Find a z-score, its percentile and normal-curve areas.
z = (x − μ) ÷ σ. A score of 85 with mean 70 and standard deviation 10 gives z = 1.5 — higher than about 93% of values.
A z-score tells you how many standard deviations a value is from the mean. Enter a raw score, the mean and the standard deviation to get the z-score, its percentile (the area to the left), the right-tail and two-tailed probabilities. Two more tools turn a z-score back into a raw score, and find the probability between any two z-values.
Z-scores let you compare values from different distributions. They feed directly into the P-Value Calculator, Confidence Interval Calculator and Sample Size Calculator. To get the mean and standard deviation use the Standard Deviation Calculator.
The calculator applies the standard method step by step.
| Item | Value |
|---|---|
| Z-score | 1.5 |
| Percentile (area to the left) | 0.9331928 (93.31928%) |
| Area to the right | 0.066807201 |
| Two-tailed probability | 0.1336144 |
| Distance from the mean | 1.5 standard deviations above |
| Within ±1 SD / ±2 SD / ±3 SD of the mean | 68.27% / 95.45% / 99.73% of values |
| Raw score | Z-score |
|---|---|
| x = 55 (z = −1.5) | -1.5 |
| x = 70 (z = 0) | 0 |
| x = 85 (z = 1.5) | 1.5 |
| x = 90 (z = 2) | 2 |
| x = 100 (z = 3) | 3 |
| z | Area to the left Φ(z) | Area to the right | Two-tailed p |
|---|---|---|---|
| 0 | 0.5000 | 0.5000 | 1.0000 |
| 0.5 | 0.6915 | 0.3085 | 0.6171 |
| 1 | 0.8413 | 0.1587 | 0.3173 |
| 1.5 | 0.9332 | 0.0668 | 0.1336 |
| 1.645 | 0.9500 | 0.0500 | 0.1000 |
| 1.96 | 0.9750 | 0.0250 | 0.0500 |
| 2 | 0.9772 | 0.0228 | 0.0455 |
| 2.326 | 0.9900 | 0.0100 | 0.0200 |
| 2.576 | 0.9950 | 0.0050 | 0.0100 |
| 3 | 0.9987 | 0.0013 | 0.0027 |
The number of standard deviations a value lies above (positive) or below (negative) the mean.
z = (x − mean) ÷ standard deviation. For x = 85, mean 70 and SD 10, z = 1.5.
The value is 2 standard deviations above the mean — higher than about 97.7% of values.
Look up the area to the left of z in a z-table, or use this calculator: z = 1.5 is the 93.3rd percentile.
About 68% of values lie within 1 SD of the mean, 95% within 2 SD and 99.7% within 3 SD.
Yes. A negative z-score means the value is below the mean.
It uses the standard method: z = (x − μ) ÷ σ x = μ + zσ percentile = Φ(z) The full working and a worked example are shown on this page.
Decimal results use standard double-precision arithmetic (about 15–16 significant digits). Tools that work with whole numbers — big numbers, factors, bases, combinations — use exact integer arithmetic, so no digits are lost.
Yes for checking your work and learning the method. The steps and formula are shown so you can understand the answer, but follow your teacher's rules about calculators in exams.
No. Everything is calculated in your browser and nothing you enter is uploaded or saved.
