Z Score Calculator

What is a z score?

A z score, also known as a standard score, expresses a raw value in units of standard deviations away from the mean. It puts every value on the same standardized scale, so a value from one data set can be compared cleanly with a value from another. Two students with raw test scores of 85 and 92 might look very different in their original scales, but if the first comes from a class with mean 75 and standard deviation 10 and the second from a class with mean 80 and standard deviation 8, both students are exactly one and a half standard deviations above their class mean. Both have a z score of +1.5.

How to calculate a z score

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

For example, if the mean is 75, the standard deviation is 10, and the raw score is 85, the deviation is 85 − 75 = 10, and the z score is 10 / 10 = +1. The raw score sits exactly one standard deviation above the mean.

What a positive z score means

A positive z score means the raw score is above the mean. The size of the z score tells you how far above. A z score of +1 is one standard deviation above the mean; a z score of +2 is two standard deviations above. Under a normal distribution, roughly 68% of values land within one standard deviation of the mean (z scores between −1 and +1), and roughly 95% land within two standard deviations (z scores between −2 and +2). A z score of +3 or higher is unusually high under that assumption.

What a negative z score means

A negative z score means the raw score is below the mean. A z score of −1.5 means the value is one and a half standard deviations below the mean, and a z score of −2 means two standard deviations below. The sign just locates the value on the lower side of the mean; it does not by itself say the value is good or bad. In a context where lower is better (golf scores, error rates), a negative z score is the desirable side.

When to use a z score

Use a z score whenever you want to compare a value against the mean of its data set on a standardized scale. Common situations:

• Comparing test scores across classes, courses, or exams that use different scales.
• Looking up probabilities under a normal distribution from a z table or a statistics function.
• Identifying outliers in a data set: very large or very small z scores point at values that are unusual for the population.
• Standardizing data in machine learning preprocessing so features sit on a common scale.
• Reporting how far a measurement is from a reference value when the reference is the mean of repeated readings rather than a fixed accepted value (for that case, see the percent error calculator instead).

The standard deviation that goes into the z score formula has to come from somewhere. If you have a data set in front of you, compute it with the standard deviation calculator and then plug that result into this calculator along with the mean and the value you want to standardize.

Common mistakes

• Mixing up the mean and the raw score in the numerator. The deviation is x − μ, not μ − x. The first gives a positive z score for values above the mean; the second flips the sign.
• Dividing by the variance instead of the standard deviation. Variance is σ²; standard deviation is σ. See the Variance vs Standard Deviation guide if it helps to keep them straight.
• Using a sample standard deviation when a population standard deviation was needed, or the other way around. Pick the one that matches your context; the Standard Deviation Formula guide covers both versions.
• Trying to compute a z score with a standard deviation of zero. The formula is undefined in that case; this calculator flags it rather than printing a misleading number.
• Treating a large negative z score as automatically bad. The sign just locates the value below the mean; whether that is good, bad, or neutral depends on what is being measured.