Z-Score Calculator
Calculate standard scores, find probabilities, and interpret statistical data
Z-Score Formulas
What is a Z-Score?
A z-score (also called standard score) tells you how many standard deviations a value is from the mean. It standardizes different datasets to a common scale, allowing meaningful comparisons. A z-score of 0 means the value equals the mean, positive scores are above average, and negative scores are below.
Z-scores are fundamental in statistics for determining how unusual or typical a value is. In a normal distribution, about 68% of values have z-scores between -1 and +1, 95% between -2 and +2, and 99.7% between -3 and +3.
Z-scores enable comparison across different scales. For example, you can compare a student's performance on different tests, or compare measurements from different populations, by converting all scores to the same standardized scale.
Z-Score Interpretation Guide
z = 0
Value equals the mean. Exactly average.
z = +1 to +2
Above average. Top 16% to 2.5% of distribution.
z > +2
Significantly above average. Unusual or exceptional.
z < -2
Significantly below average. Potentially concerning.
Common Z-Score Values
Reference table for z-scores and their corresponding percentiles:
| Z-Score | Percentile | Interpretation | 1 in X |
|---|---|---|---|
| -3.0 | 0.13% | Extremely low | 1 in 740 |
| -2.0 | 2.28% | Very low | 1 in 44 |
| -1.0 | 15.87% | Below average | 1 in 6 |
| 0.0 | 50.00% | Average | 1 in 2 |
| +1.0 | 84.13% | Above average | Top 16% |
| +2.0 | 97.72% | Very high | Top 2.3% |
| +3.0 | 99.87% | Extremely high | Top 0.13% |
Applications of Z-Scores
Academic Testing
Compare student performance across different tests or years. SAT, GRE, and IQ tests are standardized to specific z-score scales.
Quality Control
Monitor manufacturing processes. Values beyond ±3 standard deviations often trigger investigation or rejection.
Finance
Altman Z-Score predicts bankruptcy. Stock analysis uses z-scores to identify unusual price movements.
Medical Research
Compare patient measurements to population norms. Growth charts, blood tests, and vital signs use z-score interpretations.
Frequently Asked Questions
What z-score is considered significant?
In hypothesis testing, z-scores beyond ±1.96 (for 95% confidence) or ±2.58 (for 99% confidence) are considered statistically significant. Values beyond ±3 are usually considered outliers.
Can z-scores be greater than 3?
Yes, though rarely. Z-scores can theoretically be any value. Scores beyond ±3 occur in about 0.3% of normally distributed data, suggesting either an outlier or non-normal distribution.
How do I interpret a negative z-score?
A negative z-score means the value is below the mean. Z = -1.5 means the value is 1.5 standard deviations below average. It's not necessarily bad - context matters.
What's the difference between z-score and t-score?
Z-scores require known population parameters and large samples. T-scores use sample estimates and account for additional uncertainty with small samples (typically n < 30).
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