Population variance and standard deviation
Measure average squared distance from a population mean and return to original units.
Population variance and standard deviation is one of 3 spread formulas in the statistics section of this library, and it is used at high school · ap · university level.
Why population variance and standard deviation works
Distances below and above the mean would cancel if you simply added them, so each one is squared first to make it positive. Averaging those squares gives a typical squared distance, and taking the square root at the end brings the answer back into the units the data were measured in.
What each symbol means
$N$ is population size and $\mu$ population mean.
Population variance and standard deviation: when it holds
Use this denominator when the data are the complete population of interest.
When it stops applying
Dividing by N assumes your data are the complete population you care about. If they are a sample being used to guess at a bigger group, this divisor produces an estimate that is systematically too small: for the two values 1 and 3 it reports a variance of 1, while the sample formula reports 2.
Population variance and standard deviation: a worked example
For population $1,3$, $\mu=2$, $\sigma^2=1$, and $\sigma=1$.
The mistake to avoid
What people do: Reporting the variance as though it were the standard deviation.
Why it goes wrong: Variance carries squared units, so a variance of 9 square centimetres is a spread of 3 centimetres, not 9. Skipping the square root inflates the reported spread and makes the units nonsense.
Do this instead: Finish with the square root whenever the answer will be described in the same units as the data, and label variance clearly when you do report it.
Population variance and standard deviation: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $N$ is population size and $\mu$ population mean.
- Check the conditions before substituting. Use this denominator when the data are the complete population of interest.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most statistics slips.
Where this formula fits
- Subject
- Statistics formulas — 21 entries in this library
- Topic
- Spread
- Level
- High school · AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where population variance and standard deviation comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Standard Deviation — the lesson behind this formula: measure typical distance from the mean.
- Statistics Calculator — check your substitution and the value it produces.
- Study statistics — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 21 statistics formulas — the full grouped reference, or the complete formula library.
Questions about population variance and standard deviation
How do I decide whether my data are a population?
Ask whether you have measured every member of the group in your conclusion. All 30 students in one named class is a population; 30 students used to describe the whole school is a sample.
Can a standard deviation come out negative?
No, never. Every term being added is a square, so the variance is zero or positive, and the square root is taken as the non-negative one by convention.
What does this measure tell me for a normal distribution?
It sets the width of the bell. About 68.27 percent of values sit within one standard deviation of the mean, 95.45 percent within two, and 99.73 percent within three.
Why square the distances instead of taking absolute values?
Squares are smooth and they add up neatly across independent pieces, which makes almost every later formula work. The absolute-value version exists and is called the mean absolute deviation, but it lacks those properties.