Slideshow version

Before we get started, open this PDF in another tab. You’ll need to reference it during the lesson and exercises: Standard Normal Probabilities

In the previous lesson, we looked at a normal distribution and nice round numbers of standard deviations. Specifically 1, 2, and 3 (both above and below the mean). Now we’ll take that idea and extend to any standard deviation by revisiting z-scores.

For a quick refresher on calculating z-scores, subtract the mean from the given value and divide by the standard deviation.

\[z = \frac{x-\mu}{\sigma}\]

The value you end with tells you how many standard deviations the data point is above or below the mean. Now, if the z-score happened to be a nice even number, like 1, 2, or 3, you could use the empirical rule to determine percentages. But if it’s not (spoiler: it won’t) that’s where a z-score table comes in handy. What the table will tell you is what proportion lies below (or to the left) of the z-score.

The initial trick is learning how to read it. Here’s a part of the table.

z score table partial

The first column is where you start. Look at your z-score and find the row that matches the beginning of it. You then move over looking at the next decimal place. So, a z-score of −0.12 will yield a value of 0.45224. That means that 45.224% of data falls below that z-score.

For another example, a z-score of −0.85 yields 0.19766.

The table gives you what proportion falls below, so in order to get above, subtract what it gives you from 1. I don’t suggest doing percentage conversions unless specifically asked to do so.

If you need to find the proportion between two z-scores, you will need to lookup both values on the table, but subtract the smaller one from the larger.

between two z-scores

The larger one will give you everything below that point, but you want to remove the part below the lower boundary, hence the subtraction.