Calculate z-scores, probabilities, and percentiles for the standard normal distribution. Convert between raw scores and z-scores.
Z-Score
-1.0000
15.87th percentile
Z-Score
-1.0000
15.87th percentile
Z-Score
-1.0000
Raw Score
85.0000
Percentile
15.87%
P(X < x)
15.8655%
z = (X - μ) / σ
z = (85.00 - 100) / 15
z = -15.0000 / 15
z = -1.0000
| Z-Score | P(Z < z) | Percentile | Interpretation |
|---|---|---|---|
| -3 | 0.13% | 0.1th | Very low (rare) |
| -2 | 2.28% | 2.3th | Low (unusual) |
| -1 | 15.87% | 15.9th | Below average |
| 0 | 50.00% | 50.0th | Average (mean) |
| 1 | 84.13% | 84.1th | Above average |
| 1.645 | 95.00% | 95.0th | 90% one-tail |
| 1.96 | 97.50% | 97.5th | 95% one-tail |
| 2 | 97.72% | 97.7th | High (unusual) |
| 2.576 | 99.50% | 99.5th | 99% one-tail |
| 3 | 99.87% | 99.9th | Very high (rare) |
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A z-score measures how many standard deviations a value is from the mean. Formula: z = (x - mean) / standard deviation. A z-score of 0 means the value equals the mean. Positive z-scores are above the mean; negative are below. About 68% of data falls within z = +/-1, 95% within z = +/-2, 99.7% within z = +/-3.
A z-score (or standard score) indicates how many standard deviations an observation is from the mean of a distribution. Z-scores standardize data to a common scale with mean 0 and standard deviation 1, enabling comparison across different datasets. They are fundamental to statistical inference, hypothesis testing, and probability calculations.
A z-score measures how many standard deviations a value is from the mean. Formula: z = (x - mean) / standard deviation. A z-score of 0 means the value equals the mean. Positive z-scores are above the mean; negative are below. About 68% of data falls within z = +/-1, 95% within z = +/-2, 99.7% within z = +/-3.
A z-score (standard score) measures how many standard deviations a value is from the mean. Z = (X - μ) / σ. A z-score of 0 is the mean, +1 is one standard deviation above, -2 is two standard deviations below.
Z-scores indicate relative position: |z| < 1 is typical (68% of data), |z| < 2 is common (95%), |z| > 2 is unusual (5%), |z| > 3 is rare (0.3%). Positive z = above average, negative z = below average.
The standard normal distribution has mean = 0 and standard deviation = 1. Any normal distribution can be converted to standard normal using z-scores. The total area under the curve equals 1 (100%).
Use the standard normal table or calculator. P(Z < z) gives the area to the left. For P(Z > z), subtract from 1. For P(a < Z < b), calculate P(Z < b) - P(Z < a).
For 90% CI: z = 1.645. For 95% CI: z = 1.96. For 99% CI: z = 2.576. These represent the z-scores that capture the middle percentage of the distribution.
Last updated: 2025-01-15
Z-Score
-1.0000
15.87th percentile