- Find Z-Score. Enter a raw data value (X), the population mean (μ), and standard deviation (σ). The calculator returns the z-score and the percentile (cumulative probability). Example: a test score of 78, class mean 70, standard deviation 8 gives z = (78-70)/8 = 1.0, which is the 84th percentile.
- Z to Probability. Enter a z-score to find the probability that a random value from the distribution falls below, above, or between specific z values. Useful for finding what percentage of a population falls in a given range.
- Find Raw Score. Enter a z-score, mean, and standard deviation to convert back to the original scale. X = μ + (z × σ). If z = 1.5, mean = 100, SD = 15: X = 100 + (1.5 × 15) = 122.5.
Z-Score Calculator
Calculate z-scores and find the corresponding percentile in a normal distribution.
Z-Score
0.5000
| Formula | (75 - 70) / 10 |
| Percentile | 69.15th |
| P(X < 75) | 69.1462% |
| P(X > 75) | 30.8538% |
Common Z-Scores for Confidence Intervals
| Z-Score | Confidence Interval | Area in Tails |
|---|---|---|
| ±1 | 68.27% | 31.73% |
| ±1.28 | 80.0% | 20.00% |
| ±1.645 | 90.0% | 10.00% |
| ±1.96 | 95.0% | 5.00% |
| ±2 | 95.45% | 4.55% |
| ±2.326 | 98.0% | 2.00% |
| ±2.576 | 99.0% | 1.00% |
| ±3 | 99.73% | 0.27% |
How to Use the Z-Score Calculator
Z-Score Formula and Critical Values
Z = (X - μ) / σ X = individual data point μ = population mean σ = population standard deviation Reverse (find X): X = μ + Z × σ
Critical z-scores for common confidence levels:
| Confidence Level | Z (two-tailed) | Z (one-tailed) | % within ±Z |
|---|---|---|---|
| 80% | ±1.282 | 1.282 | 80% |
| 90% | ±1.645 | 1.645 | 90% |
| 95% | ±1.960 | 1.960 | 95% |
| 99% | ±2.576 | 2.576 | 99% |
| 99.9% | ±3.291 | 3.291 | 99.9% |
The empirical rule: z between -1 and +1 covers 68.27% of normal data. Between -2 and +2 covers 95.45%. Between -3 and +3 covers 99.73%.
Frequently Asked Questions
A z-score (also called a standard score) measures how many standard deviations a data point is from the mean. Z = 0 means exactly at the mean. Z = +1 means 1 standard deviation above the mean (84th percentile). Z = -2 means 2 standard deviations below the mean (2.3rd percentile). Z-scores allow comparison across different scales: a z-score of 1.5 on a math test means the same distance from average as a z-score of 1.5 on a height measurement, even though the units differ.