Z-Score Calculator: Convert a Value to z (and Back) with Normal Probabilities
Turn a raw score into a z-score (or back) and see the matching areas under the standard normal curve, with a shaded chart.
- Nothing you type is stored
- Free
- No signup
z-score
1.5
- P(Z < z), area to the left
- 0.933193
- P(Z > z), area to the right
- 0.066807
- P(−|z| < Z < |z|)
- 0.866386
- Percentile
- 93.32
Areas assume a normal (bell-shaped) distribution.
Show working
- z = (x − μ) / σ = (85 − 70) / 10 = 1.5
- P(Z < 1.5) = Φ(1.5) = 0.933193
- P(Z > 1.5) = 1 − Φ(z) = 0.066807
- P(−1.5 < Z < 1.5) = 2Φ(|z|) − 1 = 0.866386
How to use
- Choose value → z, z → value, or area for a z.
- Enter the value (or z), the mean and the standard deviation.
- Read z, the percentile and the areas.
Good to know
z = (x − μ)/σ tells how many standard deviations a value is above (+) or below (−) the mean. A mark of 85 in a test with mean 70 and SD 10 has z = 1.5.
Areas come from the standard normal cumulative function Φ(z), computed to about 15 significant digits (not a printed table). For example Φ(1.96) = 0.975002, so 95% of values lie within ±1.96.
Percentile = Φ(z) × 100: z = 1.5 is about the 93rd percentile. These areas assume the data is roughly normal (bell-shaped); for skewed data they are only a guide.
Questions
What is a good z-score?
It depends on the context. z = 0 is exactly average, z = 1 is better than about 84% and z = 2 better than about 97.7% of a normal population.
What is the area for z = 1.96?
P(Z < 1.96) = 0.975, P(Z > 1.96) = 0.025, and P(−1.96 < Z < 1.96) = 0.95.
Can a z-score be negative?
Yes. A negative z means the value is below the mean.
How do I get the value for a given z?
x = μ + zσ. Choose “z → value” and enter z, the mean and the SD.
Is my data stored or sent anywhere?
No. The calculation runs entirely in your browser; nothing you type leaves your device.