Z-Scores & Probability

When we look at a standard normal distribution, we see something like the below. Here, we have the mean at the highest point, the mean, median and mode on top of one another and we have our z-scores along the bottom. Having a standard normal distribution helps us to compare two different normal distributions, which have different means & standard deviations.

If we know the mean and standard deviation, we can calculate the z score for a datapoint, using the formula: z = (x-mean)/standard deviation

If x = 90, the mean is 65.5 and the standard deviation is 14.5, we would get a z-score of 1.69.

This can be plotted as below – I’ve also plotted a student at -1.67.

So then. Let’s calculate the probability of scoring above 1.69 or below -1.67. Let’s start with our -1.07. We look up the values using the z-score table. On the left, we find 1.0 and then we select the correct number from the columns (in this case – 1.07).

This shows as 0.1423, which as a percentage out of 1, is 14.23%. So, we can say that there is a 14.23% chance of scoring below this value.

If however, we are looking to the right of a value, to see the likelihood of scoring higher, we need to do something a little different. Instead of simply taking the value from the table, we need to first subtract it from 1.

For example, in the below, the z-score is 0.9545; which if subtracted from 1 gives us 0.0455 (or 4.55%).

If we were to want to know the probability of scoring between 50 and 90 (the two points we calculated above), we would simply do:

100-(14.55+4.55) = 81.22% probability you will score within that range.

Calculating X from the z-score.

We can also calculate X from the Z score, if we know the standard deviation and the mean, as below: