Data: 1, 3, 4, 6, 6. Remove 1. What happens to the mean and the…

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The question

Question: Data: 1, 3, 4, 6, 6.

Remove 1.

What happens to the mean and the median?

1 Mean before

We add all the numbers, from left to right.

So the total is 20.

There are 5 numbers.

The mean shares the total equally: 20 divided by 5.

So the mean is 4.

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.

4 is left in the middle.

So the median is 4.

3 Mean after

1 is gone, so the total loses 1.

20 minus 1 is 19.

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: 19 divided by 4.

19 does not share out evenly.

4 times 4 is 16.

3 is left, and 3 divided by 4 is 0.75.

So the new mean is 4.75.

4 Median after

Now the list in order, without 1.

We cross out the smallest and the biggest, one pair at a time.

Two numbers are left in the middle: 4 and 6.

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

4 plus 6 is 10.

10 divided by 2 is 5.

5 Compare

The mean went up from 4 to 4.75.

4.75 minus 4 is 0.75.

So it went up 0.75.

The median went up from 4 to 5.

5 minus 4 is 1.

So it went up 1.

Here 1 is not far from the other numbers.

So the mean moves only a little.

The median moves to the new middle of the list.

So the mean goes up 0.75, and the median goes up 1.

6 Check

Let's check the new mean: if every number were 4.75, 4 of them would make 4.75 times 4.

That is 19, 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.