Z-Scores and Percentiles: Where Does a Score Stand?
October 21, 2026
A z-score, or standard score, expresses how far a value is from the mean in units of standard deviation: z = (value − mean) ÷ standard deviation. It standardizes data from different scales so that you can compare them. A positive z-score means above average, negative below, and zero exactly average. If the data are roughly normally distributed, a z-score can be converted into a percentile, the percentage of values that fall below it.
Suppose a test has a mean of 500 and a standard deviation of 100. A student who scores 650 has z = (650 − 500) ÷ 100 = 1.5. From the standard normal distribution, a z of 1.5 corresponds to the 93.3rd percentile.
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Open the Standard Deviation Calculator →Common z-Scores and Percentiles
The values below come from the standard normal cumulative distribution. A z of 0 is the 50th percentile. A z of 1 is the 84th, and 2 is the 97.7th. Because the normal curve is symmetric, a z of −1 is the 15.9th percentile and −2 the 2.3rd.
These percentiles hold only when the data are approximately normal. For skewed data, such as incomes, they can be misleading.
| z-score | Percentile (area below) | Example (mean 500, SD 100) |
|---|---|---|
| -2.0 | 2.3% | 300 |
| -1.0 | 15.9% | 400 |
| 0 | 50.0% | 500 |
| 1.0 | 84.1% | 600 |
| 1.5 | 93.3% | 650 |
| 1.645 | 95.0% | 664.5 |
| 2.0 | 97.7% | 700 |
| 3.0 | 99.9% | 800 |
Comparing Across Different Scales
Z-scores let you compare apples with oranges. Suppose Maria scored 82 on a math exam with mean 70 and SD 8, and 76 on a history exam with mean 68 and SD 5. Her z-scores are (82 − 70) ÷ 8 = 1.5 in math and (76 − 68) ÷ 5 = 1.6 in history. She performed slightly better relative to her classmates in history, even though her raw score was lower.
In the IQ scale (mean 100, SD 15), an IQ of 130 has z = 2, and an IQ of 85 has z = −1.
Using z-Scores to Find Cutoffs
Reverse the formula to find values for a given percentile: value = mean + z × SD. To find the score needed to be in the top 5% on the test above, use z = 1.645: 500 + 1.645 × 100 = 664.5. For the top 1%, z = 2.326, giving 732.6. For the middle 95%, use ±1.96: 304 to 696.
Quality control uses z-scores to flag unusual measurements: values beyond ±3 are rare for normal data (about 0.27% of the time), so they trigger a review.
Limits and Assumptions
The percentile conversion assumes a normal distribution. Many datasets are skewed or have heavy tails. Test scores are often designed to be approximately normal, but income, response times, and web traffic are not. For small samples, estimates of the mean and standard deviation are uncertain. Also, a z-score says how unusual a value is, not whether it is good or bad. And percentiles from the actual data can be computed directly when you have the full dataset.
Practical Uses
Standardized tests report scaled scores and percentiles. Pediatric growth charts use z-scores for height and weight. Finance uses z-scores to measure how far a return is from average. Spreadsheets compute z-scores with STANDARDIZE(x, mean, sd) and percentiles with NORM.S.DIST(z, TRUE).
Frequently Asked Questions
What is a z-score?
The number of standard deviations a value is above or below the mean.
What percentile is a z-score of 2?
About the 97.7th percentile for normal data.
How do I find the score for the top 10%?
Use z = 1.28: mean + 1.28 × standard deviation.
Can z-scores be used on non-normal data?
They can be computed, but converting them to percentiles with the normal table may be inaccurate.
What is a standard normal distribution?
A normal distribution with mean 0 and standard deviation 1.