Z-score Calculator

Raw scores are hard to compare when they come from different scales. A 78 on one exam and a 152 on another say little until you know how everyone else did. A z-score puts every value on the same yardstick by measuring how many standard deviations it sits above or below the mean. The default example, a score of 5 from a distribution with mean 3 and standard deviation 2, has a z-score of exactly 1, and about 84.13% of a normal population falls below it.

The converter goes both ways between z-scores and the five common kinds of normal probability: the left tail, the right tail, the area from the center, the area between −Z and Z, and the two tails combined. The last calculator gives the area between any two z-scores.

What is z-score?

A z-score, also called a standard score, is the signed distance between a value and the mean, measured in standard deviations. A positive z-score is above the mean, a negative one is below it, and zero is exactly average. Converting every value of a normal distribution to z-scores produces the standard normal distribution, which has mean 0 and standard deviation 1, so a single table or function can answer questions about any normal variable.

Suppose a class's test scores have a mean of 70 and a standard deviation of 8. A student who scored 82 has z = (82 − 70) ÷ 8 = 1.5. On another test with mean 60 and standard deviation 5, a score of 69 gives z = 1.8, so the second result is the stronger performance relative to the class, even though the raw number is lower.

Z-scores are used to find percentiles, to flag unusual values (anything beyond ±3 is rare in a normal population) and to build test statistics in hypothesis testing. Children's growth charts report height and weight as z-scores for age, and bone density scans report T-scores and Z-scores, which follow the same idea.

z = (x − μ) ÷ σ

x

the raw score

μ

the population mean

σ

the population standard deviation

For a sample, use the sample mean x̄ and sample standard deviation s instead.

Z-table

A z-table lists the cumulative probability Φ(z), the area under the standard normal curve to the left of z. To read it, find the row for the first decimal of z and the column for the second decimal. For z = 1.96, go to the row 1.9 and the column 0.06 to find 0.9750, which means 97.5% of values lie below 1.96 standard deviations above the mean.

The table below covers non-negative z-scores. For negative values, use the symmetry of the curve: Φ(−z) = 1 − Φ(z). So the area to the left of −1.96 is 1 − 0.9750 = 0.0250. The other probabilities follow from the same table. The right tail is 1 − Φ(z), the area from the center to z is Φ(z) − 0.5, the area between −z and z is 2Φ(z) − 1, and the two tails together are 2(1 − Φ(z)).

Printed tables come in several formats. Some list the area from 0 to z instead of the area to the left, and some list only the right tail, so always check the heading or picture before reading a value. A few rows worth remembering: Φ(1) ≈ 0.8413, Φ(1.645) ≈ 0.95, Φ(2) ≈ 0.9772 and Φ(3) ≈ 0.9987.

Frequently asked questions