Z-Score Calculator
Calculate z-scores, percentiles, and areas under the standard normal curve. Free online z-score calculator.
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Z-Score
1.5000
Percentile
93.32%
Area Below
0.9332
Interpretation:1.50σ above the mean — 93.3th percentile
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Z-Score Reference Table
| Z-Score | Percentile | Interpretation |
|---|---|---|
| -3.0 | 0.13% | 3.0 standard deviations below the mean |
| -2.5 | 0.62% | 2.5 standard deviations below the mean |
| -2.0 | 2.28% | 2.0 standard deviations below the mean |
| -1.5 | 6.68% | 1.5 standard deviations below the mean |
| -1.0 | 15.87% | 1.0 standard deviations below the mean |
| -0.5 | 30.85% | 0.5 standard deviations below the mean |
| +0.0 | 50.00% | Exactly at the mean |
| +0.5 | 69.15% | 0.5 standard deviations above the mean |
| +1.0 | 84.13% | 1.0 standard deviations above the mean |
| +1.5 | 93.32% | 1.5 standard deviations above the mean |
| +2.0 | 97.72% | 2.0 standard deviations above the mean |
| +2.5 | 99.38% | 2.5 standard deviations above the mean |
| +3.0 | 99.87% | 3.0 standard deviations above the mean |
Understanding Z-Scores & Normal Distribution
A z-score tells you how far a data point is from the mean in terms of standard deviations. It transforms any normal distribution into the standard normal distribution, where the mean is 0 and the standard deviation is 1. This makes it possible to compare values from completely different datasets.
The Empirical Rule (68-95-99.7)
In a normal distribution, approximately 68% of data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three. This rule is why z-scores are so powerful: they immediately tell you how unusual a value is.
Practical Applications
- Education: Standardized test scores (SAT, IQ) are often reported as z-scores or scaled from them.
- Finance: Z-scores help measure how far a stock price deviates from its historical average.
- Medicine:Growth charts use z-scores to compare a child's measurements to population norms.
- Quality control: Manufacturing processes use z-scores to identify defects and outliers.
Frequently Asked Questions
A z-score (or standard score) measures how many standard deviations a data point is from the mean of a dataset. A z-score of 0 means the value is exactly average. Positive z-scores indicate values above the mean, while negative z-scores indicate values below the mean.