Z-Score Calculator
What Is a Z-Score?
A z-score (also called a standard score) measures how many standard deviations a data point is from the mean of a distribution. A z-score of 0 means the value equals the mean. A z-score of 1.0 means the value is one standard deviation above the mean. A z-score of -2.0 means the value is two standard deviations below the mean.
The formula is: z = (X − μ) / σ where X is the data value, μ (mu) is the population mean, and σ (sigma) is the standard deviation.
Z-scores are used across statistics, finance, medicine, and education to compare values from different distributions, identify outliers, and find percentile ranks in a normal distribution.
How to Use This Z-Score Calculator
- Z-score from a value: Enter your data value, the mean, and standard deviation. The calculator finds the z-score and percentile rank.
- Percentile from z-score: Enter a z-score to find its percentile rank in the standard normal distribution.
- Value from z-score: Enter a z-score, mean, and standard deviation to find the raw data value at that position.
Z-Score Percentile Reference Table
| Z-Score | Percentile | % of Data Below |
|---|---|---|
| -3.0 | 0.13% | 0.13% |
| -2.0 | 2.28% | 2.28% |
| -1.5 | 6.68% | 6.68% |
| -1.0 | 15.87% | 15.87% |
| -0.5 | 30.85% | 30.85% |
| 0.0 | 50.00% | 50.00% |
| 0.5 | 69.15% | 69.15% |
| 1.0 | 84.13% | 84.13% |
| 1.5 | 93.32% | 93.32% |
| 1.645 | 95.00% | 95.00% |
| 1.96 | 97.50% | 97.50% |
| 2.0 | 97.72% | 97.72% |
| 2.576 | 99.50% | 99.50% |
| 3.0 | 99.87% | 99.87% |
Worked Example: Test Score Percentile
A class exam has a mean of 72 and a standard deviation of 10. Sarah scored 88. What is her percentile?
Step 1: z = (88 − 72) / 10 = 16 / 10 = 1.6
Step 2: Look up z = 1.6 → approximately the 94.5th percentile
Sarah scored better than roughly 94.5% of the class. If there are 30 students, she placed in the top 1–2.
Common Z-Score Applications
- Standardized testing (SAT, IQ): SAT scores are scaled to mean 1000, SD 200. IQ is scaled to mean 100, SD 15. Z-scores tell you exactly where you fall in the distribution.
- Finance — Altman Z-Score: A model using z-scores from financial ratios to predict corporate bankruptcy risk. A score above 2.99 indicates low risk; below 1.81 indicates distress.
- Outlier detection: Data points with |z| > 3 are typically flagged as outliers in datasets — they fall beyond 3 standard deviations.
- Medical testing: Bone density T-scores and Z-scores compare patients to age-matched norms. A Z-score of -2.0 on bone density is considered below the expected range.
- Quality control: Six Sigma manufacturing uses z-scores to measure process defect rates — a "6 sigma" process has a z-score of 6, meaning only 3.4 defects per million.
The Empirical Rule (68-95-99.7)
- ±1σ (z = -1 to 1): Contains approximately 68.27% of data in a normal distribution
- ±2σ (z = -2 to 2): Contains approximately 95.45% of data
- ±3σ (z = -3 to 3): Contains approximately 99.73% of data
- Only about 0.27% of data falls beyond ±3 standard deviations from the mean
Frequently Asked Questions About Z-Scores
What is a good z-score?
It depends on context. For test performance, a z-score of 1.0+ (top 84%) is strong. For quality control, you want z-scores close to 0 (near average). For outlier detection, |z| > 2 or 3 flags unusual values.
What does a negative z-score mean?
A negative z-score means the value is below the mean. A z-score of -1.5 means the value is 1.5 standard deviations below average — at approximately the 6.7th percentile.
What is a z-score of 1.96?
Z = 1.96 corresponds to the 97.5th percentile — meaning 97.5% of values in a normal distribution fall below this point. It's the standard critical value for 95% confidence intervals in two-tailed hypothesis tests.
How do I find a percentile from a z-score?
Look up the z-score in a standard normal table (z-table), or use the calculator above. The table gives the area under the normal curve to the left of the z-score, which equals the percentile rank.
What is a z-score of 2.0?
Z = 2.0 corresponds to the 97.72nd percentile — your value is 2 standard deviations above the mean and exceeds roughly 97.7% of the distribution.
Can z-scores be used for non-normal distributions?
Yes — you can calculate z-scores for any distribution. However, the percentile conversion using the normal distribution table is only accurate when the data is approximately normally distributed. For skewed distributions, use non-parametric methods.
What is the difference between a z-score and a t-score?
Z-scores use the known population standard deviation. T-scores are used when the population standard deviation is unknown and must be estimated from the sample — common in small samples. As sample size increases, t-distributions approach the normal distribution.
What IQ score corresponds to a z-score of 2?
IQ has mean 100 and SD 15. At z = 2: IQ = 100 + 2×15 = 130. This is in the "very superior" range, above 97.7% of the population.