Step by step
The question
Question: samples of 36 are taken from numbers with mean 50 and SD 12: find the mean and the SD of the sample means.
μ = 50, σ = 12, n = 36: mean and SD of x̄?
The central limit theorem: the means of many samples make a bell.
many samples of 36: their means make a bell
1 The center
Its middle is the same as the numbers' mean: 50.
μx̄ = μ = 50
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 36 is 6.
12 divided by 6 is 2.
σx̄ = σ√n = 12√36 = 126 = 2
Careful!
We divide by the square root of 36, not by 36.
Careful: divide by √n, not by n
The bell of the sample means: middle 50, one SD is 2.
3 Check
Let's check: times the square root of 36 takes us back.
2 times 6 is 12, the SD σ.
So it is right!
2 × √36 = 2 × 6 = 12
Let's look back.
First: the center.
Then: the spread.
Last: check.
Answer
Careful: divide by √n, not by n