Step by step
The question
Question: find the sample standard deviation of 15, 9, 12, 18 and 6.
The standard deviation, SD, tells how far the numbers usually are from the mean.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find the mean
First the mean: add them all, then divide by how many.
The total is 60.
60 divided by 5 is 12.
2 Distances from the mean
A table: how far each number is from the mean.
15 minus 12 is 3.
15 is above the mean.
9 minus 12 is −3.
Below the mean.
And the same for the rest.
Look: the distances add up to 0.
The ones below the mean cancel the ones above.
So adding the distances tells us nothing: it is always 0.
We need every distance to count as positive.
So we square each distance.
A square is never negative.
3 Square them
A new column: each distance squared.
3 squared is 9.
−3 squared is −3 times −3.
Negative times negative is positive: 9.
And the rest.
4 Add the squares
We add the squares: 9 plus 9 plus 0 plus 36 plus 36 is 90.
5 Divide
These numbers are a sample: only part of the whole group.
A sample is usually a bit closer to its own mean than the whole group is.
So we divide by one less than how many: n minus 1, where n is how many numbers.
That makes the answer a little bigger, to make up for it.
There are 5 numbers, so we divide by 5 minus 1, which is 4.
90 divided by 4 is 22.5.
Careful!
For a sample we divide by 4, not by 5.
6 Square root
We squared the distances, so the variance is in squared units.
The square root brings it back to the units of the numbers.
The square root of 22.5 is 4.7434 and more digits.
We round to one more decimal place than the data, so the answer does not look more exact than the numbers.
So the SD is about 4.7.
So the numbers are usually about 4.7 away from the mean, 12.
7 Check
Let's check: 4.7 squared is 22.09.
That is about the variance, 22.5.
So it is right!
8 Pick the choice
Our answer is SD ≈ 4.7.
That is choice D.
9 Why not the others
A is 4.2.
Wrong: divided by n = 5: a sample divides by n − 1 = 4.
That is the Careful mistake from before.
B is 22.5.
Wrong: that is the variance: take its square root.
C is 12.
Wrong: that is the mean.
Let's look back.
First: find the mean.
Then: distances from the mean.
Then: square them.
Then: add the squares.
Then: divide.
Then: square root.
Then: check.
Then: pick the choice.
Last: why not the others.