Z-Score Calculator · Practical guide
Calculate a Z Score from Mean and Standard Deviation
One subtraction, one division: z = (x − μ) / σ tells you how many standard deviations a value sits from the mean.
At a glance
The Formula: z = (x − μ) / σ
Given a data point x, a mean μ, and a standard deviation σ, the z-score is z = (x − μ) / σ. It answers one question: how many standard deviations is this value from the mean? A z of 0 is exactly average; +1 is one standard deviation above; −2 is two below.
Interactive tool
Z-Score Calculator
Enter your own values, then compare the result with the examples below.
Results
Worked Example: A Test Score
A class's exam has a mean of 75 and a standard deviation of 8. A student scores 91:
- Subtract the mean: 91 − 75 = 16
- Divide by the standard deviation: 16 / 8 = z = 2.00
- Interpret: the score is 2 standard deviations above the mean — the calculator labels anything above +2 "Far above average (outlier)."
The percentile comes from the standard normal curve: z = 2.00 corresponds to the 97.72nd percentile, meaning the student outscored about 97.7% of the class.
Sample vs. Population Standard Deviation
The formula is identical either way — what changes is which σ you enter. Use the population standard deviation (dividing by n) when your data is the whole group, and the sample standard deviation (dividing by n − 1) when it is a subset. Enter whichever you have in the Summary Stats tab; the math is the same.
Why Z-Scores Are Useful
Z-scores put different scales on equal footing. An SAT section score (mean 500, σ 100) of 620 and a class exam score of 91 (mean 75, σ 8) are both z = 1.2 — equally impressive relative to their groups. That comparability is why z-scores show up in grading, quality control, finance, and medical reference ranges.
Frequently Asked Questions
What if my standard deviation is from a sample?
Enter it directly — the z-score formula z = (x − μ) / σ works the same with a sample or population standard deviation. Just be consistent about which one you use.
What does a negative z score mean?
The value is below the mean. A z of −1.5 means the data point sits 1.5 standard deviations under the average; the sign shows direction, the size shows how unusual it is.
Can I calculate a z score without the mean?
No — you need both the mean and standard deviation. If you only have raw data, use the Full Dataset tab, which computes both for you automatically.
Why is my z score so large?
A large z means a small standard deviation or a value far from the mean. If |z| exceeds 3, double-check your inputs — such values occur in under 0.3% of a normal distribution.