The Z-score Calculator converts a raw score into a z-score and finds normal distribution probabilities for a z-score or between two z-scores.
Z-score = 1
| Z-score | 1 |
| P(X < x) | 0.841345 |
| P(X > x) | 0.158655 |
| Percentile | 84.13th |
| P(x < Z) | 0.97725 |
| P(x > Z) | 0.02275 |
| P(0 to Z) | 0.47725 |
| P(−Z < x < Z) | 0.9545 |
| P(x < −Z or x > Z) | 0.0455 |
| P(Z₁ < x < Z₂) | 0.341345 |
| Outside the bounds | 0.658655 |
A z-score (standard score) tells you how many standard deviations a value lies above or below the mean: z = (x − μ) ÷ σ. Positive z is above the mean, negative below. Z-scores put different scales on a common footing, so a test score and a height can be compared.
Worked example (defaults): x = 5, μ = 3, σ = 2 gives z = 1. About 84.13% of a normal population lies below that value.
For normally distributed data, a z-score maps to a percentile through the standard normal distribution. z = 1.96 leaves 2.5% in each tail, which is why it is used for 95% confidence intervals. z = 2 has 97.72% below it.
Standardised testing, quality control (six sigma), finance (Altman z-score for bankruptcy risk), growth charts for children and outlier detection all use z-scores. Values beyond ±3 are rare in normal data.
Can a z-score be 0? Yes, when the value equals the mean.
Do z-scores need normal data? The calculation works for any data, but probability conversion assumes normality.
What is a t-score? A similar statistic used with small samples and unknown σ.