Standard Deviation Calculator

Written byOleh Soloninupdated 5 October 2026

Values

Mode

Result

Count of numbers8
Mean18
Std Dev5.237229
Variance27.428571
Coefficient of variation29.1%
Step by step
x x − mean (x − mean)² 10 −8 64 12 −6 36 23 5 25 23 5 25 16 −2 4 23 5 25 21 3 9 16 −2 4 Sum of squares 192 ÷ 7 = variance 27.4286; √27.4286 = 5.2372

How tightly your data clusters around its average

Calculate standard deviation, variance and mean from a list of numbers, with both the sample and population formulas.

An average alone can hide everything interesting. Two datasets can share a mean of 50 while one hovers between 49 and 51 and the other swings from 0 to 100.

Standard deviation measures that spread. Roughly two-thirds of a normal dataset falls within one deviation of the mean, and about 95% within two.

How to use

Paste the numbers separated by commas, spaces or new lines.
Choose the formula: sample (divide by n − 1) or population (divide by n).
Read the mean, variance, standard deviation, coefficient of variation and the step-by-step table.

Sample against population

For the set 10, 12, 23, 23, 16, 23, 21, 16 the mean is 18 and the sample standard deviation is 5.24. Treating the same numbers as a whole population changes the divisor from seven to eight and drops the deviation to 4.90.

The coefficient of variation turns spread into a percentage of the mean — 29.1% here. It is what makes dispersion comparable across units: a deviation in centimetres and one in currency cannot be compared directly, but their percentages can.

The set 10, 12, 23, 23, 16, 23, 21, 16. Worked out with this calculator.
Measure	Sample	Population
Mean	18	18
Variance	27.43	24.00
Standard deviation	5.24	4.90
Coefficient of variation	29.1%	27.2%

How to calculate standard deviation

1. Find the mean: add the numbers and divide by how many there are. 2. Subtract the mean from each number and square the result. 3. Add the squares and divide by n for a population or n − 1 for a sample: that is the variance. 4. Take the square root: that is the standard deviation.

For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the squared deviations add up to 32. Population variance is 32 ÷ 8 = 4, so the deviation is 2; sample variance is 32 ÷ 7 ≈ 4.57, deviation ≈ 2.14. The coefficient of variation is the deviation as a percentage of the mean: 2 ÷ 5 = 40%.

What this calculator shows

Standard deviation and variance for your dataset
The mean and how many values were counted
The coefficient of variation, the deviation as a share of the mean
A step-by-step table of deviations and their squares

What to keep in mind

Sample standard deviation divides by n−1, population divides by n.
The 68/95 rule assumes a roughly normal distribution; skewed data will not follow it.

FAQs

Should I use the sample or population formula?

Sample, unless your numbers are literally every member of the group. Measuring 30 of 500 employees is a sample; measuring all 500 is a population.

Why does the sample version divide by n−1?

Deviations in a sample are measured from its own mean, which always sits closer to its numbers than the true mean of the population does. The sum of squared deviations therefore runs low on average — by exactly (n − 1)/n — and dividing by n − 1 instead of n removes that bias from the variance.

What does a standard deviation of zero mean?

Every value is identical. There is no spread at all, so every point sits exactly on the mean.

How is variance related to standard deviation?

Variance is the square of the deviation. Deviation is more useful day to day because it is in the same units as your data.

What is the coefficient of variation?

The standard deviation divided by the mean, as a percentage, also called the relative standard deviation. For 2, 4, 4, 4, 5, 5, 7, 9 it is 2 ÷ 5 = 40%. It compares relative spread for data measured from a true zero with a positive mean that is not close to zero.

What separates the sample deviation from the population one?

The divisor. A sample divides by the count minus one, a population by the count itself. On eight numbers that is 5.24 against 4.90.

What is the coefficient of variation for?

Comparing spread across different units. A deviation of 5.24 means nothing on its own; 29.1% of the mean immediately gives it scale — close to a third of the mean. Whether that is large depends on what you are measuring.

Worked example

Eight values

Input: 2, 4, 4, 4, 5, 5, 7, 9

Output: Population σ = 2, mean = 5

Note: A textbook set: most scores sit within two points of the mean of 5, with 9 as the largest value in the set.

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Result

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