Step by step
The question
Question: Data: 5, 7, 7, 9, 12.
Add 8.
What happens to the mean and the median?
1 Mean before
We add all the numbers, from left to right.
So the total is 40.
There are 5 numbers.
The mean shares the total equally: 40 divided by 5.
So the mean is 8.
2 Median before
The median is the middle number.
For it, the numbers must be in order.
They already are, smallest first.
We cross out the smallest and the biggest, one pair at a time.
And the next pair.
7 is left in the middle.
So the median is 7.
3 Mean after
8 comes in, so the total gets 8 more.
40 plus 8 is 48.
One more number: now there are 6.
Careful!
We have 6 numbers now, not 5.
So we share by 6.
We share the new total equally: 48 divided by 6.
So the new mean is 8.
4 Median after
Now 8 goes into the list, in its place in order.
We cross out the smallest and the biggest, one pair at a time.
And the next pair.
Two numbers are left in the middle: 7 and 8.
The median is halfway between them: we add them and divide by 2.
7 plus 8 is 15.
15 divided by 2 is 7.5.
5 Compare
The mean was 8, and it is still 8.
It stays the same.
The median went up from 7 to 7.5.
7.5 minus 7 is 0.5.
So it went up 0.5.
8 is exactly the mean.
Adding the fair share does not change the fair share.
So the mean stays the same, and the median goes up 0.5.
6 Check
Let's check the new mean: if every number were 8, 6 of them would make 8 times 6.
That is 48, the new total.
So it is right!
Let's look back.
First: mean before.
Then: median before.
Then: mean after.
Then: median after.
Then: compare.
Last: check.