Standard Deviation Calculator

Standard deviation calculator

Enter a list of numbers and choose whether the data is an entire population or a sample drawn from a larger population. The calculator reports the selected standard deviation and variance, plus both population and sample values so you can compare the denominators.

For a future study estimating a mean, the sample size calculator's mean mode uses an entered population or planning SD and an absolute margin of error. A pilot sample SD remains uncertain; the z-based planning model is not a substitute for a design-specific study calculation.

Population standard deviation

Use the population formula when the values include every member of the group you want to describe. First find the mean, subtract the mean from each value, square those deviations, add them, and divide by the population size n. Standard deviation is the square root of that variance.

population variance = sum of (x - mean)^2 / n

population standard deviation = square root of population variance

Sample standard deviation

Use the sample formula when the entered observations are used to estimate variability in a larger population. The corrected sample variance divides by n - 1, often called Bessel's correction. The NIST Statistical Reference Datasets define sample standard deviation using this denominator.

sample variance = sum of (x - sample mean)^2 / (n - 1)

sample standard deviation = square root of sample variance

Worked example

For 1, 3, 4, 7, and 8, the mean is 4.6 and the sum of squared deviations is 33.2. Dividing by 5 gives a population variance of 6.64 and population standard deviation of about 2.5768. Dividing by 4 gives a sample variance of 8.3 and sample standard deviation of about 2.8810.

Variance, standard error, and margin of error

Variance is measured in squared units, while standard deviation returns to the original data units. NIST's Engineering Statistics Handbook describes standard deviation as the square root of variance and notes that squaring gives more weight to values farther from the mean.

Standard error estimates the spread of sample means and is calculated as the sample standard deviation divided by the square root of n. The displayed margin of error multiplies that standard error by the selected normal critical value. It is a normal approximation, not a substitute for a t interval or a study-specific sampling design.

How to interpret standard deviation

A smaller standard deviation means values are generally closer to the mean; a larger standard deviation means they are more dispersed. The number is not automatically good or bad. Its meaning depends on the measurement scale, context, distribution, and how the data were collected.

Frequently asked questions

Should I choose sample or population?

Choose population when the list contains the complete group of interest. Choose sample when the observations represent only part of a larger group.

Why does sample standard deviation use n - 1?

Estimating the mean from the same sample uses one degree of freedom. Dividing by n - 1 corrects the sample variance as an estimator of population variance.

Can I enter negative numbers and decimals?

Yes. Separate integers, negative values, or decimals with commas, spaces, semicolons, or line breaks.