Descriptive statistics Updated November 2026

z-score

A z-score says how many standard deviations a value sits above or below the mean: z = (x - mean) / sd. A z of 1.5 means 1.5 standard deviations above the mean; a negative z means below it.

Syntax

z = (x - mean) / sd

Part	What it is
`x`	the value you are standardizing
`mean`	the mean of the population or sample
`sd`	the standard deviation

Examples

Test score x = 85, mean = 70, sd = 10
z = (85 - 70) / 10

Result: z = 1.5

Excel: =STANDARDIZE(85, 70, 10)

x = 60, mean = 70, sd = 10
z = (60 - 70) / 10

Result: z = -1

Share of values below z = 1.5 (standard normal)
=NORM.S.DIST(1.5, TRUE)

Result: 0.9332, so the score 85 is at about the 93rd percentile

Reverse: which value has z = 2 when mean = 70, sd = 10?
x = mean + z * sd = 70 + 2 * 10

Result: x = 90

Common mistakes

Dividing by the variance instead of the standard deviation; the denominator is sd.
Using the sample mean and sd of a different group than the one the value belongs to; standardize against the group you are comparing with.
Reading a z-score as a percentile directly; a z of 1.5 is not the 150th percentile, look it up with NORM.S.DIST or a z table.

Practice z-score

These drills check your answer on live data, free, no account needed.

z-score of one valueLast week a call center agent handled 33 calls in a shift. Across the group, the mean is 70 with a standard deviation of 11.3. What is the z-score of this value? Round to 2 decimal places.

Questions

What is a good z-score?

There is no good or bad in general. z between -2 and 2 covers about 95% of a normal population, so values beyond that are unusual. In quality control or outlier screening, |z| > 3 is a common flag.

How do I convert a z-score to a percentage?

Use the standard normal cumulative distribution: in Excel =NORM.S.DIST(z, TRUE) gives the share of values below z. For z = 1 that is 0.8413, or 84.13%.

Can I compare z-scores from different tests?

Yes, that is the main use. A z of 1.2 on a math test and 0.4 on a reading test means the math result is relatively stronger, regardless of the original scales.

Standard deviationThe standard deviation formula, sample versus population versions, a worked example with exact numbers and the Excel functions STDEV.S and STDEV.P.
Normal distributionHow to find normal distribution probabilities with z-scores, read the standard normal table, find a percentile cutoff, and use NORM.DIST and NORM.INV.
PercentileHow to find a percentile of a data set and the percentile rank of one value, with exact worked examples, Excel PERCENTILE.INC and common mix-ups.
OutliersHow to detect outliers with the 1.5 IQR fence rule and the z-score rule, worked examples with exact fences, and what to do with outliers once found.