A clear answer before the details
Standard deviation is the square root of variance and describes a typical distance from the mean. Sample variance uses N − 1; population variance uses N.
Measure spread around the mean
A sample standard deviation calculator describes how far data values typically sit from their mean. It first finds the mean, subtracts that mean from every observation, squares the deviations, averages them with the sample divisor n − 1, and takes the square root. The result is in the same unit as the original dataset. A variance calculator reports the squared spread before the square root is taken.
Sample versus population standard deviation
Use the population standard deviation calculator mode when your list contains every member of the group being described; it divides by N. Use sample mode when the observations represent a larger population; it divides by N − 1 to reduce bias in the estimate. The mean and standard deviation calculator also shows the minimum, maximum, range, and variance so you can interpret spread in context instead of treating one statistic as a complete summary.
Formula guide
The key equations are separated below. Start with the calculator mode, then choose the shape or model that matches your problem.
Population standard deviation
Divide by N when the dataset is the entire population.
Sample standard deviation
Divide by n − 1 when estimating a larger population.
Variance
Variance is in squared units.
Worked example
For [1, 2, 3], the mean is 2. The sample squared deviations sum to 2, so s = √(2 ÷ 2) = 1.
Formulas for common use cases
Use the matching equation for the problem you are solving, then enter its variables in the calculator above.
Spread around the mean
A deviation shows whether a value is above or below average.
Range
Range shows the full distance between the extremes.
Standard score
A z-score expresses distance from the mean in standard deviations.
How to use this calculator
- Enter numbers separated by commas, spaces, or new lines.
- Choose sample or population mode.
- Review the deviation table and summary.
- Use the mean and range alongside standard deviation for context.
Where it helps
Standard deviation helps describe exam scores, sensor readings, experiment results, manufacturing tolerances, and survey data. The sample/population choice matters: use population mode when the list is the complete group of interest, and sample mode when it represents a larger population.
Common mistakes
Do not use sample mode with only one value. Keep the mean calculation separate from the standard-deviation divisor, and avoid interpreting a large standard deviation as automatically “bad”; its meaning depends on the scale and context of the data.