Step by step
The question
Question: samples of 25 are taken from numbers with mean 100 and SD 15: find the mean and the SD of the sample means.
μ = 100, σ = 15, n = 25: mean and SD of x̄?
The central limit theorem: the means of many samples make a bell.
many samples of 25: their means make a bell
1 The center
Its middle is the same as the numbers' mean: 100.
μx̄ = μ = 100
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 25 is 5.
15 divided by 5 is 3.
σx̄ = σ√n = 15√25 = 155 = 3
Careful!
We divide by the square root of 25, not by 25.
Careful: divide by √n, not by n
The bell of the sample means: middle 100, one SD is 3.
3 Check
Let's check: times the square root of 25 takes us back.
3 times 5 is 15, the SD σ.
So it is right!
3 × √25 = 3 × 5 = 15
Let's look back.
First: the center.
Then: the spread.
Last: check.
Answer
Careful: divide by √n, not by n