Finding z-scores given probabilities

These problems assume the standard normal distribution.

Example 1

Find the z-score corresponding to the 95th percentile.

Solution .

Recall that 95% of the scores are less than the 95th percentile. This means that on the figure below we must find the value of z at the navy blue line.

Based on the graph we can estimate that the z corresponding to the 95th percentile is a little more than 1.6. To get a better estimate we will use the table. First note that the .95 area includes two parts. The first part to the left of zero has an area of .5. The part to the right of zero must have an area of .45 to make a total of .95. We need to use the table to find the z that has an area of .4500. Using the table we find that z = 1.64 has area of .4495 and that z = 1.65 has area of .4505. In this case use z = 1.645. You'll note that if you used the table in the book, that this is one of the special values mentioned at the bottom of the table.

Example 2

Find the z-score corresponding to the 20th percentile.

Solution .

You can see from the graph below that the z-score must be negative and smaller than -0.8.

By definition of 20th percentile, 20% of the data is to the left of the navy blue line. Since 50% of the data is to the left of zero, the area between 0 and our z value must be .5 - .2 = .3000. In the table z = .84 has an area of .2995 and z = .85 has an area of .3023. Since the area at z = .84 is closer to .3000 than the area is at z = .85, we use z = -.84 as the 20th percentile.