How to Calculate a Z-score

Calculates the z-score as (value − mean) ÷ standard deviation, expressing how many standard deviations the value lies from the mean.

Explanation

A z-score tells you how many standard deviations a value lies away from the mean.

z = \dfrac{x - \bar{x}}{\sigma}

Here $x$ is the value, $\bar{x}$ the mean and $\sigma$ the standard deviation.

After the transformation the z-scores have a mean of 0 and a standard deviation of 1. This is called standardising the data.

Example

With a value of 80, a mean of 60 and a standard deviation of 10:

z = \dfrac{80 - 60}{10} = 2

The value sits two standard deviations above the mean.

Why standardise

Because it puts quantities with different units, means and spreads onto a single ruler.

Suppose you score 80 in maths (mean 60, standard deviation 10) and 80 in English (mean 70, standard deviation 5). Both raw scores are 80, and both z-scores happen to be 2.0. Had the English mean been 75, its z-score would be 1.0, and the maths result would clearly be the stronger one.

Once you have a z-score, the normal distribution turns it into a probability: $z = 2$ puts a value in roughly the top 2.3%.

Watch out

A standard deviation of zero makes the calculation impossible
The z-score itself assumes nothing about the shape of the distribution, but reading it as a probability does assume approximate normality
The Japanese deviation value is simply this z-score rescaled as $50 + 10z$