Z-Score Calculator
Convert raw score x into standard deviations z = (x - μ) / σ.
- P(X < x), the percentile
- 0.841345 (84.13th percentile)
- P(X > x)
- 0.158655
- P(−|z| < Z < |z|)
- 0.682689
- Two-tailed p-value
- 0.317311
Making different scales comparable
A z-score says how many standard deviations a value sits from the mean:z = (x − μ) / σ. A z of 0 is exactly average, +2 is two standard deviations above, −1.5 is one and a half below. Because the units cancel, a z-score lets you compare measurements from completely different scales — a test score against a height, a rainfall figure against a share price.
That is the main practical use. Scoring 85 on a test with mean 75 and standard deviation 5 gives z = 2; scoring 90 where the mean is 88 and the standard deviation is 1 gives z = 2 as well. The raw scores differ but the relative standing is identical. It is also the basis of outlier detection, where |z| above 3 is a common flag.
Converting a z-score to a percentile requires the normal distribution, and this is where care is needed: the percentile interpretation only holds if the data is roughly normal. The z-score itself is always computable, but reading "z = 2 means the 97.7th percentile" off a skewed distribution is simply wrong. Check the shape of the data before trusting the percentile.
Frequently Asked Questions
What is a Z-score?
A Z-score measures how many standard deviations a raw score x is above or below the population mean μ.
What is the Z-score formula?
z = (x - μ) / σ, where x is raw score, μ is mean, and σ is standard deviation.
What is a z-score?
How many standard deviations a value sits from the mean: z = (x - mean) / standard deviation. A z of +2 is two standard deviations above average. Because it is unitless, it lets you compare observations from completely different distributions.
What z-score counts as unusual?
Conventionally, beyond +/-2 is uncommon (about 5% of a normal distribution) and beyond +/-3 is rare (about 0.3%). The 1.96 value is the exact cut-off for a two-tailed 95% interval, which is where the familiar 95% confidence interval comes from.