Standard Deviation Calculator: Sample, Population and Variance

Standard Deviation Calculator

Calculate sample or population standard deviation, variance, mean, count, sum, range, standard error and coefficient of variation from one dataset.

Separate values with commas, semicolons, spaces or new lines. Use a period for decimals; commas are separators, not thousands marks. Maximum: 10,000 values.



Standard deviation formulas

For a population of N values, the mean is μ = Σx/N and the population standard deviation is:

σ = √[Σ(x − μ)² / N]

For a sample of n observations, the sample mean is x̄ = Σx/n and Bessel’s correction uses n − 1:

s = √[Σ(x − x̄)² / (n − 1)]

Variance is the square of the standard deviation. Sample standard error is s/√n. The coefficient of variation is |standard deviation ÷ mean| × 100%, and a z-score is (value − mean) ÷ standard deviation.

Checked examples

DatasetTypeMeanVarianceStandard deviation
2, 4, 4, 4, 5, 5, 7, 9Population542
2, 4, 4, 4, 5, 5, 7, 9Sample54.57142857142.1380899353
1, 2, 3, 4, 5Population321.4142135624
10, 10, 10Population1000
10, 14Sample1282.8284271247
−2, 0, 2Population02.66666666671.6329931619

How to use the Standard Deviation Calculator

  1. Paste or type the numbers in the dataset box.
  2. Choose Population when the values are the complete group, or Sample when they are observations from a larger group.
  3. Optionally enter one value to calculate its z-score.
  4. Select Calculate standard deviation and review the spread and descriptive statistics.

Sample or population?

Use population standard deviation when the dataset contains every member of the group you want to describe. Use sample standard deviation when the observations are used to estimate variability in a larger population. The sample formula divides by n − 1, so its result is usually larger for the same values.

Limits and interpretation

  • The calculator accepts up to 10,000 finite numbers, including negative values, decimals and scientific notation.
  • Commas separate values. Enter 1250 rather than 1,250 when you mean one thousand two hundred fifty.
  • Standard deviation has the same unit as the original observations; variance has squared units.
  • The coefficient of variation is not defined when the mean is zero and may be misleading when the mean is close to zero.
  • A z-score is not defined when every value is identical because the standard deviation is zero.
  • Results describe the entered data. They do not by themselves establish normality, causation or statistical significance.

Frequently asked questions

What does a low standard deviation mean?

It means the values are clustered relatively close to their mean. A larger standard deviation indicates greater spread, in the dataset’s original unit.

Why do sample and population results differ?

The sample formula divides the squared deviations by n − 1 to reduce bias when estimating a larger population’s variance.

Can I calculate a z-score?

Yes. Enter an optional value and the calculator reports how many standard deviations it lies above or below the mean.

Does the calculator support repeated values?

Yes. Enter each observation, including repetitions, because frequency affects the mean and standard deviation.

Related calculators

Use the Weighted Average Calculator when observations have unequal weights, or the Percentage Calculator for proportions and percentage change.

Reference

The formula definitions follow the NIST/SEMATECH e-Handbook of Statistical Methods.