🔔 Z-Score Calculator
Turn a raw score into a z-score and percentile, look up normal-distribution probabilities, or work backwards from a percentile to the value.
Quick answer: A score of 85 with mean 70 and standard deviation 10 has z = (85 − 70) ÷ 10 = 1.5, which is the 93.32nd percentile: 93.32% of values fall below it and 6.68% above.
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Z-Score Calculator inputs
Result
z-score of 85
z = 1.5
1.5 standard deviations above the mean · 93.32nd percentile
- Percentile
- 93.32
- P(Z < z) left tail
- 0.933193
- P(Z > z) right tail
- 0.066807
| Two-tailed P(|Z| > |z|) | 0.133614 |
| P(−1.5 < Z < 1.5) | 0.866386 |
| P(0 < Z < |z|) | 0.433193 |
| Share below / above | 93.32% / 6.68% |
Step by step
- z = (x − μ) ÷ σ = (85 − 70) ÷ 10 = 15 ÷ 10 = 1.5
- Left tail: P(Z < 1.5) = Φ(1.5) = 0.933193 → 93.32nd percentile
- Right tail: P(Z > 1.5) = 1 − 0.933193 = 0.066807
- Two-tailed: 2 × P(Z > 1.5) = 0.133614
Common z-scores
| z | Percentile (left tail) | Two-tailed p | Within ±z |
|---|---|---|---|
| 0 | 50% | 1 | 0% |
| 0.5 | 69.15% | 0.617075 | 38.29% |
| 1 | 84.13% | 0.317311 | 68.27% |
| 1.282 | 90% | 0.200001 | 80% |
| 1.5 | 93.32% | 0.133614 | 86.64% |
| 1.645 | 95% | 0.100001 | 90% |
| 1.96 | 97.5% | 0.05 | 95% |
| 2 | 97.72% | 0.0455 | 95.45% |
| 2.326 | 99% | 0.02 | 98% |
| 2.576 | 99.5% | 0.01 | 99% |
| 3 | 99.87% | 0.0027 | 99.73% |
The z-score formula
A z-score (standard score) measures the distance between a value and the mean in units of standard deviation.
z = (x − μ) ÷ σ · x = μ + z·σ
Here x is the value, μ the mean and σ the standard deviation. Converting to z lets you compare values from different scales – an exam score and a height, say – on one common scale.
Worked example
A test has a mean of 70 and a standard deviation of 10. You scored 85.
- z = (85 − 70) ÷ 10 = 1.5.
- Area to the left: Φ(1.5) = 0.9332, so you are at the 93.32nd percentile.
- Area to the right: 1 − 0.9332 = 0.0668, so 6.68% scored higher.
Working backwards: the 95th percentile is z = 1.6449, so x = 70 + 1.6449 × 10 ≈ 86.45.
The empirical rule
| Range | Share of values |
|---|---|
| μ ± 1σ | 68.27% |
| μ ± 1.96σ | 95% |
| μ ± 2σ | 95.45% |
| μ ± 3σ | 99.73% |
How to use the calculator
Choose what you want to find. Enter a value with the mean and standard deviation to get its z-score and percentile, enter a percentile to find the matching value, or enter a z-score directly to read off the probabilities.
Which probability do I need?
- Left tail P(Z < z): the percentile – share of values below.
- Right tail P(Z > z): share of values above – a one-tailed p-value for a “greater than” test.
- Two-tailed: chance of being at least this far from the mean in either direction – the p-value for a two-sided z-test.
The probabilities assume the data is normally distributed. For strongly skewed data, percentiles from a z-score can be misleading.
Frequently asked questions
What does a z-score tell you?
How many standard deviations a value lies from the mean. z = 0 is exactly average, z = 1 is one standard deviation above, and z = −2 is two below. In a normal distribution about 68% of values have |z| < 1, 95% have |z| < 1.96 and 99.7% have |z| < 3.
How do I convert a z-score to a percentile?
The percentile is the area under the standard normal curve to the left of z, Φ(z), times 100. z = 1.5 gives Φ(1.5) = 0.9332, the 93.32nd percentile. The calculator computes Φ precisely, so you don’t need a z-table.
Can a z-score be negative?
Yes – a negative z-score means the value is below the mean. z = −1.5 is the 6.68th percentile, the mirror image of z = 1.5.
What is a “good” z-score?
It depends on context. In testing, z above 1 (top ~16%) is above average and above 2 (top ~2.3%) is exceptional. In quality control or statistics, |z| above 1.96 or 3 is often treated as unusual or an outlier.