Standard Deviation Calculator
Last updated: November 4, 2026
Measure the spread of a numeric list using an explicit sample or population divisor, with variance and mean.
On this page
Quick summary: Measure the spread of a numeric list using an explicit sample or population divisor, with variance and mean.
Read the complete guide
Updated reference sections in this guide are in English.
Using this calculator
Enter the raw observations and choose what the list represents. Select population when these are all the values whose spread you want to describe. Select sample when using the list to estimate a wider population’s variability. The tool does not decide whether sampling was representative or independent.
Calculation method
Squared-deviation total = Σ(x − mean)²
Population variance = squared-deviation total / n
Sample variance = squared-deviation total / (n − 1)
Standard deviation = square root of variance
The implementation first computes the mean, then sums squared deviations. This avoids subtracting two large, nearly equal raw square totals. Each observation has equal weight; grouped frequencies are not inferred from the text.
Worked examples
Compare both divisors
For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the squared-deviation total is 32. Population variance is 32 ÷ 8 = 4 and population standard deviation is 2. Sample standard deviation is √(32 ÷ 7), approximately 2.13809.
All observations agree
For 6, 6, 6, every deviation is zero. Both supported standard deviations are zero.
Check this scenario in the calculator:
- Treat this list as
- Sample: divide by n − 1
- Observations
- 2, 4, 4, 4, 5, 5, 7, 9
Understanding the result
Standard deviation is in the same unit as the measurements. Variance is in squared units. Adding the same constant to every observation changes the mean but leaves spread unchanged; multiplying all observations by a factor multiplies standard deviation by the factor’s absolute value.
Boundaries and assumptions
Supply 1–1,000 observations between −10¹² and 10¹². Sample mode requires at least two. Missing observations are not silently dropped: remove blank comma-separated entries. Large offsets with extremely small differences can exceed ordinary browser precision. This is descriptive arithmetic, not a confidence interval, a normality check or a prediction range.
Questions and reference
Why is sample mode larger?
For the same nonconstant list, dividing by n − 1 gives a larger variance than dividing by n. The choice follows what the data represent.
Can one observation have a population standard deviation?
Yes, zero. A one-observation sample cannot provide the sample variance used here.
Reference: NIST: measures of scale.