What is a z-score?
A z-score says how many standard deviations a value is above or below the mean. 0 is exactly average, 1 is one standard deviation above it, and −2 is two below. When results follow a normal (bell) curve, the z-score also tells you what share of results are lower, which is what a z table lists.
On a normal curve, about 68.3% of results fall within 1 standard deviation of the mean, 95.0% within 1.96, 95.5% within 2 and 99.7% within 3. If your results are clearly lopsided, the z-score still says how far from the mean a value is, but the percentages won’t fit them well.
Source Microsoft Support · STANDARDIZE function ↗CDC Principles of Epidemiology · Lesson 2, Section 7: Measures of spread ↗
How to calculate a z-score
Here it is with your numbers:
- z = (85 − 75) ÷ 10
- z = 1
- The bell curve’s area below 1 is 84.13%.
It means about 84.13% of results are lower than 85, and 15.87% are higher.
In Excel or Google Sheets, =STANDARDIZE(x, mean, standard deviation) gives the z-score and =NORM.S.DIST(z, TRUE) gives the share of the curve below it.
Source Microsoft Support · NORM.S.DIST function ↗
Z table
The percent of a normal curve below each z-score, and between −z and z. For a negative z, the percent below is the percent above the positive one: 15.87% for −1.
| z | Percent below | Between −z and z |
|---|---|---|
| 0.00 | 50.00% | 0.00% |
| 0.50 | 69.15% | 38.29% |
| 1.00 | 84.13% | 68.27% |
| 1.28 | 89.97% | 79.95% |
| 1.50 | 93.32% | 86.64% |
| 1.96 | 97.50% | 95.00% |
| 2.00 | 97.72% | 95.45% |
| 2.50 | 99.38% | 98.76% |
| 2.58 | 99.51% | 99.01% |
| 3.00 | 99.87% | 99.73% |
A printed z table rounds z to 2 decimals before looking it up, so it can differ in the last digit. For 42 with a mean of 40 and a standard deviation of 1.5, the exact z is 1.3333 and 90.88% of the curve is below it; the table row for 1.33 says 90.82%. Excel’s NORM.S.DIST also uses the exact z.
FAQs
What is a z-score?
A z-score is how many standard deviations a value is above or below the mean. Excel calls it a normalized value: 42 with a mean of 40 and a standard deviation of 1.5 has a z-score of 1.33.
How do you calculate a z-score?
Subtract the mean from the score, then divide by the standard deviation. A test score of 85 in a class with a mean of 75 and a standard deviation of 10 has a z-score of (85 − 75) ÷ 10, which is 1.
How do I find a z-score?
From a score, subtract the mean and divide by the standard deviation. From a percentile, find the z-score with that percent of the bell curve below it: Excel’s NORM.S.INV(0.908789) gives 1.3333347, so the 90.8789th percentile has a z-score of 1.33. Choose “Percentile” above to do the same here.
Sources
- Microsoft Support: STANDARDIZE function
- Microsoft Support: NORM.S.DIST function
- Microsoft Support: NORM.S.INV function
- CDC Principles of Epidemiology: Lesson 2, Section 7: Measures of spread