The data are 10, 12, 13, 15 and 60. The value 60 is removed. How do… Step by Step

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The question

Question: Data: 10, 12, 13, 15, 60.

Remove 60.

What happens to the mean and the median?

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Mean before

We add all the numbers, from left to right.

total = 10 + 12 + 13 + 15 + 60

So the total is 110.

total = 110

There are 5 numbers.

The mean shares the total equally: 110 divided by 5.

mean = 1105

Mean: the total divided by how many.

mean = total ÷ how many

So the mean is 22.

mean = 22

2 Median before

The median is the middle number.

median = the middle one, in order

For it, the numbers must be in order.

They already are, smallest first.

in order : 10, 12, 13, 15, 60

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

And the next pair.

13 is left in the middle.

So the median is 13.

median = 13

3 Mean after

60 is gone, so the total loses 60.

110 minus 60 is 50.

new total = 110 − 60 = 50

One number less: now there are 4.

how many = 5 − 1 = 4

Careful!

We have 4 numbers now, not 5.

So we share by 4.

Careful: not 50 ÷ 5 = 10 ✗

We share the new total equally: 50 divided by 4.

new mean = 504

50 does not share out evenly.

4 times 12 is 48.

2 is left, and 2 divided by 4 is 0.5.

4 × 12 = 48, 2 left

2 ÷ 4 = 0.5

So the new mean is 12.5.

new mean = 12.5

4 Median after

Now the list in order, without 60.

in order : 10, 12, 13, 15

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

Two numbers are left in the middle: 12 and 13.

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

12 plus 13 is 25.

25 divided by 2 is 12.5.

new median = 12 + 132 = 252 = 12.5

5 Compare

The mean went down from 22 to 12.5.

22 minus 12.5 is 9.5.

So it went down 9.5.

mean : 22 → 12.5: 22 − 12.5 = down 9.5

The median went down from 13 to 12.5.

13 minus 12.5 is 0.5.

So it went down 0.5.

median : 13 → 12.5: 13 − 12.5 = down 0.5

60 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.

60 is far from the others: it pulls the mean

the median only looks at the middle

So the mean goes down 9.5, and the median goes down 0.5.

mean: down 9.5, median: down 0.5

6 Check

Let's check the new mean: if every number were 12.5, 4 of them would make 12.5 times 4.

That is 50, the new total.

So it is right!

12.5 × 4 = 50

7 Pick the choice

Our answer is mean: down 9.5, median: down 0.5.

That is choice D.

D) mean: down 9.5, median: down 0.5

8 Why not the others

A is mean: down 60, median: down 0.5.

Wrong: the mean does not go down by the value taken away.

A) mean: down 60, median: down 0.5 ✗ the mean does not go down by the value taken away

B is mean: down 9.5, median: no change.

Wrong: with 4 values the median is halfway between 12 and 13.

B) mean: down 9.5, median: no change ✗ with 4 values the median is halfway between 12 and 13

C is mean: down 0.5, median: down 9.5.

Wrong: swapped: the mean moves much more.

C) mean: down 0.5, median: down 9.5 ✗ swapped: the mean moves much more

Let's look back.

First: mean before.

Then: median before.

Then: mean after.

Then: median after.

Then: compare.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer