Mean, median and mode
Find mean, median, all repeated modes, range, sum and a sorted list. See how an outlier changes the average compared with the middle value.
Standard deviation calculator: sample and population
See both common standard-deviation definitions from the same data. Use the population value to describe the complete set you supplied, or the sample estimate when that set is being used to estimate a larger population’s variance.
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Worked result
Population SD 2; sample SD approximately 2.13808993530.
- Calculate the mean, 5.
- Sum squared distances from that mean to obtain 32.
- Divide by 8 for the population variance or 7 for the sample variance, then take square roots.
Compute both deviations for eight values
For 2, 4, 4, 4, 5, 5, 7 and 9, the mean is 5 and the squared deviations sum to 32.
n = 8; mean = 5
Squared deviations = 9, 1, 1, 1, 0, 0, 4, 16
Sum of squared deviations = 32
Population variance = 32/8 = 4
Population standard deviation = 2
Sample variance = 32/7 ≈ 4.57142857143
Sample standard deviation ≈ 2.13808993530The difference is a denominator choice, not a rounding discrepancy. Standard deviation has the same units as the original values; variance has squared units.
Make the denominator explicit
Population variance divides the squared-deviation total by n. The conventional sample variance divides by n−1 because the sample mean has already been estimated from the same data. Taking a square root yields the corresponding standard deviation.
The n−1 correction makes sample variance an unbiased estimator under the usual independent, identically distributed sampling assumptions; it does not make the sample standard deviation itself exactly unbiased or repair a biased sampling process.
SS = Σ(x − mean)²
population variance = SS/n
sample variance = SS/(n − 1), n ≥ 2
standard deviation = square root of the corresponding varianceInput limits and common mistakes
At least one observation is needed for the population calculation and two for the sample calculation. For a one-value input, the sample result is undefined rather than zero. All-equal data have zero spread.
Up to 5,000 values are supported. Floating-point calculations can lose detail when tiny differences are supplied alongside a huge common offset. Re-expressing such measurements relative to a sensible reference can help; this tool does not infer measurement precision.
Questions about standard deviation
Which standard deviation should I use?
Use the population definition to describe the complete collection of interest. Use the sample estimate when your observations are a sample used to estimate a larger population variance.
Does n−1 fix a biased sample?
No. It corrects the conventional variance estimator under sampling assumptions; it does not make an unrepresentative sample representative.
Is this the standard error?
No. Standard deviation describes spread among values. A standard error describes uncertainty of an estimator and requires additional assumptions.
Sources and verification
These references explain the formulas or formats. The worked examples above were created for this tool and checked against independent calculations or format round trips. AI-assisted development does not imply a named human expert review.
- NIST: measures of scale — Sample variance, standard deviation and their units.
Published 2026-10-23. Examples are illustrations, not measurements of a service or claims about search rankings.
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Further reading
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