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Z-table for proportions

In statistics, we apply z-table for proportions, for calculating proportions below, above and between certain values. The z-table is also known as the standard normal table.

Looking up in a z-table can contribute to understanding the procedures, but all statistical software has the table and all other common statistical tables embedded.

 

Example: Z-table for proportions above

Say that a HR in a certain company runs a test for applicants. The test scores for all the applicants are normally distributed with a mean of 39 points and a standard deviation of 3. The applicant, Dari, is doing well in the interviews, and HR wish to see how she has done in the test. She has scored 47.

HR wish to see what percentage of the test scores are greater than Dari’s score?

First, we calculate the z-score: 47 – 39 / 3 = 2.666667 and look this score up in the z-table. But before that, let’s have a look at the graph:

Z-table for proportions. Curve

So, we are looking for the area to the right of the σ=2.67. From the z-table we find the exact example:

Z-table for proportions

 

For our z-score value of 2.67, the value is 0.99621. As the z-score table gives the area to the left, this is the proportion of scores less than Dari’s. So, the rest are the ones above Dari’s. We therefore subtract 1 from the this and get the area to the right: 1-0.99621 = 0.00379. Only 0.38% of the test scores are higher than Dari’s.

 

Proportion below

Another way of using the z-table for proportions: The calculation for proportions below a given z-score is the value directly read in the z-table. So, for the example above, the proportion below Dari’s score was 0.99621 ≈ 99.62% which, of course could have been found directly by subtracting the 0.38 from 1.00.

 

Proportion between

Using the same logic as above, we can also calculate the area between two values. Say, we are to calculate the proportion that have scored more than 39 and less than Dari’s 47. We would then subtract the area up to 39 from the larger area up to 47.

The area to the left of Dari’s score was 0.9962 and as 39 is the mean, the area to the left of this is 0.5. Now: .9962-0.5 = .4962 = 49.62%.

 

Z-table for proportions in Excel

The NORM.S.INV gives the critical z-value in and the NORM.S.DIST can give the p-value:

Z-table for proportions in Excel

 

Learning statistics

Khan Academy offer some videos with worked examples on Z-table for proportions:

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