Test scores: 70, 75, 80, 85, 90. Add 20. What happens to the mean…

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The question

Question: Test scores: 70, 75, 80, 85, 90.

Add 20.

What happens to the mean and the median?

1 Mean before

We add all the numbers, from left to right.

So the total is 400.

There are 5 numbers.

The mean shares the total equally: 400 divided by 5.

So the mean is 80.

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.

80 is left in the middle.

So the median is 80.

3 Mean after

20 comes in, so the total gets 20 more.

400 plus 20 is 420.

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: 420 divided by 6.

So the new mean is 70.

4 Median after

Now 20 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: 75 and 80.

The median is halfway between them: we add them and divide by 2.

75 plus 80 is 155.

155 divided by 2 is 77.5.

5 Compare

The mean went down from 80 to 70.

80 minus 70 is 10.

So it went down 10.

The median went down from 80 to 77.5.

80 minus 77.5 is 2.5.

So it went down 2.5.

20 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 down 10, and the median goes down 2.5.

6 Check

Let's check the new mean: if every number were 70, 6 of them would make 70 times 6.

That is 420, 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.