Calculate standard score (z-value), normal distribution p-values, percentiles, and outlier detection.
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z = (x - μ) / σzxμσ“A student scores 85 on a standardized test where the national average is 70 with a standard deviation of 10.”
Subtract the population mean from the individual score.
Divide the raw difference by the standard deviation.
Under standard normal distribution, 93.32% of participants scored below 85.
The student scored 1.50 standard deviations above the average, outperforming 93.3% of test takers.
| Z-Score Range | Percentage of Distribution | Two-Tailed Probability | Designation |
|---|---|---|---|
| |z| < 1.0 | 68.27% | 31.73% outside | Within 1 standard deviation |
| |z| < 1.96 | 95.00% | 5.00% outside | Standard 95% confidence boundary |
| |z| < 2.0 | 95.45% | 4.55% outside | Within 2 standard deviations |
| |z| < 2.58 | 99.00% | 1.00% outside | 99% confidence boundary |
| |z| < 3.0 | 99.73% | 0.27% outside | Within 3 standard deviations (Statistical outlier) |
Provide your numeric inputs into the designated fields in Z Score Calculator.
Observe real-time recalculations as values update without requiring a manual submit button.
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Yes. A negative z-score indicates that the data point is below the distribution mean. For example, a z-score of -2.0 means the value is two standard deviations below the average.
In standard practice, any observation with an absolute z-score |z| > 3.0 is considered a significant outlier, occurring in fewer than 0.27% of cases in a normal distribution.
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