Step by step
The question
Question: find the sample standard deviation of 10, 12 and 14.
The standard deviation, SD, tells how far the numbers usually are from the mean.
1 Find the mean
First the mean: add them all, then divide by how many.
The total is 36.
36 divided by 3 is 12.
2 Distances from the mean
A table: how far each number is from the mean.
10 minus 12 is −2.
Negative: 10 is below the mean.
12 minus 12 is 0.
14 minus 12 is 2.
Above the mean.
Look: the distances add up to 0.
The ones below the mean cancel the ones above.
3 Square them
A new column: each distance squared.
−2 squared is −2 times −2.
Negative times negative is positive: 4.
And the rest.
4 Add the squares
We add the squares: 4 plus 0 plus 4 is 8.
5 Divide
These numbers are a sample: only part of the whole group.
For a sample we divide by one less than how many, n minus 1.
There are 3 numbers, so we divide by 3 minus 1, which is 2.
8 divided by 2 is 4.
Careful!
For a sample we divide by 2, not by 3.
6 Square root
The variance is in squared units.
The square root brings it back.
The square root of 4 is 2.
So the numbers are usually about 2 away from the mean, 12.
7 Check
Let's check: 2 squared is 4.
That is the variance, 4.
So it is right!
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.
Last: check.