Step by step
The question
Question: X is normal, with mean 12 and SD 1.5: which value has 60% below it?
A value with 60 percent below it is also called the 60th percentile.
1 Shade the area
The bell, with the mean 12 in the middle.
x is the cut-off we want.
60 percent is below x: we shade it, on the left of x.
2 Area to the left
The z-table works with the area to the left of z.
60 percent is 0.6.
It is already on the left.
Careful!
More than half the bell is on the left.
So x is above the mean, and z is positive.
3 Find it inside the table
Now we look for 0.6000 inside the table.
The closest number is 0.5987: row 0.2, column .05.
So z is about 0.25.
4 Back to x
Start at the mean, then go z SDs: x is the mean plus z times the SD.
0.25 times 1.5 is 0.375.
12 plus 0.375 is 12.375.
5 Check
Let's check: the z of 12.375 is 12.375 minus 12, over 1.5.
That is 0.25.
The table at 0.25 gives 0.5987: about 0.6 to the left.
So it is right!
Let's look back.
First: shade the area.
Then: area to the left.
Then: find it inside the table.
Then: back to x.
Last: check.