Standard Deviation Calculator — Sample and Population

Deviations from the mean (18)
x	x − x̄	(x − x̄)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

Deviations from the mean (18)

x − x̄

(x − x̄)²

-8

-6

-2

How to calculate standard deviation

Standard deviation measures how far numbers typically are from their mean. For the data 10, 12, 23, 23, 16, 23, 21, 16:

Mean: add the numbers and divide by how many there are: 144 ÷ 8 = 18.
Deviations: subtract the mean from each number: −8, −6, 5, 5, −2, 5, 3, −2.
Squares: square each deviation and add them up: 64 + 36 + 25 + 25 + 4 + 25 + 9 + 4 = 192.
Variance: divide by n − 1 = 7 for a sample: 192 ÷ 7 = 27.43 (or by n = 8 for a population: 24).
Standard deviation: take the square root: √27.43 = 5.24 (population: √24 = 4.90).

The calculator shows these steps with your numbers and lists every deviation in a table that you can download as a CSV file.

Sample or population?

The two formulas differ only in the divisor. A sample standard deviation divides by n − 1 because a sample’s own mean sits closer to its values than the true population mean would, which makes the spread look smaller; dividing by one less corrects for that (Bessel’s correction). Use it for survey answers, lab measurements, or one class’s test scores when you want to describe a larger group. A population standard deviation divides by n and is right only when the numbers are the whole group — every employee’s salary at a company, every game a team played this season. With many values the two are almost the same; with few values they differ noticeably.

Symbol	Name	Divide the squares by
s	sample standard deviation	n − 1
σ (sigma)	population standard deviation	n
s², σ²	variance	the same, without the square root

Reading the result

For data shaped roughly like a bell curve, 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 18 and a standard deviation of 5.24, most values in the example lie between about 12.8 and 23.2. The result also gives the median and quartiles, which are better summaries when the data are skewed or have outliers, the coefficient of variation (standard deviation as a percent of the mean, for comparing spread across different scales), and the standard error of the mean, s ÷ √n.

Accuracy and very large numbers

The calculator works out the mean first and then the squared deviations — the two-pass method — so it stays accurate even for values like 1,000,000,004 and 1,000,000,007, where a shortcut formula loses precision. Results are shown to about ten significant figures; round them for reporting with the rounding calculator. For percentages of a total or changes between two values, use the percentage calculator; for square roots and powers, the scientific calculator.

Frequently asked questions

How do you calculate standard deviation?

Find the mean, subtract it from each number and square the result, add the squares, divide by n − 1 for a sample (or n for a whole population), and take the square root.

Should I use sample or population standard deviation?

Use the sample formula, which divides by n − 1, when your numbers are part of a larger group — a survey, an experiment, test scores from one class. Use the population formula only when you have every value you care about.

What is the difference between variance and standard deviation?

Variance is the average squared distance from the mean; standard deviation is its square root, so it is in the same units as the data. For the example, the sample variance is 27.43 and the standard deviation 5.24.

What does a standard deviation tell you?

How spread out the numbers are around the mean. For roughly bell-shaped data, about 68% of values lie within one standard deviation of the mean and about 95% within two.

What is the standard error of the mean?

The sample standard deviation divided by the square root of n. It measures how much the mean itself would vary from sample to sample: 5.24 ÷ √8 = 1.85 for the example.

How do I enter data from a spreadsheet?

Copy the column and paste it into the box. Numbers on separate lines, or separated by commas, spaces or tabs, all work; anything that is not a number is skipped and listed under the box.