Sample or population?

Sample, almost always. Use the population divisor only when your data genuinely is every member of the group rather than a draw from it. On small datasets the two differ noticeably.

Is a standard deviation of 5 large?

It depends entirely on the mean. Against a mean of 10 it is enormous and against 10,000 it is negligible, which is what the coefficient of variation is for.

Math & Statisticsstatistics

Standard Deviation Calculator

Standard deviation on both conventions, with the coefficient of variation that makes it comparable across datasets. A deviation of 5 means nothing until you know whether the mean is 10 or 10,000.

Also called: sd calculator, variance calculator.

Standard deviation

5.636193

5.636193 on the sample convention, against a mean of 13.83. That is a coefficient of variation of 40.74%. About two thirds of the data should lie between 8.2 and 19.47. On the population convention it would be 5.1451, a difference of 9.5%. A value of 0 is 2.45 deviations below the mean, which is unusual for this data.

Variance

31.77

Mean

13.83

On the other convention

5.15

Coefficient of variation

40.74%

One deviation below the mean

8.2

One deviation above the mean

19.47

Two deviations below

2.56

Two deviations above

25.11

Z score of your value

-2.45

How many values

Convention used

sample

On the two conventions

On the population convention it would be 5.1451, a difference of 9.5%.

On the value you scored

A value of 0 is 2.45 deviations below the mean, which is unusual for this data.

How this is calculated

Method and background

How this is calculated

Variance is the average squared distance from the mean, and the standard deviation is its root, which returns it to the units of the data. The divisor is the whole convention argument: a sample uses n-1 because the mean was estimated from the same data and using it consumes a degree of freedom, and on six values the two answers differ by ten percent. The coefficient of variation divides by the mean, which makes a spread comparable between datasets measured in different units or at different scales.

the sample form divides by n-1 because the mean was estimated from the same data

n: How many values

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

six values as a sample

Numbers: 12, 18, 7, 22, 15, 9
This data is: A sample (n−1)
A value to score against the data: 0

Standard deviation5.636193

root of the sum of squares over five

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the population divisor gives a smaller figure

Numbers: 12, 18, 7, 22, 15, 9
This data is: The whole population (n)
A value to score against the data: 22

Standard deviation5.145116

boundary: the same data over six rather than five, and 22 scored against it

Open this example

Method and limits

What it assumes

Numeric values, with anything unparseable ignored.

What it deliberately does not model

The empirical rule intervals assume a roughly normal distribution and mislead badly on a skewed one.
Standard deviation is sensitive to outliers, since the distances are squared.
The coefficient of variation is meaningless when the mean is near zero.

Formula version 1.0.0 · definition 1.0.0 · India · Report a problem with this calculator

Frequently asked questions

Sample or population?: Sample, almost always. Use the population divisor only when your data genuinely is every member of the group rather than a draw from it. On small datasets the two differ noticeably.
Is a standard deviation of 5 large?: It depends entirely on the mean. Against a mean of 10 it is enormous and against 10,000 it is negligible, which is what the coefficient of variation is for.