to use the Standard Deviation Calculator
- 01
Paste observations separated by spaces, commas, semicolons, or new lines. Omit headers and thousands separators.
- 02
Choose population for a complete group or sample when estimating spread from a subset.
- 03
Calculate and inspect the summary. Expand the first 20 deviations to check entries, or download the report containing every entered value.
Examples
Sample correction
Using those same eight observations as a sample divides 32 by 7. Sample variance is about 4.57143 and sample standard deviation is about 2.13809.
Which denominator to use
Population variance divides the sum of squared deviations from the mean by n. Sample variance divides by n − 1 to estimate population variance from a sample. The sample formula requires at least two observations; one observation is sufficient for population variance, which is then zero.
Interpreting the units
Variance has squared data units. Taking its square root returns standard deviation to the same units as the observations, so a deviation calculated from lengths in centimeters is also in centimeters. A large value indicates more spread around the mean but does not, by itself, describe distribution shape.
Outliers and numerical care
Squaring deviations gives distant observations substantial influence. Inspect unusual values before interpreting the result. The calculator accumulates mean and squared deviations incrementally to reduce avoidable cancellation, although all numeric results remain subject to floating-point rounding.
What to know
- Up to 10,000 equally weighted observations, with magnitude no greater than 1e100.
- The on-screen deviation table shows the first 20 entries; exported text includes all observations.
Common questions
Are repeated observations counted?
Yes. Each entry is one equally weighted observation. Repeated values are retained rather than deduplicated.
Can I enter a frequency table?
No. Expand frequencies into repeated observations or use a weighted statistics tool. This calculator does not interpret a second column as weights.
What happens when every observation is equal?
All deviations from the mean are zero, so both supported variance formulas return zero when their minimum sample size is met.
Does standard deviation imply a normal distribution?
No. The quantity can be computed for these finite data without a normality assumption. Normal-distribution probability rules need separate justification.