Z-Score Calculator
A z-score counts how many standard deviations a value is from the mean: a score of 85 with mean 70 and SD 10 has z = 1.5, higher than about 93 % of scores.
1.5
- P(Z < z), percentile
- 93.3193 %
- P(Z > z), right tail
- 6.68072 %
- P(−|z| < Z < |z|)
- 86.6386 %
Step-by-step solution
- z = (x − μ) ÷ σ
- z = (85 − 70) ÷ 10 = 1.5
- Area to the left from the standard normal table: Φ(1.5) = 0.933193
Z-Score Formula
z = (x − μ) ÷ σ
x value, μ mean, σ standard deviation
Worked Example
A test score of 85 when the mean is 70 and the standard deviation is 10:
- z = (x − μ) ÷ σ
- z = (85 − 70) ÷ 10 = 1.5
- Area to the left from the standard normal table: Φ(1.5) = 0.933193
P(Z > z), right tail: 6.68072 % · P(−|z| < Z < |z|): 86.6386 %
What a z-score tells you
A z-score is the number of standard deviations a value lies from the mean: z = (x − μ) ÷ σ. In a normal distribution about 68 % of values lie within ±1, 95 % within ±1.96 and 99.7 % within ±3. The probabilities shown assume the data are approximately normal; for skewed data a large z-score is less unusual than the table suggests. Use the population standard deviation when you know it and the sample standard deviation otherwise.
Frequently asked questions
How do I convert a z-score to a percentile?
Look up Φ(z) in the standard normal table (or use =NORM.S.DIST(z, TRUE) in Excel, pnorm(z) in R). z = 1.5 gives 0.9332, the 93.32nd percentile.
What does a negative z-score mean?
The value is below the mean. z = −1 is one standard deviation below, about the 16th percentile.
What z-score is unusual?
About 95 % of a normal distribution lies within z = ±1.96, and 99.7 % within ±3, so |z| > 2 or 3 is commonly treated as unusual.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology .
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Z-Score Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/z-score-calculator/
IEEE
Scientific Converters, "Z-Score Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/z-score-calculator/