Standard deviations from the mean

All normal distributions have the same shape on a scale of 'standard deviations from the mean'.

Expressing an x-value in terms of standard deviations from the mean gives a z-score for the value. The z-score describes where the value lies in the above diagram.

This equation can also be written in the form

x  =  μ  +  z × σ

Probabilities and z-scores

The properties of normal distributions on the previous page give the probabilites that a z-score will be within ±1, ±2 and ±3:

Any other probability (area) relating to a normally distributed random variable, X, can be found in terms of z-scores:

  1. Translate the x-values whose probabilities are needed into a z-score, then
  2. Use the z-score to identify the relevant area in the above diagram

(We will explain how to accurately obtain probabilities from z-scores in the following page.)

Weights of apples

A factory packing apples has observed that Fuji apples from its supplying farms have approximately a normal distribution with mean of 180 grams and standard deviation 10 grams.

The diagram above translates the chosen apple weight, x, into a z-score that describes how many standard deviations (in units of 10g) it is above the mean (180g).

The z-score defines the position of the apple weight on the lower of the two axes. The highlighted area is the probability of a lower value.