Z Score Calculator
Calculate a z-score from a value, mean and standard deviation — with percentile, probability and interpretation from the standard normal distribution.
About the Z Score Calculator
A z-score (standard score) tells you how many standard deviations a value sits from the mean: z = (x − μ) ÷ σ. This z score calculator takes your raw value, the mean and the standard deviation, and instantly returns the z-score, a plain-language interpretation, and the percentile — the share of a normal population falling below your value — computed from the standard normal cumulative distribution.
For example, scoring 85 on a test with mean 70 and standard deviation 10 gives z = (85 − 70) ÷ 10 = 1.5, which corresponds to roughly the 93rd percentile: you outscored about 93% of test takers. Z-scores make different scales comparable — a z of +1.5 means the same relative standing whether the underlying test was out of 100 or 800 — which is why they underpin standardized testing, growth charts, quality control and finance.
Statistics students use z-scores for hypothesis tests and normal-probability questions; analysts use them to flag outliers (|z| > 2 or 3 is a common rule); and the built-in reference table of common z values makes textbook lookups unnecessary.
How to Use the Z Score Calculator
- 1Enter the raw value x you want to standardize.
- 2Enter the population mean and standard deviation.
- 3Read the z-score, interpretation and percentile instantly.
- 4Use the reference table to compare against common z values.
Frequently Asked Questions
How do I calculate a z-score with an example?
z = (value − mean) ÷ standard deviation. A test score of 85 with mean 70 and SD 10 gives z = (85 − 70) ÷ 10 = 1.5 — the score is 1.5 standard deviations above average, around the 93rd percentile of a normal distribution.
How do I convert a z-score to a percentile?
Use the standard normal CDF, Φ(z). Φ(1.5) ≈ 0.9332, so z = 1.5 is about the 93rd percentile. Landmarks worth memorising: z = 0 is the 50th percentile, z = 1 the 84th, z = 2 the 98th, and z = −1 the 16th. The calculator computes Φ(z) for any z automatically.
What does a negative z-score mean?
The value is below the mean. z = −0.8 on a test with mean 70 and SD 10 corresponds to a score of 62, at roughly the 21st percentile. Negative z-scores are not 'bad' by definition — for golf scores or delivery times, below the mean is better.
What z-score counts as an outlier?
Common rules flag |z| > 2 (outside ~95% of a normal population) as unusual and |z| > 3 (outside ~99.7%, per the 68-95-99.7 rule) as an outlier. A data point with z = 3.5 would occur by chance only about 2 times in 10,000 under normality — worth investigating.
What is the difference between a z-score and a t-score?
A z-score assumes you know the population standard deviation (or have a large sample); a t-score uses the sample's own standard deviation and the heavier-tailed t-distribution, which matters for small samples (roughly n < 30). For everyday standardization and percentile lookups, the z-score is the tool.