Population Standard Deviation Calculator

Working

1Mean
$\bar{x} = \frac{\sum x}{n}$
$\frac{40}{8} = 5$
2Squared deviations
$\sum (x - \bar{x})^2$
$32$
Subtract the mean from each value, square it and add them up. See the table below.
3Population variance
$\sigma^2 = \frac{SS}{n}$
$\frac{32}{8} = 4$
4Standard deviation
$\sigma = \sqrt{\sigma^2}$
$\sqrt{4} = 2$

x	x - mean	(x - mean)^2
2	-3	9
4	-1	1
4	-1	1
4	-1	1
5	0	0
5	0	0
7	2	4
9	4	16
Total	0	32

x - mean

(x - mean)^2

-3

-1

Total

About Population Standard Deviation Calculator

This calculator finds the population standard deviation, written with the Greek letter sigma, for data that is every value in the group you are describing. It divides the sum of squared deviations by n. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

Use it when your numbers are the whole population: all the marks of one class when the class is the subject, or every item in a production batch. For data that is only a sample, use the sample version, which divides by n - 1.

How to use Population Standard Deviation Calculator

Paste or type your numbers, separated by commas, spaces or new lines. Use a dot for decimals.
Pick Sample (n - 1) if the numbers are a sample, or Population (n) if they are the whole group.
For grouped data, switch Data entry to Frequency table and enter one "value, frequency" pair per line. A class interval such as 10-20, 4 is counted at 15.
Read the standard deviation at the top, then the variance, mean, sum, count, standard error and margin of error.
Open the working and the deviations table to see each (x - mean) and its square, and the totals that feed the formula.

Features

Sample and population standard deviation side by side, with the one you chose as the headline.
Variance, mean, count, sum, range, standard error and margin of error at 90%, 95% or 99% confidence.
Step-by-step working in written-out formulas, with a table of deviations and squared deviations.
Frequency table and class interval input, with a weighted calculation instead of an expanded list.
Coefficient of variation as a percentage of the mean.
Copy the results as text. Works with long lists and on a phone.

Population standard deviation formula

Add up the squared deviations from the mean, divide by n and take the square root.

sigma = sqrt( sum((x - mean)^2) / n )

Worked examples

Data: 2, 4, 4, 4, 5, 5, 7, 9 gives sigma = 2
The mean is 5. The squared deviations add up to 32, so the variance is 4 and its square root is 2.

Privacy: how it works

All the arithmetic happens in your browser tab. The numbers you paste are not sent to Ulkaa or to any other server.

Tips

One extreme value can inflate the standard deviation a lot. Check the deviations table for a very large squared deviation.
Standard deviation has the same unit as your data. Variance is in squared units, which is harder to interpret.
To compare spread between data sets with different means or units, use the coefficient of variation.
A sample needs at least two values. With one value the sample standard deviation is undefined.

Frequently asked questions

Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Does it run in my browser?

Yes. Your numbers are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Your numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

When do I use the population formula?

When your data contains every member of the group you want to describe, not just a sample of it.

Is population standard deviation always smaller?

Yes, for the same numbers it is slightly smaller than the sample version, because n is larger than n - 1.