Z-Score Calculator

What Is the Z-Score Calculator?

A z-score (standard score) tells you how many standard deviations a value is above or below the mean of its data set -- a way to compare values from different distributions on a common scale.

How to Use This Tool

Enter the value, the mean, and the standard deviation of the distribution. The z-score appears instantly.

The Formula

Worked Example

Reading the Sign and Size of z

The z-score has two pieces of information. The sign tells you the direction: positive means the value is above the mean, negative means below. The size tells you the distance in standard deviations. A z of 0.3 is only slightly away from average, while a z of 2.8 is far out in the tail of the distribution.

For example, with a mean of 70 and a standard deviation of 10, a value of 55 gives z = (55 - 70) / 10 = -1.5. That is exactly as far from the mean as 85 is, but on the low side.

Comparing Values From Different Scales

The main strength of the z-score is that it removes units, so you can compare results that were measured differently. Suppose a student scored 82 on Exam A, which had a mean of 75 and a standard deviation of 5, and 90 on Exam B, which had a mean of 80 and a standard deviation of 10. These figures are hypothetical.

Although the raw score on Exam B is higher, the performance on Exam A was stronger relative to that exam's group, because 1.4 is further above average than 1.0.

Working Backwards

Sometimes you know the z-score and want the original value. Rearrange the formula: value = mean + z × standard deviation. With a mean of 100, a standard deviation of 15, and a z of -2, the value is 100 + (-2 × 15) = 70. The calculator itself computes z from the three inputs, but this rearrangement is useful for checking your work or converting a target z back to a raw score.

What z Does and Does Not Tell You

• The z-score describes distance from the mean regardless of the shape of the data. Turning it into a percentile or a probability additionally requires the assumption that the data is roughly normally distributed, and a standard normal table or software.
• This tool does not convert z into a percentile. It reports only the z-score itself.
• Many people treat values beyond about 3 standard deviations from the mean as unusual outliers, but that is a convention, not a rule. In heavy-tailed data, values that far out occur more often than intuition suggests.

Where the Mean and Standard Deviation Come From

You must supply the mean and standard deviation yourself. These might come from a published test, a manufacturer's specification, or your own data. If they come from your own list of numbers, calculate them with the Mean, Median and Mode Calculator and the Standard Deviation Calculator on this site. Make sure you use the standard deviation that matches your situation. Using the sample version for sampled data is the usual choice, and mixing up sample and population gives a slightly different z.

Common Mistakes

• Entering a variance as the standard deviation. If you have the variance, take its square root first.
• Entering a zero standard deviation. Division by zero is not defined, and the tool asks for a non-zero standard deviation. A standard deviation is also conceptually positive, so enter its absolute size.
• Comparing z-scores from very different distributions. If one distribution is heavily skewed, equal z-scores do not mean equally rare results.
• Forgetting the reference group. A z-score is only meaningful relative to the specific mean and standard deviation you used.

Frequently Asked Questions