Z Score Calculator

Find how many standard deviations a score is above or below the mean, from a mean and standard deviation you enter.

The value you want to standardise, for example 85. It can be negative or a decimal.

The average of the group the score belongs to, for example 70. You enter it; it is not worked out here.

The spread of that group, greater than 0, for example 10. You enter it too.

Worked example

Inputs: score x = 85, mean = 70, standard deviation = 10. Formula: z = (x − mean) ÷ standard deviation. Substitution: (85 − 70) ÷ 10 = 15 ÷ 10. Result: z = 1.5, so 85 is 1.5 standard deviations above the mean.

How it works

A z score says how many standard deviations a score lies from the mean: z = (x − mean) ÷ standard deviation. A positive z is above the mean, a negative z is below it, and 0 is exactly at the mean. You supply the mean and the standard deviation; this calculator does not work them out from data (the Standard Deviation Calculator does). The arithmetic is the same whether they are population values or the sample mean and sample standard deviation of the group the score belongs to. The standard deviation must be greater than 0, because a standard deviation of 0 means every value is the same. A result that is a finite decimal, such as 1.5, is exact and is shown as a plain number. A repeating result, such as 1 ÷ 3, is shown with a leading "≈" and rounded to 6 decimal places, with its exact fraction underneath. The working writes the result as an equation with "=" when it is exact and "≈" when it is rounded. This gives one z score for one value: it does not look up probabilities or percentiles from a z-table.

What this result means

The z score is the number of standard deviations the score lies from the mean: positive above it, negative below it and 0 exactly on it. It lets you compare scores from groups with different means and spreads.

Common mistakes

  • Entering the wrong measure of spread. The standard deviation is needed, not the range or the variance. The Standard Deviation Calculator finds it from a list of numbers.
  • Reading the z score as a percentile or a probability. A z score of 1.5 means 1.5 standard deviations above the mean; turning it into a percentage needs a z-table or a normal distribution, which this tool does not provide.

Assumptions and limitations

  • One score at a time, using a mean and standard deviation you enter. There is no z-table, p-value or confidence interval, and no mode for the z score of a sample mean.
  • The standard deviation must be greater than 0.

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