Calculate z-scores, find raw scores from z-scores, and look up cumulative probabilities using the standard normal distribution.
| Formula | Expression |
|---|---|
| Z-score | z = (x − μ) / σ |
| Raw score | x = μ + z · σ |
| Percentile | P = Φ(z) × 100% |
| Property | Value |
|---|---|
| Raw score (x) | 72 |
| Mean (μ) | 60 |
| Std Dev (σ) | 10 |
| Z-score | 1.2 |
| Percentile (CDF) | 88.49th |
| % of data below x | 88.49% |
| % of data above x | 11.51% |
| Z-score | % Below (CDF) | Use |
|---|---|---|
| 0 | 50% | |
| 0.5 | 69.15% | |
| 1 | 84.13% | |
| 1.28 | 89.97% | |
| 1.5 | 93.32% | |
| 1.645 | 95% | 90% CI (one-tail) |
| 1.96 | 97.5% | 95% CI (two-tail) |
| 2 | 97.72% | |
| 2.326 | 99% | |
| 2.576 | 99.5% | 99% CI (two-tail) |
| 3 | 99.87% | |
| 3.5 | 99.98% |
Two students both score 72, yet the one in the class that averaged 65 with a spread of 4 outranks the one in the class that averaged 60 with a spread of 10: z = 1.75 against z = 1.2, roughly the 96th percentile against the 88th. The raw gap above the mean is smaller for the first student, 7 points against 12, but it is measured in much smaller units. That is the whole point of a z-score, and this calculator converts in three directions: raw score to z, z back to a raw score, and a percentile to both.
Start with the page defaults: mean μ = 60, standard deviation σ = 10 and a raw score x = 72.
First find how far the score sits from the average: 72 − 60 = 12 points above it.
Then express that distance in standard deviations. The formula is z = (x − μ) ÷ σ, so z = 12 ÷ 10 = 1.2. Run it backwards with x = μ + z × σ and you get 60 + 1.2 × 10 = 72.
Next comes the percentile. The page evaluates the standard normal cumulative distribution, Φ(z), with a polynomial approximation published by Abramowitz and Stegun, and Φ(1.2) = 0.8849. So 88.49% of values fall below 72 and 11.51% above it, assuming the data follow a bell curve.
The third tab runs the other way. Enter 95 as the percentile and the page returns z = 1.6449 and a raw score of 60 + 1.6449 × 10 = 76.4485. Z-scores and raw scores are rounded to four decimals, percentages to two.
The z-score itself is valid for any data, but the percentile comes from the normal curve. For skewed data such as incomes or waiting times it can be well off, and ranking the real values is more honest.
A standard deviation of zero would mean dividing by zero, and a 0th or 100th percentile corresponds to z of minus or plus infinity. The page returns no result in those cases instead of printing a meaningless number.
1.96 leaves 2.5% in each tail, which is a 95% two-sided interval. 1.645 leaves 5% in the upper tail, which is the 95th percentile or a 90% two-sided interval. The table labels that row 90% CI (one-tail), so read it as the 95th percentile.
Beyond z = 3 lies about 0.135% of a normal distribution, roughly 1 value in 740 on each side. In 100,000 records that is around 135 values above 3, so rare for one person, routine for a large dataset.