Step by step
The question
Question: Data: 1, 2, 3, 4, 90.
Remove 3.
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.
3 is left in the middle.
So the median is 3.
3 Mean after
3 is gone, so the total loses 3.
100 minus 3 is 97.
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: 97 divided by 4.
97 does not share out evenly.
4 times 24 is 96.
1 is left, and 1 divided by 4 is 0.25.
So the new mean is 24.25.
4 Median after
Now the list in order, without 3.
We cross out the smallest and the biggest, one pair at a time.
Two numbers are left in the middle: 2 and 4.
The median is halfway between them: we add them and divide by 2.
2 plus 4 is 6.
6 divided by 2 is 3.
5 Compare
The mean went up from 20 to 24.25.
24.25 minus 20 is 4.25.
So it went up 4.25.
The median was 3, and it is still 3.
It stays the same.
The mean moved more than the median.
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 4.25, and the median stays the same.
6 Check
Let's check the new mean: if every number were 24.25, 4 of them would make 24.25 times 4.
That is 97, 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.