The Lumenary · Grades 3–5

Data displays and the mean as a fair share

Name: ______________________

Date: ______________________

You will learn to

Draw and read bar graphs with scales, and line plots of measurements in fractions of a unit
Answer "how many more" and "how many in all" questions from displays, and check the scale before comparing bars
Recognise a misleading graph whose axis does not start at zero, and explain why it exaggerates differences
Understand the mean as the amount each person gets when the total is shared out equally, and use mean × number of values = total

1.
On a bar graph, each grid line stands for 5 children. The bar for "football" reaches exactly the 4th grid line. How many children chose football?
2.
On another bar graph the grid lines are labelled 0, 2, 4, 6, 8, … The bar for "cycling" stops exactly halfway between the lines for 8 and 10. How many children cycle?
3.
A fruit seller in Accra makes two pictographs. In the mango chart, one picture stands for 5 mangoes, and the row has 6 pictures. In the banana chart, one picture stands for 2 bananas, and the row has 9 pictures. How many more mangoes than bananas did she sell?
4.
Nadia scored 6, 9, 3 and 10 points in four basketball games. If her points were shared out equally over the four games, how many points would each game get? (This is the mean.)
5.
Five friends have a mean of 8 stickers each. Four of them have 6, 9, 5 and 10 stickers. How many stickers does the fifth friend have?
6.
A bar graph about two buses has an axis that starts at 90. The bar for Bus A (100 passengers) looks twice as tall as the bar for Bus B (95 passengers). What is true?
- A.Bus A carried twice as many passengers as Bus B.
- B.Bus A carried only 5 more passengers — the raised axis makes the difference look big.
- C.Bus B carried more passengers than Bus A.
7.
Rahul says: "The mean of 2, 4 and 9 can't be 5, because 5 isn't one of the numbers." Is he right?
- A.Yes — the mean must be one of the data values.
- B.No — the mean is the fair share, 15 ÷ 3 = 5, and it need not be one of the values.
- C.No — the mean is the middle number, 4.
8.
Help Margaret. Complete fair_share(towers) so it returns the mean (total ÷ number of towers), and can_level(towers) so it returns True exactly when the towers can be levelled with whole cubes.
Computational — Practice
9.
A list of data values are all between 3 and 9 (some may be equal to 3 or 9). Which of these could not be the mean?
- A.10
- B.3½
- C.9
- D.6
10.
Lean proved fair_share: if moves turn the towers d into level towers of height m, then d.length * m = d.sum. (d.length is the number of towers and d.sum is the total number of cubes.) Which sentence says the same thing?
- A.Fair share × number of towers = total number of cubes.
- B.Every set of towers can be levelled.
- C.The fair share equals the tallest tower.
- D.The fair share is always one of the towers.

Answer key — Data displays and the mean as a fair share

1.
Each grid line stands for 5 children, and the bar reaches the 4th line, so 20 children chose football: $4 \times 5 = 20$ .
2.
The lines go up in twos. Halfway between 8 and 10 is 9, so 9 children cycle.
3.
Mangoes: $6 \times 5 = 30$ . Bananas: $9 \times 2 = 18$ . She sold $30 - 18 =$ 12 more mangoes. The banana row has more pictures, but fewer bananas — always check the key.
4.
The total is $6 + 9 + 3 + 10 = 28$ points. Shared equally over 4 games: $28 \div 4 =$ 7 points per game. Check: $7 \times 4 = 28$ , the total.
5.
With a mean of 8 for 5 friends, the total is $5 \times 8 = 40$ stickers. The four friends have $6 + 9 + 5 + 10 = 30$ . So the fifth friend has $40 - 30 =$ 10 stickers.
6.
The bars are drawn from 90, so they are $100 - 90 = 10$ and $95 - 90 = 5$ units tall: one looks twice the other. But the real numbers are 100 and 95, a difference of only 5 passengers. An axis that does not start at zero exaggerates differences.
7.
The total is $2 + 4 + 9 = 15$ , and $15 \div 3 = 5$ . So the mean is 5, even though no one started with 5. Rahul is not right: the mean is the fair share, and it need not be one of the data values.
8.
```
def fair_share(towers):
    return sum(towers) / len(towers)


def can_level(towers):
    return sum(towers) % len(towers) == 0
```
For 3, 7, 2, 6 the total 18 leaves remainder 2 when divided by 4, so the towers cannot be levelled with whole cubes, and the fair share is 4.5.
9.
The mean is the level you reach by sharing the cubes out equally. That level can never be taller than the tallest tower (9): level towers of 10 would need more cubes than all the towers have together. In the same way it can never be shorter than the shortest tower (3). So 10 is impossible; 3½, 6 and 9 are all possible means.
10.
d.length * m = d.sum reads: (number of towers) × (level) = (total number of cubes). That is fair share × number of towers = total, which is the same as "fair share = total ÷ number of towers" whenever the towers can be levelled.