Z Score Calculator

A Z score calculator helps determine how far a value is from the mean of a dataset in terms of standard deviations. Z scores are commonly used in statistics, probability, data analysis, and research to compare values that come from the same distribution or from different datasets.

What Is a Z Score?

A Z score, also called a standard score, describes the position of a value relative to the mean. It tells you how many standard deviations the value is above or below the average.

Z Score Formula

The standard formula is:

z = (x − μ) ÷ σ

Here, x is the individual value, μ is the population mean, and σ is the population standard deviation.

Z Score Example

Suppose a test has a mean score of 70, a standard deviation of 10, and a student scores 85.

z = (85 − 70) ÷ 10

z = 15 ÷ 10 = 1.5

The student's Z score is 1.5. This means the score is 1.5 standard deviations above the mean.

How to Interpret a Z Score

A positive Z score means the value is above the mean, while a negative Z score means it is below the mean. A Z score of zero means the value is exactly equal to the mean.

Why Are Z Scores Useful?

Z scores make it easier to compare values relative to their own distributions. For example, a test score that looks high on its own may be easier to evaluate when you know how far it is from the average score.

They are also useful when working with standard normal distributions, probabilities, percentiles, and statistical comparisons.

Positive and Negative Z Scores

If the original value is greater than the mean, the Z score will be positive. If the value is lower than the mean, the Z score will be negative.

For example, with a mean of 50 and a standard deviation of 5:

Value 60: z = (60 − 50) ÷ 5 = 2

Value 40: z = (40 − 50) ÷ 5 = -2

Both values are two standard deviations from the mean, but they are on opposite sides of it.

Z Score and Percentiles

For data that follows a normal distribution, a Z score can be used with a standard normal table or statistical software to find the approximate percentile or probability associated with a value.

For example, a Z score of approximately 1.96 corresponds to about the 97.5th percentile in a standard normal distribution.

Common Mistakes to Avoid

Make sure the mean and standard deviation are calculated or selected correctly. Also check whether the problem gives a population standard deviation or a sample standard deviation, because statistical formulas may differ depending on the situation.

A standard deviation of zero cannot be used in the ordinary Z score formula because division by zero is undefined.

Quick Summary

A Z score shows how many standard deviations a value is above or below the mean. The basic formula is z = (x − μ) ÷ σ. Positive values are above the mean, negative values are below it, and a score of zero represents the mean.