Step by step
The question
Question: which number has a z-score of −0.75, when the mean is 80 and the SD is 8?
z = −0.75: mean 80, SD 8. x?
A z-score of −0.75 means 0.75 SDs below the mean.
z = −0.75: 0.75 SDs below the mean
1 Write the formula
Start at the mean, then go z SDs: x is the mean plus z times the SD.
x = mean + z × SD
Careful!
A negative z-score means below the mean.
So x comes out smaller than the mean.
Careful: a negative z means below the mean: x is smaller
2 Put the numbers in
−0.75 times 8 is −6.
80 minus 6 is 74.
x = 80 + (−0.75) × 8 = 80 − 6 = 74
3 Read it
74 on the number line: 0.75 SDs below the mean.
4 Check
Let's check: the z-score of 74 is 74 minus 80, over 8.
That is −0.75.
So it is right!
z = 74 − 808 = −68 = −0.75
Let's look back.
First: write the formula.
Then: put the numbers in.
Then: read it.
Last: check.
Answer
x = 80 + (−0.75) × 8 = 80 − 6 = 74