A z-score, or standard score, rescales a value into units of standard deviations from the mean. That simple change lets you compare an SAT score with an ACT score, judge whether a measurement is unusual, and look up probabilities on the standard normal curve. This calculator does the conversion in either direction and replaces a printed z-table with exact tail areas and a shaded graph.
How to use the z-score calculator
- Choose where to start: a raw score with its mean and standard deviation, a z-score you already have, or a percentile.
- Fill in the fields that appear.
- Pick the probability to highlight: left tail, right tail, two tails or the middle area. The tape lists all four either way.
- The normal curve below shades the selected area and marks your score.
Z-score formula
Here x is the raw score, μ the mean and σ the standard deviation. To go back from a z-score to a raw score:
Probabilities come from the standard normal cumulative distribution Φ(z), the area to the left of z:
Worked example
A student scores 85 on a test where the mean is 70 and the standard deviation is 10.
- Z-score: (85 − 70) ÷ 10 = 1.5. The score is one and a half standard deviations above average.
- Left tail: Φ(1.5) ≈ 0.933193, so about 93.3% of scores are lower — roughly the 93rd percentile.
- Right tail: 1 − 0.933193 ≈ 0.066807; about 6.7% of students scored higher.
- Two-tailed: 2 × 0.066807 ≈ 0.133614.
Working backward, the 95th percentile of the same test is z ≈ 1.6449, which is 70 + 1.6449 × 10 ≈ 86.45 points.
Common z-values
| z | Left-tail area Φ(z) | Use |
|---|---|---|
| 1.0 | 0.8413 | One standard deviation above the mean |
| 1.282 | 0.9000 | 90th percentile |
| 1.645 | 0.9500 | 90% two-sided / 95% one-sided critical value |
| 1.960 | 0.9750 | 95% confidence interval, two-tailed α = 0.05 |
| 2.326 | 0.9900 | 99th percentile |
| 2.576 | 0.9950 | 99% confidence interval |
| 3.0 | 0.99865 | Three-sigma limit in quality control |
By symmetry, Φ(−z) = 1 − Φ(z), so negative z-scores mirror these values.
Using z-scores well
Comparing across scales
Suppose one applicant scored 1,300 on a test with mean 1,050 and SD 200 (z = 1.25), and another scored 28 on a test with mean 21 and SD 5 (z = 1.4). The second result is relatively stronger even though the numbers look very different. Real admissions comparisons use published concordance tables, but the z-score captures the idea.
Hypothesis tests
In a z-test the test statistic is a z-score, and the p-value is the tail area beyond it. A two-tailed p-value below 0.05 corresponds to |z| > 1.96.
Check the normality assumption
Tail probabilities are only as good as the normal model behind them. For strongly skewed data, such as incomes or wait times, a z-score still measures distance from the mean, but its percentile can be badly off. In that case use the percentile calculator on the actual data. For probabilities between two raw values, or to find a cutoff for a given probability, use the normal distribution calculator.
Frequently asked questions
What does a z-score tell you?
How many standard deviations a value is from the mean. A z-score of 1.5 means the value is 1.5 standard deviations above average; −2 means two standard deviations below. It lets you compare values measured on different scales.
What is a good or unusual z-score?
For roughly normal data, about 95% of values have z-scores between −1.96 and 1.96, and 99.7% fall between −3 and 3. Values beyond ±2 are often called unusual and beyond ±3 rare.
How do I get a p-value from a z-score?
For a one-tailed test, the p-value is the tail area beyond z: P(Z > z) for an upper-tail test or P(Z < z) for a lower-tail test. For a two-tailed test, double the smaller tail: z = 1.96 gives p ≈ 0.05.
Can I use the sample standard deviation?
Yes, as an estimate of σ when the sample is reasonably large. For small samples drawn from a population with unknown σ, test statistics follow a t distribution rather than the normal, so normal probabilities are only approximate.
How accurate are the probabilities?
The calculator evaluates the normal cumulative distribution with a double-precision algorithm, accurate to around 15 digits, so it agrees with printed z-tables to every digit they show and handles far tails that tables omit.