Step by step
The question
Question: samples of 100 are taken from numbers with mean 64 and SD 20: find the mean and the SD of the sample means.
μ = 64, σ = 20, n = 100: mean and SD of x̄?
The central limit theorem: the means of many samples make a bell.
many samples of 100: their means make a bell
1 The center
Its middle is the same as the numbers' mean: 64.
μx̄ = μ = 64
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 100 is 10.
20 divided by 10 is 2.
σx̄ = σ√n = 20√100 = 2010 = 2
Careful!
We divide by the square root of 100, not by 100.
Careful: divide by √n, not by n
The bell of the sample means: middle 64, one SD is 2.
3 Check
Let's check: times the square root of 100 takes us back.
2 times 10 is 20, the SD σ.
So it is right!
2 × √100 = 2 × 10 = 20
Let's look back.
First: the center.
Then: the spread.
Last: check.
Answer
Careful: divide by √n, not by n