Standard Deviation Calculator

Paste a list of numbers to find the standard deviation, variance and mean. Switch between sample and population and follow every step of the calculation.

Separate numbers with commas, spaces or new lines.
Data is a
Sample standard deviation (s)
–
Variance–
Mean–
Count (n)–
Sum of squares–
xx − mean(x − mean)²

What standard deviation tells you

Standard deviation measures how spread out numbers are around their average. If every value is close to the mean, the standard deviation is small. If values are scattered far above and below the mean, it is large. It is one of the most widely used numbers in statistics, quality control, finance and science.

The formulas

Population: σ = √( Σ(x − μ)² ÷ N )
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )

Step by step

Using the data 2, 4, 4, 4, 5, 5, 7, 9:

  1. Find the mean: the sum is 40 and there are 8 values, so the mean is 5.
  2. Subtract the mean from each value: −3, −1, −1, −1, 0, 0, 2, 4.
  3. Square each difference: 9, 1, 1, 1, 0, 0, 4, 16.
  4. Add them up: the sum of squares is 32.
  5. Divide: by N = 8 for a population (variance 4), or by n − 1 = 7 for a sample (variance 4.571).
  6. Take the square root: population SD = 2, sample SD ≈ 2.138.

Interpreting the result

In a roughly bell-shaped data set, about 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three. If test scores have a mean of 75 and a standard deviation of 8, about two-thirds of students scored between 67 and 83.

Only need the mean, median or mode? Use the average calculator.

Last reviewed October 2026.

Frequently asked questions

Should I use sample or population standard deviation?

Use population standard deviation when your data includes every member of the group you care about, such as the scores of every student in one class. Use sample standard deviation when your data is a sample used to estimate a larger group, which is the usual case in research and surveys.

Why does the sample formula divide by n − 1?

A sample tends to be a little less spread out than the whole population, so dividing by n would underestimate the true spread. Dividing by n − 1, called Bessel's correction, fixes that bias.

What does a large standard deviation mean?

It means the values are spread widely around the mean. A small standard deviation means most values sit close to the mean. Two classes can have the same average score but very different standard deviations.

What is the difference between variance and standard deviation?

Variance is the average squared distance from the mean. Standard deviation is its square root, which puts it back in the same units as your data, making it easier to interpret.

What is the 68-95-99.7 rule?

For data that follows a normal (bell-shaped) distribution, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.