Worked examples
The question
Question: Data: 4, 6, 7, 9, 29.
Remove 29.
What happens to the mean and the median?
1 Mean before
We add all the numbers, from left to right.
total = 4 + 6 + 7 + 9 + 29
So the total is 55.
total = 55
There are 5 numbers.
The mean shares the total equally: 55 divided by 5.
mean = 555
mean = total ÷ how many
So the mean is 11.
mean = 11
2 Median before
The median is the middle number.
For it, the numbers must be in order.
They already are, smallest first.
in order : 4, 6, 7, 9, 29
We cross out the smallest and the biggest, one pair at a time.
And the next pair.
7 is left in the middle.
So the median is 7.
median = 7
median = the middle one, in order
3 Mean after
29 is gone, so the total loses 29.
55 minus 29 is 26.
new total = 55 − 29 = 26
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 26 ÷ 5 = 5.2 ✗
We share the new total equally: 26 divided by 4.
new mean = 264
26 does not share out evenly.
4 times 6 is 24.
2 is left, and 2 divided by 4 is 0.5.
4 × 6 = 24, 2 left
2 ÷ 4 = 0.5
So the new mean is 6.5.
new mean = 6.5
4 Median after
Now the list in order, without 29.
in order : 4, 6, 7, 9
We cross out the smallest and the biggest, one pair at a time.
Two numbers are left in the middle: 6 and 7.
The median is halfway between them: we add them and divide by 2.
6 plus 7 is 13.
13 divided by 2 is 6.5.
new median = 6 + 72 = 132 = 6.5
5 Compare
The mean went down from 11 to 6.5.
11 minus 6.5 is 4.5.
So it went down 4.5.
mean : 11 → 6.5: 11 − 6.5 = down 4.5
The median went down from 7 to 6.5.
7 minus 6.5 is 0.5.
So it went down 0.5.
median : 7 → 6.5: 7 − 6.5 = down 0.5
29 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.
29 is far from the others: it pulls the mean
the median only looks at the middle
So the mean goes down 4.5, and the median goes down 0.5.
= mean: down 4.5, median: down 0.5
6 Check
Let's check the new mean: if every number were 6.5, 4 of them would make 6.5 times 4.
That is 26, the new total.
So it is right!
6.5 × 4 = 26
Let's look back.
First: mean before.
Then: median before.
Then: mean after.
Then: median after.
Then: compare.
Last: check.
Answer = mean: down 4.5, median: down 0.5
▶ Watch it solved
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.
total = 70 + 75 + 80 + 85 + 90
So the total is 400.
total = 400
There are 5 numbers.
The mean shares the total equally: 400 divided by 5.
mean = 4005
mean = total ÷ how many
So the mean is 80.
mean = 80
2 Median before
The median is the middle number.
For it, the numbers must be in order.
They already are, smallest first.
in order : 70, 75, 80, 85, 90
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.
median = 80
median = the middle one, in order
3 Mean after
20 comes in, so the total gets 20 more.
400 plus 20 is 420.
new total = 400 + 20 = 420
One more number: now there are 6.
how many = 5 + 1 = 6
Careful!
We have 6 numbers now, not 5.
So we share by 6.
Careful: not 420 ÷ 5 = 84 ✗
We share the new total equally: 420 divided by 6.
new mean = 4206
So the new mean is 70.
new mean = 70
4 Median after
Now 20 goes into the list, in its place in order.
in order : 20, 70, 75, 80, 85, 90
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.
new median = 75 + 802 = 1552 = 77.5
5 Compare
The mean went down from 80 to 70.
80 minus 70 is 10.
So it went down 10.
mean : 80 → 70: 80 − 70 = down 10
The median went down from 80 to 77.5.
80 minus 77.5 is 2.5.
So it went down 2.5.
median : 80 → 77.5: 80 − 77.5 = 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.
20 is far from the others: it pulls the mean
the median only looks at the middle
So the mean goes down 10, and the median goes down 2.5.
= mean: down 10, median: 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!
70 × 6 = 420
Let's look back.
First: mean before.
Then: median before.
Then: mean after.
Then: median after.
Then: compare.
Last: check.
Answer = mean: down 10, median: down 2.5
▶ Watch it solved
The question
Question: Data: 2, 3, 3, 4, 5, 7.
Add 4.
What happens to the mean and the median?
1 Mean before
We add all the numbers, from left to right.
total = 2 + 3 + 3 + 4 + 5 + 7
So the total is 24.
total = 24
There are 6 numbers.
The mean shares the total equally: 24 divided by 6.
mean = 246
mean = total ÷ how many
So the mean is 4.
mean = 4
2 Median before
The median is the middle number.
For it, the numbers must be in order.
They already are, smallest first.
in order : 2, 3, 3, 4, 5, 7
We cross out the smallest and the biggest, one pair at a time.
And the next pair.
Two numbers are left in the middle: 3 and 4.
The median is halfway between them: we add them and divide by 2.
3 plus 4 is 7.
7 divided by 2 is 3.5.
median = 3 + 42 = 72 = 3.5
median = the middle one, in order
3 Mean after
4 comes in, so the total gets 4 more.
24 plus 4 is 28.
new total = 24 + 4 = 28
One more number: now there are 7.
how many = 6 + 1 = 7
Careful!
We have 7 numbers now, not 6.
So we share by 7.
Careful: not 28 ÷ 6 ✗
We share the new total equally: 28 divided by 7.
new mean = 287
So the new mean is 4.
new mean = 4
4 Median after
Now 4 goes into the list, in its place in order.
in order : 2, 3, 3, 4, 4, 5, 7
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 new median is 4.
new median = 4
5 Compare
The mean was 4, and it is still 4.
It stays the same.
mean : 4 → 4: stays the same
The median went up from 3.5 to 4.
4 minus 3.5 is 0.5.
So it went up 0.5.
median : 3.5 → 4: 4 − 3.5 = up 0.5
4 is exactly the mean.
Adding the fair share does not change the fair share.
4 = the mean: the fair share does not change
So the mean stays the same, and the median goes up 0.5.
= mean: stays the same, median: up 0.5
6 Check
Let's check the new mean: if every number were 4, 7 of them would make 4 times 7.
That is 28, the new total.
So it is right!
4 × 7 = 28
Let's look back.
First: mean before.
Then: median before.
Then: mean after.
Then: median after.
Then: compare.
Last: check.
Answer = mean: stays the same, median: up 0.5
▶ Watch it solved
