The formula
z = (x − μ) / σ
What it means
Subtracting the mean and dividing by the deviation puts any value on a common scale where zero is the mean and one unit is one standard deviation. That is what makes it possible to compare things measured in different ways: a 7 on a hard exam can beat a 9 on an easy one, and the z-score says so. If the distribution is also roughly normal, each z translates straight into a percentile.
How to work it out by hand
- Subtract the mean from the value
- Divide that difference by the standard deviation
- The result is how many deviations separate you from the mean
- If the distribution is normal, look that z up in the table for the percentile
What is worth knowing
The percentile shown here assumes the distribution is normal, and that assumption has to be earned: for incomes, which are heavily skewed, a z of 2 is nowhere near the 97.7th percentile. The z-score itself assumes nothing and is always correct as a measure of distance; it is the translation into a percentile that needs normality. In a normal distribution, two thirds of the data fall between −1 and 1, 95 % between −2 and 2 and 99.7 % between −3 and 3.