Step by step
The question
Question: samples of 9 are taken from numbers with mean 40 and SD 6: find the mean and the SD of the sample means.
μ = 40, σ = 6, n = 9: mean and SD of x̄?
The central limit theorem: the means of many samples make a bell.
many samples of 9: their means make a bell
1 The center
Its middle is the same as the numbers' mean: 40.
μx̄ = μ = 40
2 The spread
Means of samples are closer together than single numbers.
Their SD is σ divided by the square root of n.
means are less spread out: SD = σ ÷ √n
The square root of 9 is 3.
6 divided by 3 is 2.
σx̄ = σ√n = 6√9 = 63 = 2
Careful!
We divide by the square root of 9, not by 9.
Careful: divide by √n, not by n
The bell of the sample means: middle 40, one SD is 2.
3 Check
Let's check: times the square root of 9 takes us back.
2 times 3 is 6, the SD σ.
So it is right!
2 × √9 = 2 × 3 = 6
Let's look back.
First: the center.
Then: the spread.
Last: check.
Answer
Careful: divide by √n, not by n