Level them off
Ana has 2 stickers, Ben has 6 and Cal has 4. The three piles are different heights. Now suppose they make it fair, so that everyone has the same number. How many does each of them get?
Ben has the most, so he passes stickers to Ana, who has the fewest. When Ben gives Ana 2 stickers, Ben is down to 4, Ana is up to 4, and Cal already has 4. All three piles are level at 4.
The dashed line at 4 is where the three bars settle once the stickers are passed around: Ben's bar is 2 above it and Ana's is 2 below it.
Nothing is added or lost
Passing stickers around does not change how many there are. Before, there were 2 + 6 + 4 = 12. After, there are 4 + 4 + 4 = 12. The stickers are only shared out differently.
The level they settle at has a name. It is the mean of 2, 6 and 4, and the mean is 4.
After the sharing, Ana, Ben and Cal each have 4 stickers: 12 in all, the same as before.
Add, then share
Passing things along by eye works for small numbers, but there is a method that always works. Put everything into one pile, then share the pile out equally.
The pile is the total: 2 + 6 + 4 = 12. Sharing it equally between the 3 people is a division: 12 ÷ 3 = 4. So the mean is the total divided by how many numbers there are.
12 ÷ 3 asks how many each friend gets: hand the pile round in turn, one to each friend per round
Deal out the whole pile, one to each friend in turn
The 12 stickers are drawn as 12 counters in one pile. Deal them out one at a time to the 3 friends, and each friend ends up with 4.
The three amounts before sharing: 2, 6 and 4, a total of 12.
Where the mean sits
The mean is never bigger than the biggest number, and never smaller than the smallest, because sharing takes from the big piles and gives to the small ones. For 2, 6 and 4, the mean 4 lies between 2 and 6.
It does not have to be one of the numbers. The mean of 3, 5 and 10 is (3 + 5 + 10) ÷ 3 = 18 ÷ 3 = 6, and nobody had 6 to start with.
Two ways to go wrong
The total, 12, is not the mean. It is the whole pile before it has been shared out, so it still has to be divided by 3.
The tallest bar, 6, is not the mean either. It is only Ben's pile. The mean uses every number, and it sits where all the bars level off.
Worked example: Attendance Table: The Mean by Leveling
Question A table shows how many children came to a reading club in one week: 18 on Monday, 22 on Tuesday, 20 on Wednesday, 17 on Thursday and 23 on Friday. (a) Find the mean number of children for a day by leveling the five days. (b) How many children came in the whole week?
1.Leveling means moving children from the bigger days to the smaller days until every day is the same. Try 20, the number on Wednesday.
Try to make every day 20, the number on Wednesday. 2.Tuesday has 2 more than 20 and Monday has 2 fewer. Move 2 from Tuesday to Monday, and both days are 20.
Move 2 children from Tuesday to Monday. Both days are now 20. 3.Friday has 3 more than 20 and Thursday has 3 fewer. Move 3 from Friday to Thursday, and both days are 20.
Move 3 children from Friday to Thursday. Both days are now 20. 4.(a) Every day is now 20, so the mean is 20 children.
(a) Every day is 20, so the mean is 20 children. 5.(b) Leveling does not change the total, so 5 × 20 = 100 children came in the week. Check: 18 + 22 + 20 + 17 + 23 = 100.
(b) 5 × 20 = 100 children, and 18 + 22 + 20 + 17 + 23 = 100.
Answer: (a) 20 children; (b) 100 children
Common mistakes
- Taking the number in the middle of the table as the mean without checking. Wednesday's 20 is the mean here only because the days above 20 have 5 extra children and the days below 20 are 5 short.
- Adding children to a small day without taking them from a big day, so that Monday becomes 20 and Tuesday stays 22. Every child moved to one day must leave another day, or the total changes.