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Z-Score Calculator
A z-score calculator for z = (x − mean) / sd, the reverse raw score x = mean + z × sd, and the mean and standard deviation of a list.
The value you want to standardize
The average of the distribution
Must be greater than 0
Formula
z = (x − mean) / sd
Enter the raw score, the mean, and the standard deviation to see z = (x − mean) / sd with your numbers, then the z-score.
This z-score calculator finds how many standard deviations a raw score sits from the mean. Enter the score, the mean, and a standard deviation greater than zero, and the page writes z = (x − mean) / sd with those numbers filled in. Reverse mode starts from a z-score and returns the raw score using x = mean + z × sd. List mode computes the mean and either the population standard deviation, dividing by n, or the sample standard deviation, dividing by n − 1, then the z-score of a value you choose.
How to calculate a z-score
The z-score formula is z = (x − mean) / sd, also written z = (x − μ) / σ. Subtract the mean from the raw score, then divide by the standard deviation. A score of 85 with a mean of 70 and a standard deviation of 10 is (85 − 70) / 10, which equals 1.5, or 1.5 standard deviations above the mean.
Rearrange the formula to find the raw score: x = mean + z × sd. A z-score of 1.5, a mean of 70, and a standard deviation of 10 gives 70 + 1.5 × 10, which is 85. The standard deviation must be greater than 0, because a value of 0 means there is no spread and division by zero leaves the z-score undefined. From a list, the population standard deviation divides the sum of squared deviations by n, and the sample standard deviation divides by n − 1.
Frequently Asked Questions
What is the z-score formula?
The z-score formula is z = (x − mean) / sd. Subtract the mean from the raw score, then divide by the standard deviation. Textbooks also write it as z = (x − μ) / σ. The standard deviation has to be greater than 0. If it is 0, the z-score is undefined because the formula would divide by zero.
How do you calculate a z-score?
Take the raw score, subtract the mean, and divide by the standard deviation. A score of 85 with a mean of 70 and a standard deviation of 10 is (85 − 70) / 10, which equals 1.5. That score is 1.5 standard deviations above the mean. A negative z-score is that far below the mean.
How do you find the raw score from a z-score?
Multiply the z-score by the standard deviation and add the mean: x = mean + z × sd. A z-score of 1.5, a mean of 70, and a standard deviation of 10 gives 70 + 1.5 × 10, which is 85.
How do population and sample standard deviations differ?
Both start from the squared distances between each number and the mean of the list. The population standard deviation divides that sum by n, the count of values. The sample standard deviation divides by n − 1, so the list needs at least two numbers. The calculator names the version it used before it reports the z-score.