Calculate Z-scores, percentiles, and probabilities from the standard normal distribution.
Select a calculation mode and enter your values to see results.
A Z-score (standard score) measures how many standard deviations a data point is from the mean. It standardizes values from different distributions onto a common scale, making comparisons possible.
Formula: Z = (x - μ) / σ
Standardization: Compare values from different distributions (e.g., SAT vs ACT scores)
Outlier Detection: Identify unusual values (typically |Z| > 3)
Probability: Calculate probabilities using the standard normal distribution
Hypothesis Testing: Foundation for many statistical tests
For normal distributions:
68% of data falls within ±1 standard deviation (|Z| ≤ 1)
95% of data falls within ±2 standard deviations (|Z| ≤ 2)
99.7% of data falls within ±3 standard deviations (|Z| ≤ 3)
| Z-Score | Percentile | Interpretation |
|---|---|---|
| -3.0 | 0.13% | Extremely low |
| -2.0 | 2.28% | Significantly low |
| -1.0 | 15.87% | Below average |
| 0.0 | 50% | Mean/Average |
| 1.0 | 84.13% | Above average |
| 1.645 | 95% | 90% CI critical value |
| 1.96 | 97.5% | 95% CI critical value |
| 2.0 | 97.72% | Significantly high |
| 2.576 | 99.5% | 99% CI critical value |
| 3.0 | 99.87% | Extremely high |
Value is above the mean
Example: Z = 1.5 means the score is 1.5 standard deviations above average
Value is below the mean
Example: Z = -1.5 means the score is 1.5 standard deviations below average
Value equals the mean
Example: Z = 0 means the score is exactly average
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A Z‑score (standard score) tells how many standard deviations a value is above or below the mean: Z = (x − μ) ÷ σ. Positive Z means above average; negative Z means below.
Enter the raw score, mean (μ), and standard deviation (σ), then click Calculate under the “Score → Z” tab. The tool also shows the percentile and tail probabilities.
Use Z‑scores to compare values from the same normal (or approximately normal) distribution, find tail probabilities, or convert between scores and percentiles.
Left tail is P(Z ≤ z), right tail is P(Z ≥ z), and two‑tail often refers to the probability outside ±|z|. Choose the area type that matches your hypothesis or interpretation.