EXPLANATION OF PROCEDURE USED WITH Z-TABLE NUMBER 3.

You want to enter a z-score of -2 but you can't because Z-TABLE NUMBER 3 does not allow negative z-score entries (top left picture). You can see that the z-table entry you are looking for is .022750. This is the area to the left of a z-table entry of -2. You can't get this directly if you are using Z-TABLE NUMBER 3.

You enter a z-score of 2 instead. The area to the left of a z-table entry of 2 shows as .477250 (top middle picture). This is not the whole area to the left of a z-score entry of 2. It is only the area to the mean. The bottom half of the area which represents the bottom half of the population is missing.

You add .5 to the z-table entry of .477250 which makes it .977250. This adds the missing half of the population. You now have the area to the left of the z-score of 2 that includes the entire population (top right picture)

You subtract .977250 from 1 to get .022750. This is the area to the right of a z-score entry of 2 (bottom left picture).

When you add .5 to the area to the left of the z-score provided by z-table 3, you get the complete area to the left of the z-score. Once you have the complete area to the left of the z-score, you can subtract it from 1 to get the complete area to the right of the z-score. The area to the right of a z-score that is positive will always be exactly the same as the area to the left of the same z-score that is negative. This is why the prcedure used for z-table 3 works.