Step by step
The question
Question: Which group is more spread out?
Group 1: 3, 6, 9.
Group 2: 1, 4, 6, 8, 11.
1 Find each mean
The standard deviation, SD, tells how far the numbers usually are from the mean.
So first we need each mean.
Group 1: add them, then divide by how many.
The total is 18.
18 divided by 3 is 6.
Now Group 2: add them, then divide by how many.
The total is 30.
30 divided by 5 is 6.
2 Distances from the mean
Under each number of Group 1, we write how far it is from the mean, 6.
We add the distances: 3 plus 0 plus 3 is 6.
Now the same for Group 2.
Its mean is 6.
We add the distances: 5 plus 2 plus 0 plus 2 plus 5 is 14.
3 Compare
The sets have 3 and 5 numbers.
More numbers make a bigger total, so the totals are not fair to compare.
We share each total by its count: the usual distance.
6 divided by 3 is 2.
14 divided by 5 is 2.8.
Group 2's numbers are further from the mean, 2.8 against 2.
Further from the mean means a bigger standard deviation.
We did not work out the SD itself.
But it grows with these distances, so the bigger distance goes with the bigger SD.
Careful!
Both means are the same, but the spread is not.
The SD is about the distances.
So Group 2 has the bigger standard deviation.
4 Check
Let's check with the range, biggest minus smallest.
Group 1: 6.
Group 2: 10.
Group 2 is spread wider too.
So it is right!
Let's look back.
First: find each mean.
Then: distances from the mean.
Then: compare.
Last: check.