Z Score Formula
Learn how to calculate z score values by hand with the z score formula, three fully worked examples, and a practice problem. Then check your answer with our free z score calculator.
What each symbol means
- x — the raw score. The single value you are evaluating: a test score, a measurement, a data point.
- μ — mu, the mean. The average of the whole group. Add every value, divide by how many there are.
- σ — sigma, the standard deviation. How spread out the values are around the mean. The standard deviation formula takes each value's distance from the mean, squares those distances, averages them, and takes the square root — the result is the typical deviation you plug into the z-score formula.
The formula does two things in order: the subtraction (x − μ) finds how far your value sits from average in raw units, and the division by σ rescales that distance into standard-deviation units. That rescaling is the whole trick — it is what lets you compare a test score out of 100 with a measurement in kilograms.
Why the formula works
The z-score formula does something clever in two moves. First, (x − μ) recenters your data: it shifts the whole distribution so the mean sits at zero. A score of 90 in a class averaging 80 becomes +10 — ten points above center, with the original units (points) still attached.
Then dividing by σ rescales everything by the group's spread. Ten points above average means something very different in a tight class (σ = 2) versus a scattered one (σ = 20): z = 5.00 in the first, z = 0.50 in the second. After the division, the units cancel out entirely — points divided by points — leaving a pure, unit-free number. That is why a z-score from a spelling test can be compared directly with a z-score from a marathon time: both live on the same standard scale, where 0 is average and ±1, ±2, ±3 mark the familiar bands of the bell curve.
The 5-step method
- Write down x, μ, and σ. Label them so you do not mix up the mean and the standard deviation mid-calculation.
- Subtract the mean: compute (x − μ). This is your raw distance from average — it can be negative.
- Divide by the standard deviation: (x − μ) / σ. The result is your z-score.
- Convert to a percentile (optional): look the z-score up in a z-table and multiply the left-tail probability by 100.
- Sanity-check: above-average values must give positive z; |z| beyond 3 deserves a second look; and a quick run through the z score calculator confirms the arithmetic.
Worked example 1 — an above-average test score
You scored 90 on a test. The class mean was 80, standard deviation 5.
Two standard deviations above the mean — a strong score, beaten by only about 2.28% of the class.
Worked example 2 — a below-average score (negatives are fine)
You scored 68. The class mean was 75, standard deviation 4.
The negative sign is correct — 68 is below the mean of 75. A z-score of −1.75 sits at the 4.01st percentile: about 96% of the class scored higher. Negatives are not errors; they are simply below-average values.
Worked example 3 — small decimals and real units
A newborn weighs 2.4 kg. The reference mean birth weight is 2.0 kg with a standard deviation of 0.5 kg. (Illustrative numbers for demonstration.)
The formula works the same with decimals and with any unit — kilograms, seconds, dollars. The units cancel out in the division, which is exactly why z-scores are unit-free.
Practice problem — try it yourself
An IQ-style test has a mean of 100 and a standard deviation of 15. You scored 115. What is your z-score, and what percentile is it? Work it out on paper, then reveal the answer.
One standard deviation above the mean — the 84.13th percentile. If you got z = 1.00, your method is solid.
Population vs sample standard deviation
The textbook z score formula uses the population standard deviation (σ) — the spread of the entire group. In real life you often have only a sample, so you estimate spread with the sample standard deviation (s), which divides by (n − 1) instead of n. For large samples the two are nearly identical; for small samples the difference matters, so always note which one you used in your working. (Strictly speaking, tiny samples call for t-scores rather than z-scores — but that is a topic for another page.)
Frequently asked questions
What is the formula for a z-score?
z = (x − μ) / σ. Subtract the mean (μ) from your value (x), then divide by the standard deviation (σ). The result tells you how many standard deviations the value sits from the mean — positive above it, negative below it, zero exactly at it.
How do you calculate a z-score by hand?
Write down x, μ, and σ; subtract the mean from your score; divide the result by the standard deviation. Example: (90 − 80) / 5 = 2.00. Then look up 2.00 in a z-table (0.9772) for the percentile. Five minutes with a calculator, no software needed.
What do x, μ, and σ mean in the z-score formula?
x is your raw score — the value being evaluated. μ (mu) is the population mean, the average of the whole group. σ (sigma) is the standard deviation, measuring how spread out the values are. Together they locate any single value on the standard scale.
Do you use population or sample standard deviation?
The textbook formula uses the population standard deviation (σ). In practice you often only have a sample, so you estimate with the sample standard deviation (s) — just say which one you used. For large samples the difference is negligible; for small samples it matters.
How do you check a hand-calculated z-score?
Verify the sign (above-average values must give positive z), the rough size (|z| beyond 3 deserves a double-check), and the percentile conversion in a table or with the calculator on our homepage. Most hand errors come from subtracting in the wrong order or dividing by variance instead of standard deviation.