Z-Score Calculator

How to use this calculator

Enter a raw score, the mean and the standard deviation to standardize it. The calculator returns the z-score, the percentile (area under the standard normal curve to the left of z) and the two-tailed p-value used in hypothesis tests. Every field except the p-value can also be an input: enter a percentile with the mean and SD to find the cut-off score, or a raw score with its z-score and the mean to back out the standard deviation.

The p-value is calculated from z but not the other way round, because a two-tailed p-value corresponds to both +z and −z. To go from a p-value to z, enter the corresponding percentile instead (for p = 0.05 two-tailed, the 97.5th percentile gives z = 1.96).

The formula

z = (x − μ) / σ

P(Z < z) = Φ(z)

p = 2 × Φ(−|z|)

x: raw score
μ: population mean
σ: standard deviation
z: z-score
P(Z<z): percentile (area to the left)
p: two-tailed p-value

Worked example

A student scores 85 on an exam with a mean of 70 and a standard deviation of 10.

z = (85 − 70) / 10 = 1.5.
Percentile = Φ(1.5) = 93.3%.
Two-tailed p = 2 × (1 − 0.9332) = 0.134.

The score is 1.5 standard deviations above the mean, better than about 93% of test-takers if scores are normally distributed.

Assumptions and limits

The data are approximately normally distributed. For skewed data, percentiles from a z-score can be badly off.
μ and σ are the population values; with a small sample, a t-distribution is more appropriate for tests.

Frequently asked questions

What is a good z-score?

It depends on context. |z| above 2 is unusual (about 5% of values), and above 3 is rare (0.3%) in a normal distribution.

Can a z-score be negative?

Yes. A negative z-score means the value is below the mean; z = −1 is at about the 16th percentile.

How is Φ(z) calculated?

Φ(z) = ½ erfc(−z/√2). SolveKit evaluates the complementary error function with W. J. Cody’s rational approximations, accurate to about 15 significant digits.