Standard deviation calculator
Paste a list of numbers. The page shows the mean, every deviation, and both the sample and the population standard deviation.
- Sample and population
- Free, no sign-up
Separate the numbers with commas or spaces. The sample figure divides by n − 1, and the population figure divides by n.
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How to use it
- Enter the numbers, separated by commas or spaces.
- Read the mean and the table of deviations in the working.
- Use the sample result for a sample, and the population result for a whole group.
How the two results differ
Both results start from the same mean and the same squared deviations. The population variance divides that sum by n, the count of values in the list. The sample variance divides by n − 1, which is the Bessel correction used when the list is a sample from a larger group. The standard deviation is the square root of the matching variance.
With one value the population figure is zero, because that value is the mean. The sample figure needs at least two values so that n − 1 is at least 1. The worked example 2, 4, 4, 4, 5, 5, 7, 9 has mean 5, a squared-deviation total of 32, a population standard deviation of 2 and a sample standard deviation of 2.138 to three decimal places.
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Questions
Which result belongs on a homework sheet?
Use the sample standard deviation when the numbers are a sample, and the population standard deviation when the list is the entire group. The working shows both.
Are the steps the full calculation?
Yes. You get the sum, the mean, each value minus the mean, each square, the sum of squares, both variances and both standard deviations.
Can I paste a column from a spreadsheet?
Yes. Commas, spaces and line breaks all separate numbers. Use a dot for a decimal, as in 2.5.
What does n − 1 change?
It makes the sample variance a little larger than the population variance. For the eight-value example, 32/8 is 4 and 32/7 is about 4.571.