Standard Deviation Calculator

Paste a list of numbers to get the standard deviation, variance, mean and quartiles, with every step of the working shown in a table.

Examples

Separate values with commas, spaces or new lines. You can paste a column straight from a spreadsheet.

What standard deviation means

Standard deviation measures how spread out numbers are from their mean. A small standard deviation means the values sit close to the average; a large one means they are scattered. Two classes can both average 80 on a test, but a class with a standard deviation of 3 scored very consistently, while one with 15 had a wide range of scores.

Sample vs. population standard deviation

Use the population standard deviation (σ) when your data includes every member of the group you care about, such as the heights of all 25 students in one class. Use the sample standard deviation (s) when your data is a sample used to describe a larger group, such as 25 students chosen to represent a whole school. Most science labs, surveys and AP Statistics problems use the sample version.

The formulas differ only in the denominator:

Formula
Populationσ = √[ Σ(x − μ)2 ÷ N ]
Samples = √[ Σ(x − x̄)2 ÷ (n − 1) ]

Dividing by n − 1 (Bessel’s correction) makes the sample variance an unbiased estimate of the population variance. A sample tends to sit closer to its own mean than to the true population mean, so dividing by n would underestimate the spread.

How to calculate standard deviation by hand

Take the data 2, 4, 4, 4, 5, 5, 7, 9.

  1. Find the mean: (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) ÷ 8 = 40 ÷ 8 = 5.
  2. Subtract the mean from each value: −3, −1, −1, −1, 0, 0, 2, 4.
  3. Square each result: 9, 1, 1, 1, 0, 0, 4, 16.
  4. Add them up: the sum of squared deviations 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: σ = 2 and s ≈ 2.138.

Interpreting the result

For data that follows a roughly normal (bell-shaped) distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. With a mean of 80 and a standard deviation of 5, roughly two-thirds of scores would fall between 75 and 85.

The variance is the standard deviation squared; it is useful in formulas but is in squared units. The standard error of the mean, s ÷ √n, estimates how much the sample mean itself would vary from sample to sample. The coefficient of variation expresses the standard deviation as a percentage of the mean, which helps compare spread between data sets with different units or scales.

Frequently asked questions

Can standard deviation be negative?

No. It is a square root of an average of squares, so it is always zero or positive. It is zero only when every value is the same.

Which one does my calculator show as sx and σx?

On most handheld calculators, sx (or Sx) is the sample standard deviation and σx is the population standard deviation. This page shows both.

How are the quartiles calculated?

The median splits the sorted data in half, and Q1 and Q3 are the medians of the lower and upper halves, leaving out the middle value when there is an odd count. This matches most handheld calculators; spreadsheets sometimes use a different method and give slightly different quartiles.

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Last updated September 27, 2026.

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