A z-score tells you how many standard deviations a value sits above or below the mean. It turns any measurement into the same scale, which is why you can compare an SAT score to an ACT score, or a height to a weight. You’ll need the mean and standard deviation, covered in the mean, median, mode and standard deviation guide.
The formula
z = (x − μ)/σ
- x is the value
- μ (mu) is the mean
- σ (sigma) is the standard deviation
Example A test has mean 70 and standard deviation 8. Find the z-score of 86.
z = (86 − 70)/8 = 16/8 = 2. The score is 2 standard deviations above the mean.
Example Same test. Find the z-score of 64.
z = (64 − 70)/8 = −6/8 = −0.75. Negative means below the mean.
A z-score of 0 is exactly the mean.
Going backwards: from z to x
Solve the formula for x:
x = μ + zσ
Example Heights have mean 165 cm and standard deviation 7 cm. What height has z = −1.5?
x = 165 + (−1.5)(7) = 165 − 10.5 = 154.5 cm.
The normal distribution
Many measurements, like heights, test scores and measurement errors, pile up in a bell shape: most values near the mean, fewer and fewer further out, symmetric on both sides. That shape is the normal distribution.
For a normal distribution:
| Range | Share of values |
|---|---|
| Within 1 standard deviation (z between −1 and 1) | about 68% |
| Within 2 (z between −2 and 2) | about 95% |
| Within 3 (z between −3 and 3) | about 99.7% |
Using the rule
Because the curve is symmetric, you can split those percentages in half.
Example IQ scores are normal with mean 100 and standard deviation 15. What percent score above 130?
- 130 is z = (130 − 100)/15 = 2.
- 95% lie between z = −2 and 2, so 5% lie outside.
- Half of that is above: 2.5%.
Example Same scores. What percent fall between 85 and 115?
85 is z = −1 and 115 is z = 1, so about 68%.
Example What percent score between 100 and 115?
That is half of the middle 68%, so 34%.
Comparing values from different scales
Example Maya scored 82 on a test with mean 74 and SD 4. Leo scored 90 on a test with mean 80 and SD 8. Who did better compared to their class?
- Maya: z = (82 − 74)/4 = 2.
- Leo: z = (90 − 80)/8 = 1.25.
- Maya’s score is further above her class average.
Percentiles and z-tables
For z-scores that aren’t whole numbers, a z-table or calculator gives the exact share of values below that z. A few values worth knowing:
| z | Share below |
|---|---|
| −2 | 2.3% |
| −1 | 15.9% |
| 0 | 50% |
| 1 | 84.1% |
| 1.645 | 95% |
| 2 | 97.7% |
On a TI-84, normalcdf(lower, upper, μ, σ) gives the share between two values.
Using the rule on data that isn’t normal
The 68-95-99.7 rule only applies to bell-shaped data. Incomes, for example, are skewed right, and the rule gives the wrong answers there. A z-score still measures distance from the mean in any data set, but the percentages don’t carry over.
Try these
Mean 50, SD 5. Find the z-score of 62.
Show answer
2.4.
Mean 200, SD 25. Find the z-score of 175.
Show answer
−1.
Mean 12, SD 3. What value has z = 1.5?
Show answer
16.5.
Scores are normal with mean 60 and SD 10. About what percent score between 40 and 80?
Show answer
About 95%. Those are z = −2 and z = 2.
Same scores. About what percent score below 50?
Show answer
About 16%. Below z = −1 is half of the 32% outside the middle.
A 640 on a test with mean 520 and SD 100, or a 26 on a test with mean 21 and SD 5. Which is better relative to its test?
Show answer
The 640. Its z-score is 1.2, compared with 1.0.