Step by step
The question
Question: Data: 6, 8, 9, 13.
Add 34.
What happens to the mean and the median?
1 Mean before
We add all the numbers, from left to right.
So the total is 36.
There are 4 numbers.
The mean shares the total equally: 36 divided by 4.
So the mean is 9.
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.
Two numbers are left in the middle: 8 and 9.
The median is halfway between them: we add them and divide by 2.
8 plus 9 is 17.
17 divided by 2 is 8.5.
3 Mean after
34 comes in, so the total gets 34 more.
36 plus 34 is 70.
One more number: now there are 5.
Careful!
We have 5 numbers now, not 4.
So we share by 5.
We share the new total equally: 70 divided by 5.
So the new mean is 14.
4 Median after
Now 34 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.
9 is left in the middle.
So the new median is 9.
5 Compare
The mean went up from 9 to 14.
14 minus 9 is 5.
So it went up 5.
The median went up from 8.5 to 9.
9 minus 8.5 is 0.5.
So it went up 0.5.
34 is far from the other numbers.
A far number pulls the mean a lot.
Only the middle numbers matter for the median.
So the far number does not pull it, and it moves just a little.
So the mean goes up 5, and the median goes up 0.5.
6 Check
Let's check the new mean: if every number were 14, 5 of them would make 14 times 5.
That is 70, 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.