Z-Score Calculator

A z-score (also called a standard score) tells you how many standard deviations a value is from the mean. Zero is exactly average, positive is above and negative is below. Because every normal (bell-shaped) distribution looks the same once it is measured in z-scores, one table of areas works for test marks, heights, IQ scores, machine parts or anything else that follows a bell curve.

The formula

z = (value − mean) ÷ standard deviation
value = mean + z × standard deviation
area below z = Φ(z), the normal cumulative probability
area above z = 1 − Φ(z)
area between −z and z = 1 − 2 × (1 − Φ(|z|))
area between a and b = Φ(z of b) − Φ(z of a)
two-tailed p-value = 2 × (1 − Φ(|z|))
z-test: z = (sample mean − μ₀) ÷ (σ ÷ √n)

Worked example

A test has a mean of 70 and a standard deviation of 10, and you scored 85. The gap from the mean is 85 − 70 = 15, so z = 15 ÷ 10 = 1.5: your score is one and a half standard deviations above average.

The area under the bell curve to the left of z = 1.5 is Φ(1.5) = 0.93319. In the z-table, that is row 1.5, column 0.00 (0.9332). So you scored higher than 93.32% of people, about the 93rd percentile. The area above is 1 − 0.93319 = 0.06681 (6.68%).

To six places, Φ(1.5) = 0.933193 and the area above is 0.066807, so the area between −1.5 and 1.5 is 0.933193 − 0.066807 = 0.866386 (86.64%), and the two tails outside it add up to 2 × 0.066807 = 0.133614 (13.36%). That is also the two-tailed p-value for z = 1.5, which is above 0.05, so it would not count as significant.

Going the other way, the 95th percentile is z = 1.6449, which is a score of 70 + 1.6449 × 10 = 86.449. The chance of a score between 60 and 85 is Φ(1.5) − Φ(−1) = 0.933193 − 0.158655 = 0.774538, or 77.45%, because 60 is z = (60 − 70) ÷ 10 = −1.

For a z-test, a sample of 36 with a mean of 103, tested against μ₀ = 100 with σ = 15, has a standard error of 15 ÷ √36 = 15 ÷ 6 = 2.5, so z = (103 − 100) ÷ 2.5 = 1.2 and the two-tailed p-value is 2 × 0.11507 = 0.23014. That is above α = 0.05, so the difference is not significant.

Common z-scores

The 68–95–99.7 rule

On a bell curve, 68.27% of values are within 1 standard deviation of the mean, 95.45% within 2 and 99.73% within 3. So a z-score beyond ±2 happens about 1 time in 22, and beyond ±3 about 1 time in 370. That is why |z| above 2 is often called unusual and above 3 a possible outlier.

How to read a z-table

Most z-tables give the area to the left of z. Find the row for the first decimal of z (1.5) and the column for the second (0.00 for 1.50, 0.06 for 1.96), and read the area. For negative z, use the negative table or work out 1 − the area for the positive z. This calculator works the area out exactly instead of rounding to 4 decimal places, and shows the slice of the table around your z for checking.

Sample or population SD for a data set

When you paste a data set, the z-scores use the data's own mean and standard deviation. Pick Population when the numbers are every value you care about (divide by n), or Sample when they are part of a bigger group (divide by n − 1, like Excel STDEV.S). Spreadsheets give the same z-scores with STANDARDIZE(x, mean, sd), the area with NORM.S.DIST(z, TRUE) and the inverse with NORM.S.INV(p).

When z-scores and these areas apply

• A z-score can be worked out for any data, but the percentages only hold when the data is roughly bell-shaped (normal). For skewed data, such as incomes, use the real percentile rank instead.
• The z-test needs the population standard deviation σ. If you only have the sample SD and a small sample (under about 30), a t-test is the better choice.
• The area is worked out with an error-function series and continued fraction accurate to about 15 decimal places, and the inverse uses Acklam's method polished with Halley's method, so very small tails such as z = 6 (p ≈ 1.97 × 10⁻⁹) stay accurate.