The z-score formula
z = x − μσ
- x is the value you're looking at
- μ is the mean of the group it comes from
- σ is the standard deviation of that group
With sample data, use the sample mean and sample standard deviation instead: z = (x − x) / s. The steps are identical.
Step by step
A class's exam scores have a mean of 70 and a standard deviation of 10. You scored 85. What is your z-score?
-
Subtract the mean: how far above or below average is the value?
85 − 70 = 15
-
Divide by the standard deviation: how many standard deviations is that?
z = 1510 = 1.5
Your score is 1.5 standard deviations above the class average.
Turn the z-score into a percentile
If the scores are roughly normal (bell-shaped), the z-score tells you what share of the class scored below you. Find the area to the left of z = 1.5 under the standard normal curve:
- In a z-table: row 1.5, column .00 gives 0.9332.
- In Excel or Google Sheets: =NORM.S.DIST(1.5,TRUE) also returns 0.9332.
So about 93% of the class scored lower: you're at roughly the 93rd percentile.
Z-scores in Excel and Google Sheets
With the value in A2, the mean in B2 and the standard deviation in C2:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | Score | Mean | SD | z | Percentile |
| 2 | 85 | 70 | 10 | 1.5 | 0.9332 |
To give every value in a column its z-score, compute the mean and standard deviation once and lock them with $ signs:
- =STANDARDIZE(A2,AVERAGE($A$2:$A$31),STDEV.S($A$2:$A$31)), then fill it down the column.
- Or simply =(A2-AVERAGE($A$2:$A$31))/STDEV.S($A$2:$A$31).
Compare scores on different scales
Z-scores really earn their keep when two numbers aren't directly comparable. Say you scored 1,250 on the SAT (mean 1,050, SD 200) and a friend scored 27 on the ACT (mean 21, SD 5.5):
- Your SAT: z = (1,250 − 1,050) / 200 = 1.00
- Your friend's ACT: z = (27 − 21) / 5.5 = 1.09
Relative to other test-takers, your friend did very slightly better, even though 27 looks nothing like 1,250. (These means and SDs are round illustrative figures; use the current published norms for real comparisons.)
How to interpret a z-score
- z = 0: exactly average.
- Between −1 and 1: typical; about 68% of values fall here in a normal distribution.
- Beyond ±2: unusual; only about 5% of values are this far out.
- Beyond ±3: rare, about 0.3%. Check whether it's an error or a genuine outlier.
These benchmarks come from the 68-95-99.7 rule and assume the data are roughly bell-shaped. For skewed data, the z-score still measures distance from the mean, but its percentile can be off.
To skip the arithmetic, the z-score calculator gives the z-score, percentile and tail areas with a shaded normal curve.