Z Score Formula

A z score expresses a raw value in units of standard deviations away from the mean. It is sometimes called a standard score because the result lives on the same standardized scale no matter what data set the original value came from. Two values with very different raw sizes can both have a z score of, say, +1.5, and that tells you they sit the same distance above their respective means in standardized terms.

That is the practical point of a z score: it strips out the units and the scale of the original data and leaves behind a comparable measure of position relative to the mean.

The formula has three pieces: the raw score, the mean, and the standard deviation. Subtract the mean from the raw score, then divide by the standard deviation.

The numerator x − μ is the raw deviation of the value from the mean, in the original units. Dividing by σ converts that deviation into a count of standard deviations, which is what makes z scores comparable across data sets that use different units or scales.

The arithmetic is short. Two steps get you from the three inputs to the final z score.

1. Subtract the mean from the raw score to get the deviation: x − μ.
2. Divide that deviation by the standard deviation: (x − μ) / σ.

That is the whole formula. The interpretation is what carries the meaning: a positive result means the value is above the mean, a negative result means it is below, and a result of zero means it sits exactly at the mean.

The sign of the z score locates the value on one side of the mean or the other; the magnitude tells you how far.

• Positive z score. The raw score is above the mean. A z score of +1 is one standard deviation above; +2 is two above; +0.5 is half a standard deviation above.
• Negative z score. The raw score is below the mean. A z score of −1.5 sits one and a half standard deviations below the mean; −2 sits two below.
• Zero. The raw score equals the mean exactly.

Under a normal distribution, roughly 68% of values fall within one standard deviation of the mean (z scores between −1 and +1) and roughly 95% fall within two (z scores between −2 and +2). A z score of +3 or more (or −3 or less) is unusually extreme under that assumption.

Take a raw score of 85 from a data set with mean 75 and standard deviation 10.

• z = (85 − 75) / 10
• z = 10 / 10
• z = 1

The z score is +1. In plain English: the value is one standard deviation above the mean. The same arithmetic runs in the z score calculator, which prints both worked steps with your numbers in place.

Same class as the previous example, but a different student scored a raw 60. The class mean is still 75, the standard deviation is still 10.

• z = (60 − 75) / 10
• z = −15 / 10
• z = −1.5

The z score is −1.5. In plain English: the value is one and a half standard deviations below the mean. Negative z scores are not bad on their own; the sign just says which side of the mean the value sits on.

Z scores really earn their keep when you need to compare values that come from different distributions. Suppose two students take final exams in different courses:

• Student A scored 85 in a class with mean 75 and standard deviation 10.
• Student B scored 92 in a class with mean 80 and standard deviation 8.

Compute each z score:

• Student A: z = (85 − 75) / 10 = +1.0
• Student B: z = (92 − 80) / 8 = +1.5

Student B's raw score (92) is higher in absolute terms, and Student B also has the higher z score. Both pieces line up here, but the z score adds the context: Student A is one standard deviation above their class mean, while Student B is one and a half. If the two classes had the same mean but very different spreads, you could easily see two students with the same raw score but different z scores. The z score is what tells you how each performance compares to its own distribution.

A few traps that catch people:

• Flipping x and μ in the numerator. The formula is x − μ, not μ − x. The first gives a positive z score for values above the mean; the second flips the sign and reverses the meaning.
• Dividing by the variance instead of the standard deviation. Variance is σ²; standard deviation is σ. They are easy to confuse, especially when both are reported on the same line. See the Variance vs Standard Deviation guide if it helps to keep them straight.
• Using sample standard deviation when population was wanted, or vice versa. Pick the one that matches your context. The Standard Deviation Formula guide covers both versions.
• Treating a large negative z score as automatically bad. The sign just locates the value below the mean. In contexts where lower is better (golf scores, error rates, response times), a negative z score is the desirable side.
• Confusing z score with percent error. Z score uses the mean of a distribution as the reference; percent error uses a fixed accepted value. They look similar on paper but answer different questions. See the percent error calculator and the Percent Error Formula guide when the reference is a fixed accepted value.

The z score formula divides by the standard deviation, so it is undefined when σ = 0. Practically, σ = 0 would mean every value in the data set is identical, in which case the concept of a z score does not really apply: there is no spread to measure against. The z score calculator flags this case rather than printing a misleading number.

Standard deviation also cannot be negative. It is, by definition, a non-negative measure of spread. If a calculation produces a negative value, something has gone wrong upstream and the result should not be used in the z score formula. Compute σ from the data first using the standard deviation calculator, confirm it is positive, and only then standardize values against it.

A few calculations look similar but answer different questions. Knowing which one you want avoids reporting the wrong number.

• Z score standardizes a value against the mean and standard deviation of its data set. It tells you how many standard deviations from the mean the value sits. z = (x − μ) / σ.
• Percent error compares a measurement to a fixed accepted reference value and is reported as a positive percentage. |experimental − accepted| / |accepted| × 100. See the percent error calculator.
• Percent change compares two values where neither is a fixed reference, and the sign tells you the direction of the change. ((new − old) / |old|) × 100. See the percentage increase calculator.
• Standard deviation describes the spread of a data set around its mean, without picking out any single value. Z score is one level up: it uses the standard deviation as the unit of measure. See the standard deviation calculator.

• Formula: z = (x − μ) / σ.
• Subtract the mean from the raw score, then divide by the standard deviation.
• Positive z score means the value is above the mean; negative means below; zero means equal to the mean.
• Roughly 68% of values fall within one standard deviation of the mean (z between −1 and +1) under a normal distribution; roughly 95% fall within two.
• Undefined when σ = 0. Standard deviation cannot be negative.
• Compute σ first with the standard deviation calculator, then standardize values with the z score calculator.