z-score is one of 1 standardization formula in the statistics section of this library, and it is used at high school · ap level.
Why z-score works
Subtracting the mean shifts the centre of the data to zero, and dividing by the standard deviation swaps the measuring unit for standard deviations. After both steps the value carries no units at all, so a score from one test can be compared directly with a score from a completely different one.
What each symbol means
$x$ is value, $\mu$ population mean, and $\sigma$ population standard deviation.
z-score: when it holds
$\sigma>0$; a negative score lies below the mean.
When it stops applying
A z-score by itself gives no percentile unless the distribution is roughly normal. In a strongly skewed data set a score of 2 may be nowhere near the 97.7th percentile that the normal curve would suggest; without a shape assumption all you can say is that at most a quarter of values lie more than 2 standard deviations from the mean.
z-score: a worked example
If $x=85,\mu=70,\sigma=10$, then $z=1.5$.
The mistake to avoid
What people do: Dividing by the variance rather than the standard deviation.
Why it goes wrong: With a value of 85, a mean of 70 and a standard deviation of 10, the correct score is 1.5. Using the variance of 100 gives 0.15, which makes an unusual value look ordinary.
Do this instead: Check the units: the denominator must be in the same units as the numerator, which means the square-rooted spread, never the squared one.
z-score: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $x$ is value, $\mu$ population mean, and $\sigma$ population standard deviation.
- Check the conditions before substituting. $\sigma>0$; a negative score lies below the mean.
- 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
- Standardization
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where z-score 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 z-score
Does a z-score of 2 always mean the top 2.5 percent?
Only under a normal distribution, where the area above 2 is 0.02275. For an unknown shape that figure is a guess, and for a skewed distribution it can be badly wrong.
Can a z-score be larger than 3?
Yes, and in large data sets it should occasionally happen. Under a normal model only about 0.27 percent of values fall beyond three standard deviations either way, so such a value is rare rather than impossible.
What do I use when I only have sample statistics?
Substitute the sample mean and sample standard deviation. The result is an estimate rather than an exact standardisation, which is one reason small-sample inference uses t rather than z.
What are the mean and spread of the z-scores themselves?
Standardising always produces a set with mean 0 and standard deviation 1. That is the whole point: it puts every variable on one shared ruler.