Step by step
The question
Question: Data: 15, 18, 20, 22, 25.
Remove 15.
What happens to the mean and the median?
1 Mean before
We add all the numbers, from left to right.
So the total is 100.
There are 5 numbers.
The mean shares the total equally: 100 divided by 5.
So the mean is 20.
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.
20 is left in the middle.
So the median is 20.
3 Mean after
15 is gone, so the total loses 15.
100 minus 15 is 85.
One number less: now there are 4.
Careful!
We have 4 numbers now, not 5.
So we share by 4.
We share the new total equally: 85 divided by 4.
85 does not share out evenly.
4 times 21 is 84.
1 is left, and 1 divided by 4 is 0.25.
So the new mean is 21.25.
4 Median after
Now the list in order, without 15.
We cross out the smallest and the biggest, one pair at a time.
Two numbers are left in the middle: 20 and 22.
The median is halfway between them: we add them and divide by 2.
20 plus 22 is 42.
42 divided by 2 is 21.
5 Compare
The mean went up from 20 to 21.25.
21.25 minus 20 is 1.25.
So it went up 1.25.
The median went up from 20 to 21.
21 minus 20 is 1.
So it went up 1.
15 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 1.25, and the median goes up 1.
6 Check
Let's check the new mean: if every number were 21.25, 4 of them would make 21.25 times 4.
That is 85, 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.