Step by step
The question
Question: which number has a z-score of −1, when the mean is 30 and the SD is 3?
z = −1: mean 30, SD 3. x?
A z-score of −1 means 1 SD below the mean.
z = −1: 1 SD 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
−1 times 3 is −3.
30 minus 3 is 27.
x = 30 + (−1) × 3 = 30 − 3 = 27
3 Read it
27 on the number line: 1 SD below the mean.
4 Check
Let's check: the z-score of 27 is 27 minus 30, over 3.
That is −1.
So it is right!
z = 27 − 303 = −33 = −1
Let's look back.
First: write the formula.
Then: put the numbers in.
Then: read it.
Last: check.
Answer
x = 30 + (−1) × 3 = 30 − 3 = 27