ExplanationFrom Z-Scores to Probabilities
When we studied z-scores, we saw that you could represent the distance between a value and the mean of a sample in terms of how many standard deviations apart they are. This is called a z-score and it allows us to say how many standard deviations above or below the mean any value lies. Z-scores can also be used to see how far apart any two values are, for example there might be 1 standard deviation between my height and yours.
It would be even more useful if you could know what proportion of a sample or a population lay between the mean and a given value. You might also want to know what proportion of a sample or population lies above or below a given value. Converting from z-scores to proportions when the underlying distribution is normal is simple and this page shows how it is done.
If you know the proportion of a population that lies in a certain range, then you know the probability of a randomly selected value falling into that range. For example, if I knew that 10% of all men were taller than 2 meters, I would know that if I picked a man at random, the probability of him being taller than 2 meters would be 0.1. If this is new to you, there is a help section on probability below. Many statistical techniques require their results to be reported in terms of probabilities so the ability to convert from z-scores to probabilities is very useful and widely used.
Converting From Z-Scores to Probabilities
Converting from z-scores to probabilities is done using z-tables, which we will show you at level two of this topic. Z-tables tell you the proportion of data that is below a given z-score in a normally distributed population.
Here is an example. Let's say that the mean of a sample of people's height is 180cm, with a standard deviation of 30cm and your own height is 190cm. The z score for 190cm (your height) is 0.33 (see the section on z-scores for more on this) so you are a third of a standard deviation above the mean. Looking up 0.33 in a z-table reveals that 63% of people in a normal distribution fall below (are shorter than) you. Clearly, this only works if the data is normally distributed.