What Is a Z-Score and How to Calculate It (With a Worked Example)

A z-score is a number that tells you how many standard deviations a value sits above or below the mean of its group — you calculate it by subtracting the mean from your value and dividing by the standard deviation. If you're staring at a raw score from your data and wondering whether it's "high", "low", or just average, that question is exactly what a z-score answers. It converts messy raw numbers into a common scale so you can compare, rank, and spot outliers with confidence.

Key Takeaways

• A z-score measures distance from the mean in standard-deviation units. A z of +1 means the value is one standard deviation above average; a z of −2 means two standard deviations below.
• The formula is z = (x − μ) / σ for a population, or z = (x − x̄) / s when you're working with a sample.
• A z-score of 0 equals the mean. About 68% of values in a normal distribution fall between z = −1 and z = +1, and roughly 95% fall between z = −2 and z = +2.
• Z-scores are commonly used to detect outliers — a value with |z| > 3 is often flagged as unusually extreme.
• You can convert a z-score into a percentile using the standard normal distribution, so a z of +1.64 corresponds to roughly the 95th percentile.

What Is a Z-Score in Simple Terms?

A z-score (also called a standard score) tells you where a single value falls relative to the average of a dataset, measured in standard deviations. Instead of saying "this student scored 82 on the exam", a z-score lets you say "this student scored 1.3 standard deviations above the class average" — which is far more informative because it accounts for how spread out the scores are.

The intuition is comparison. A test score of 82 means nothing on its own. If the class averaged 60 with tight scores, 82 is exceptional. If the class averaged 78 with wide scatter, 82 is barely above average. The z-score captures that context in one number.

Because z-scores put everything on the same scale, they let you compare values that were originally measured differently — like comparing someone's height (in cm) against their weight (in kg) relative to their respective group averages.

What Is the Z-Score Formula?

The z-score formula standardizes a raw value by centering it on the mean and scaling it by the standard deviation:

z = (x − μ) / σ

Where:
- x = the raw value you're evaluating
- μ (mu) = the population mean
- σ (sigma) = the population standard deviation

When you only have a sample (which is almost always the case in real research), you use the sample mean (x̄) and sample standard deviation (s):

z = (x − x̄) / s

The result is unitless. Whether your data was in dollars, milliseconds, or IQ points, the z-score strips away the units and leaves you with pure "distance from average".

How Do You Calculate a Z-Score Step by Step?

Calculating a z-score takes three steps: find the mean, find the standard deviation, then apply the formula. Here's a worked example with real numbers.

Suppose you're a graduate student analyzing exam scores for a class of 10 students:

72, 68, 85, 90, 78, 65, 88, 74, 80, 70

Step 1 — Calculate the mean (x̄).
Sum = 770. Divide by 10 → x̄ = 77.0

Step 2 — Calculate the standard deviation (s).
Subtract the mean from each score, square the differences, sum them (= 668), divide by n − 1 (= 9) to get the variance (74.22), then take the square root → s ≈ 8.61

Step 3 — Apply the formula for a chosen value.
Say you want the z-score for the student who scored 90:

z = (90 − 77) / 8.61 = 13 / 8.61 ≈ +1.51

That student scored 1.51 standard deviations above the class average — comfortably in the top ~7% of the class if the scores are roughly normal.

For the student who scored 65:

z = (65 − 77) / 8.61 = −12 / 8.61 ≈ −1.39

That student sits about 1.39 standard deviations below the mean, placing them near the bottom ~8%.

Running this in StatRyx skips the manual arithmetic — you upload your data, and it standardizes every value and returns the z-scores instantly, so you don't risk a math slip in your dissertation appendix.

How Do You Interpret a Z-Score?

A z-score's sign tells you direction and its magnitude tells you distance. Positive means above the mean; negative means below; zero means exactly average. The bigger the absolute value, the more unusual the value is. This table shows what common z-scores mean in a normal distribution:

Under the empirical rule, about 68% of values fall within ±1 SD, 95% within ±2 SD, and 99.7% within ±3 SD of the mean. That's why a |z| greater than 3 is often treated as a potential outlier worth investigating.

When Should You Use a Z-Score?

Use a z-score whenever you need to compare a value against a group, standardize variables measured on different scales, or flag outliers before running an analysis. Three of the most common uses:

• Outlier detection: Screening for values with |z| > 3 before running a t-test or ANOVA, since extreme scores can distort results.
• Standardizing variables: Converting predictors to z-scores (z-standardization) so coefficients in a regression are comparable and easier to interpret.
• Finding percentiles and probabilities: Once a value is a z-score, you can look up exactly what proportion of the distribution falls below it.

Note that z-scores assume your data is roughly normally distributed when you translate them into percentiles or probabilities. If your distribution is heavily skewed, a z-based percentile can mislead — that's a moment to check your data's shape first.

How Do You Report a Z-Score in APA 7?

In APA 7 style, the z statistic is italicized, lowercase, and reported without a leading zero on the p value. For example: z = 1.51, p = .066 (two-tailed). When you're using a z-test to compare a value against a known population, report the test statistic, the exact p value, and — where relevant — the effect size and confidence interval. StatRyx produces this notation for you automatically, so the write-up drops straight into your results section correctly formatted.

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Frequently Asked Questions

What does a negative z-score mean?

A negative z-score means the value is below the mean of its group. For example, a z-score of −1.5 tells you the value sits one and a half standard deviations below average. The sign only indicates direction — a z of −1.5 and a z of +1.5 are equally far from the mean, just on opposite sides.

Is a high or low z-score better?

Neither is universally "better" — it depends entirely on what you're measuring. A high positive z-score is good for something like exam performance or income,