Z-Score Calculator
A z-score places a value inside its distribution with one number: how many standard deviations it lies above or below the mean. Because the units cancel, a z-score lets you compare an exam mark with a waiting time, or this month’s sales with last year’s. On a normal curve it also gives the share of cases below and above the value.
The calculator opens on a score of 82 against a mean of 70 and a standard deviation of 8. Enter one data point or a list of them, a sample mean with its size, or a pasted data sample. Each answer shows the working, the areas below and above and a shaded normal curve. The fourth option runs the other way and turns a z-score back into a value.
How it works
Sample mean: z = (x̄ − μ) / (σ / √n)
Back to a value: x = μ + z × σ
Area below z = Φ(z), the standard normal cumulative probability. Area above = 1 − Φ(z)
Here x is a value, μ the mean, σ the standard deviation, x̄ a sample mean and n the sample size. Use the population mean and standard deviation when you know them, or the sample’s own mean and standard deviation to place a value inside that sample.
A worked example. A score of 82 with μ = 70 and σ = 8 gives z = (82 − 70) / 8 = 1.5. The score lies one and a half standard deviations above the mean, with 93.32% of a normal distribution below it and 6.68% above it. A sample mean divides by the standard error, so the same 3-point gap counts for more: a mean of 73 from 25 values gives z = 3 / (8 / √25) = 1.875.
The areas assume a normal distribution. For a skewed variable the z-score still measures distance from the mean, and the percentages drift from the truth. The bands shown, under 2 typical, 2 to 3 unusual and 3 or more extreme, follow the empirical rule: about 95% of a normal distribution lies within 2 standard deviations of the mean and about 99.7% within 3. The areas agree with standard normal tables and with R’s pnorm().
Frequently Asked Questions
What is a z-score?
A z-score states how many standard deviations a value lies from the mean. A z-score of 1.5 is one and a half standard deviations above the mean, and a z-score of 0 is the mean itself. It has no units, so values on different scales can be compared.
How do you calculate a z-score?
Subtract the mean from the value and divide by the standard deviation: z = (x − μ) / σ. For a score of 82 with a mean of 70 and a standard deviation of 8, z = (82 − 70) / 8 = 1.5.
How do you calculate the z-score of a sample mean?
Divide by the standard error in place of the standard deviation: z = (x̄ − μ) / (σ / √n). A sample mean of 73 from 25 values, with μ = 70 and σ = 8, gives z = 3 / 1.6 = 1.875.
What does the area below a z-score mean?
It is the share of a normal distribution that lies below that value, which is also its percentile. A z-score of 1.5 has 93.32% of the curve below it and 6.68% above it.
What z-score counts as unusual?
A common convention treats values within 2 standard deviations of the mean as typical, values 2 to 3 away as unusual and values 3 or more away as extreme. About 95% of a normal distribution lies within 2 standard deviations and about 99.7% within 3.
How do I turn a z-score back into a value?
Multiply the z-score by the standard deviation and add the mean: x = μ + z × σ. With a mean of 70 and a standard deviation of 8, a z-score of 1.5 is a value of 82.