How to Calculate a Z-Score

A z-score of 1.5 means the value is 1.5 standard deviations above the mean. On a normal distribution that is the 93.32nd percentile: 93.32% of values fall below it and only 6.68% fall above it. It is a strong score, but not an outlier — outliers usually start beyond |z| = 2 or 3. The diagram above shades exactly this area: everything to the left of z = +1.50. If you want the picture rather than the table, the bell curve generator draws the curve from any mean and SD.

Z-Score vs Percentile

A z-score and a percentile describe the same position in two different units. The z-score counts standard deviations from the mean; the percentile counts the percentage of values below it. Converting between them needs the normal distribution — use a z-score to percentile converter or a z-score table.

This table is the empirical rule in numbers: 68.27% of values fall within one standard deviation of the mean (z between −1 and +1), 95.45% within two, and 99.73% within three. To go further, work out percentiles from a mean and SD with the empirical rule percentile calculator.

One warning: percentiles from a z-score only work when the data is roughly normal. You can calculate a z-score for any dataset, but turning it into a percentile assumes a bell-shaped distribution. If your data is skewed, report the z-score on its own.

How to Find X from a Z-Score

Sometimes you know the z-score and need the original value. Rearrange the formula to x = μ + zσ. For example, what test score sits at z = +1.28 with a mean of 70 and a standard deviation of 10?

x = 70 + 1.28 × 10 = 70 + 12.8 = 82.8. A score of 82.8 is at the 90th percentile. To check the reverse direction — starting from a probability instead — use the probability to z-score tool, which finds the z-score for any percentile.

How to Calculate a Z-Score in Excel

Excel has a built-in function for exactly this: =STANDARDIZE(x, mean, standard_dev). For the worked example above, =STANDARDIZE(85,70,10) returns 1.5. To get the percentile as well, use =NORM.S.DIST(1.5,TRUE), which returns 0.9332 — the 93.32nd percentile.

If you are starting from raw data, calculate the mean with =AVERAGE(A:A), the population standard deviation with =STDEV.P(A:A) and the sample standard deviation with =STDEV.S(A:A). Then any value’s z-score is =(A2-AVERAGE(A:A))/STDEV.P(A:A) for a population, or the same formula with STDEV.S for a sample. STANDARDIZE itself always uses the population standard deviation, so for sample data use the manual formula instead.

Population vs Sample Z-Score

Use the population z-score when you have every value in the group — a whole class, a full production run, all the data there is. Use the sample z-score when your data is only part of a larger group, such as 50 survey responses from a population of thousands. The sample formula divides by N − 1 instead of N, which gives a slightly larger standard deviation and slightly smaller z-scores, as the five-score example above shows (1.41 vs 1.26 for the top score). If a textbook or exam question just says “mean 70, SD 10”, it means the population values, so use z = (x − μ) / σ.

Common Mistakes When Calculating a Z-Score

• Using the variance instead of the standard deviation. The formula divides by σ, not σ². Take the square root first.
• Mixing up population and sample standard deviations. Dividing by N when you should divide by N − 1 (or the reverse) changes every z-score.
• Dropping the minus sign. A z-score of −1.50 is below the mean; +1.50 is above it. The sign is half the answer.
• Reading the z-table backwards. Most tables give the area to the left of z. The area to the right is 1 minus that value — for z = +1.50, the upper tail is 1 − 0.9332 = 0.0668.
• Rounding the standard deviation too early. Keep at least two extra digits until the final division, or your z-score will drift.
• Turning a z-score into a percentile for skewed data. The percentile reading assumes a normal distribution; for anything else, check the shape first — or shade the standard deviations on a bell curve to see the areas.

Can a Z-Score Be Negative?

Yes. A negative z-score simply means the value is below the mean. A z-score of −1.50 is 1.5 standard deviations below the mean, at the 6.68th percentile. Negative z-scores are just as common as positive ones in any real dataset — about half of all values sit below the mean.

What Is a Good Z-Score?

There is no universal “good” z-score — it depends on what you are measuring. For a test, a z-score of +1 or higher (84th percentile and up) is a strong result; for a manufacturing measurement, a z-score near 0 means the part is close to target, and a large |z| means it is out of spec. As a rough guide, |z| under 1 is typical, |z| between 1 and 2 is unusual, and |z| above 2 deserves a second look.

What Z-Score Is Considered an Outlier?

Values beyond |z| = 2 are often flagged as possible outliers, and values beyond |z| = 3 almost always are: only 0.27% of values in a normal distribution fall outside three standard deviations. Before deleting an outlier, check whether it is a data-entry error or a genuine extreme value — a z-score flags the value, it does not explain it. For the statistical background, see the NIST/SEMATECH e-Handbook of Statistical Methods.