Charts, graphs and time series

(a) The tallest bar is Bus (9 students), so the mode is Bus.

(b) Walk = 8, Cycle = 3, so $8 - 3 = 5$ more.

(a) Count each place: castle 11, farm 7, museum 6. Check: $11 + 7 + 6 = 24$.

(b) The mode is the most common choice: castle.

(c) Label the horizontal axis with the three places and the vertical axis 0 to 12 in equal steps. Draw bars of height 11, 7 and 6 with equal widths.

Saturday has 4 symbols and each symbol stands for 2 hours.

$4 \times 2 = 8$ hours

(a) Months above 16°C: May, Jun, Jul, Aug, Sep, Oct, so 6 months (April is exactly 16°C, not above).

(b) May and October are both at 20°C.

Check each bar against the table.

Blue: 10 and 10. Green: 4 and 4. Brown: the table says 15 but the bar reaches 16.

So the brown bar is one too tall.

(a) Bars above the 1.4 line: years 4, 5, 6, 7 and 8, so 5 years.

(b) Years 5 and 6 both reach exactly 1.5 million.

(a) Tennis is 12 and football is 17. Cricket $= 40 - 12 - 17 = 11$, so draw the bar up to 11.

(b) $\frac{12}{40} = \frac{3}{10}$

(c) There are $5 + 4 = 9$ parts and 5 are large: $\frac{5}{9}$

(d) Divide both sides by 4: $5 \div 4 = 1.25$, so $1.25 : 1$

(a) Key: one square = 4 houses. January = $4 + 4 + 2 = 10$ (half a square is 2).

(b) $11 = 4 + 4 + 3$, so draw 2 full squares and $\frac{3}{4}$ of a square.

(c) February $= 4 + 4 + 1 = 9$, March $= 12$, April $= 11$.

May $= 60 - (10 + 9 + 12 + 11) = 60 - 42 = 18$

(a) Each circle is 12 cakes, so $24 \div 12 = 2$ circles for banana.

(b) Chocolate $5 \times 12 = 60$; vanilla $3\frac{1}{4} \times 12 = 39$; lemon $1\frac{1}{2} \times 12 = 18$; banana 24.

Year 7 total $= 60 + 39 + 18 + 24 = 141$, which is less than 150, so Year 8 sold more.

(a) Draw a bar up to 35 above May and a bar up to 20 above June, the same width as the others.

(b) The vertical scale goes up 5 cm per square, so half a square is $5 \div 2 = 2.5$ cm.

February's bar is at $15 + 2.5 = 17.5$ cm, not 15.5 cm.

Label the horizontal axis with the five days and the vertical axis with a linear scale from 0 to 10.

Draw bars of equal width: Monday 9, Tuesday 10, Wednesday 8, Thursday 6, Friday 3.

Put in the given values: total 198, without mattress 59, double total 45, single without 17, king total 83, king with 67.

King without: $83 - 67 = 16$. Single total: $198 - 45 - 83 = 70$, so single with $= 70 - 17 = 53$.

Double without: $59 - 17 - 16 = 26$, so double with $= 45 - 26 = 19$.

With mattress total: $198 - 59 = 139$ (check: $53 + 19 + 67 = 139$).

Each symbol is 8 games, so a quarter symbol is 2.

(a) Monday has 2 symbols: $2 \times 8 = 16$

(b) Tuesday: $8 + 8 + 6 = 22$. Wednesday: $8 + 2 = 10$. Difference: $22 - 10 = 12$

(c) Thursday: $\frac{1}{4} \times 32 = 8$, so draw 1 symbol. Friday: $32 - 8 = 24$, so draw 3 symbols.

Put in the given values: total 56, plastic total 32, blue total 14, blue plastic 5, red not plastic 8, red total 12.

Red plastic: $12 - 8 = 4$. Blue not plastic: $14 - 5 = 9$.

Black total: $56 - 12 - 14 = 30$. Black plastic: $32 - 4 - 5 = 23$.

Black not plastic: $30 - 23 = 7$. Not plastic total: $56 - 32 = 24$.

Monday shows $1\frac{1}{2}$ ovals for 18 eggs, so one oval is $18 \div 1.5 = 12$ eggs.

Tuesday: $24 \div 12 = 2$ ovals.

Wednesday: $27 \div 12 = 2\frac{1}{4}$ ovals (two whole ovals and a quarter).

Key: one oval represents 12 eggs.

Males: $30 - 20 = 10$. Two chose tennis, leaving 8 split equally: 4 cricket, 4 swimming.

Females: cricket $5 - 4 = 1$, swimming 11, so tennis $20 - 1 - 11 = 8$.

Totals: cricket 5, tennis $2 + 8 = 10$, swimming $4 + 11 = 15$ (sum 30).

Check the chart against good practice:

- The vertical axis has no label (e.g. 'Mark').
- The scale jumps from 0 straight to 71, so the bar heights mislead.
- Tom's bar is twice as wide as the other bars.

Any two of these earn the marks.

(a) January has 3 whole squares: $3 \times 8 = 24$

(b) $12 = 8 + 4$, so draw one whole square and a half square for April.

(c) February $3.5 \times 8 = 28$, March $2.5 \times 8 = 20$

Total $= 24 + 28 + 20 + 12 = 84$

(a) From the chart: red 8, white 10, yellow 5.

Blue $= 30 - 8 - 10 - 5 = 7$

(b) White has the most flowers (10), so the mode is white.

(a)(i) 3 circles: $3 \times 8 = 24$

(a)(ii) 2 circles and a quarter: $2 \times 8 + 2 = 18$

(b) Wednesday is 1 part and Thursday 8 parts, so 9 parts make 36: $36 \div 9 = 4$

Wednesday 4, Thursday $8 \times 4 = 32$

Draw half a circle for Wednesday ($4 = \frac{1}{2} \times 8$) and 4 full circles for Thursday ($32 \div 8 = 4$).

(a) Count each pet: dog 8, rabbit 3, cat 5, hamster 2 (total 18).

(b) Put the pets along the bottom, a frequency scale going up in 1s, and draw bars of heights 8, 3, 5 and 2.

(c) Dog has the tallest bar, so it is the most popular.

Spanish Year 11: $67 - 34 = 33$

French Year 11: $113 - 45 - 33 = 35$

Year 10 total: $207 - 113 = 94$

German Year 10: $94 - 33 - 34 = 27$

French total: $33 + 35 = 68$; German total: $27 + 45 = 72$ (check: $68 + 72 + 67 = 207$)

Draw a dual bar chart: days along the horizontal axis, hours (0 to 10) up the vertical axis.

For each day draw two bars side by side, e.g. Wednesday: Lena 6, Pavel 7.

Add a key showing which bar is Lena and which is Pavel, and label the vertical axis 'Hours'.

Total people: $30 + 10 + 50 = 90$

Each person: $360 \div 90 = 4°$

Coffee $30 \times 4 = 120°$, Hot chocolate $10 \times 4 = 40°$, Tea $50 \times 4 = 200°$

Draw each angle with a protractor and label each sector with the drink.

Use the tens-and-hundreds digits as the stem (16, 17, 18, 19) and the units as leaves.

16 | 0 2 8

17 | 2 2 3 7 8

18 | 0 0 3 4 6 8

19 | 1 7

Key: 16 | 0 means 160 cm. Check there are 16 leaves in total.

(a) The largest sector (150°) is Merit.

(b) Distinction is 105° for 7 students, so each student is $105 \div 7 = 15°$.

Total $= 360 \div 15 = 24$ students.

(a) The men bar for 2000 reaches 60.

(b) In 2010 the bar goes from 160 to 210: $210 - 160 = 50$ children.

(c) 2020: men and women together reach 200; children go from 200 to 280, so 80.

Ratio $80 : 200 = 2 : 5$

Use the tens digit as the stem and the units digit as the leaf, then put the leaves in order:

1 | 7 9

2 | 5 5 6 7 7 7 8 9

3 | 3 7 7

4 | 5 7

Key: 2 | 5 means 25 years

Total $= 90 + 105 + 105 = 300$ potatoes, so each potato is $360 \div 300 = 1.2°$.

Correct angles: $90 \times 1.2 = 108°$, $105 \times 1.2 = 126°$ twice. Salim's sectors are clearly not these sizes (one is a right angle), so they are wrong.

Also, the sectors are not labelled with the types of potato.

Frequencies read from the chart: 6, 4, 5, 8, 7, 5.

Total throws: $6+4+5+8+7+5 = 35$.

Each student: $35 \div 5 = 7$ throws.

Use the tens digit as the stem and the units digit as the leaf.

Put the leaves in order:

1 | 5 7 8 9 9

2 | 0 2 2 4 5 5 8

3 | 2 3 5

Key: 2 | 5 means 25 cm. Check there are 15 leaves.

Total goals: $50 + 45 + 25 = 120$

Each goal is $360 \div 120 = 3^\circ$.

City $50 \times 3 = 150^\circ$, Rovers $45 \times 3 = 135^\circ$, United $25 \times 3 = 75^\circ$

Draw and label each sector with a protractor.

Key: one square = 4 trees.

Apple 3 squares = 12, cherry $1\frac{1}{4}$ squares = 5, pear $1\frac{1}{2}$ squares = 6.

Plum $= 30 - (12 + 5 + 6) = 7$ trees.

$7 \div 4 = 1\frac{3}{4}$, so draw one full square and three quarters of a square.

(a) The desktop section in 2015 reaches 100.

(b) Laptops: $260 - 100 = 160$, $340 - 120 = 220$, $440 - 160 = 280$

Total $= 160 + 220 + 280 = 660$

(c) Tablets: $320 - 260 = 60$ in 2015 up to $780 - 440 = 340$ in 2017, a rise of 280. Tablets grew most.

(d) No: we know how many were sold, not the profit on each item.

Total fans: $11 + 17 + 8 = 36$

Each fan is $360° \div 36 = 10°$

Burger $11 \times 10 = 110°$, pie $17 \times 10 = 170°$, hot dog $8 \times 10 = 80°$

Check: $110 + 170 + 80 = 360°$. Draw each sector with a protractor and label it.

Each person is worth $360 \div 60 = 6°$

Peas: $24 \times 6 = 144°$; Carrots: $16 \times 6 = 96°$; Mushrooms: $20 \times 6 = 120°$

Check: $144 + 96 + 120 = 360°$. Draw the sectors with a protractor and label each one.

A dual bar chart works well.

- Put the four sports along the horizontal axis.
- Use a frequency scale up to 16 on the vertical axis, going up in equal steps (e.g. 2s).
- For each sport draw a boys bar and a girls bar side by side: Hockey 3, 6; Tennis 8, 14; Football 15, 7; Golf 9, 1.
- Add a key for boys and girls, and label both axes.

(a) 12 | 3 5 9

13 | 0 3 3 5 7 8

14 | 7 7 8 9

15 | 0 1

Key: 12 | 3 means 123 cm

(b) Heights over 140: 147, 147, 148, 149, 150, 151, which is 6 children.

$P = \frac{6}{15} = \frac{2}{5}$

Midpoints: 130, 150, 170, 190, 210

Plot (130, 12), (150, 28), (170, 30), (190, 22), (210, 8).

Join neighbouring points with straight lines. Do not join the last point back to the first.

Plot each frequency above the middle of its class: 2.5, 7.5, 12.5, 17.5, 22.5.

Points: $(2.5, 8), (7.5, 24), (12.5, 13), (17.5, 11), (22.5, 4)$

Join them in order with a ruler. Do not join the last point back to the first.

Adults: $160 - 85 = 75$. Adult romance: $33 - 19 = 14$. Child adventure: $76 - 34 = 42$.

Adult horror: $75 - 14 - 34 - 7 = 20$, so 20 children chose thriller.

Child horror: $85 - 19 - 42 - 20 = 4$

Total horror: $20 + 4 = 24$

German: $200 - 104 - 70 = 26$, so boys choosing German $= 26 - 18 = 8$

Boys: 90, of which 60 chose French and 8 chose German.

Boys choosing Spanish $= 90 - 60 - 8 = 22$

Look at the frequency axis: it goes 30 then 50, so 40 is missing and the scale is not even.

Points should be at class midpoints (5, 15, 25, 35, 45). The last point is at (50, 5) but should be at (45, 5).

(a) Black is a right angle, $90^\circ$, for 135 cars, so $1^\circ$ is $135 \div 90 = 1.5$ cars.

White: $80 \times 1.5 = 120$ cars.

(b) Total cars: $360 \times 1.5 = 540$

$P(\text{grey}) = \frac{50}{540} = \frac{5}{54}$

Check each point against the table at the class midpoints.

The point at 15 is plotted at 8, but the frequency is 6.

The first and last points are joined; a frequency polygon should not be closed.

The vertical axis also has no 'Frequency' label.

Things wrong with the graph:

1. The points are joined with curves; a time series should use straight lines.
2. The vertical axis starts at 6, which exaggerates the changes.

Other valid points: the first quarter of each year is missing, and it is not clear what 2, 3, 4 on the horizontal axis mean.

(a) Put the tens digits as stems and the units in order as leaves:

6 | 4 7 9 9
7 | 0 0 1 5 6 6 7
8 | 0 0 1 1 2 4 7
9 | 1 4

Key: 6 | 4 means 64 marks.

(b) Marks below 71: 64, 67, 69, 69, 70, 70, so 6 students failed.

$P(\text{fail}) = \frac{6}{20} = \frac{3}{10}$, but $\frac{1}{4} = \frac{5}{20}$, so Omar is wrong.

Plot each frequency above the middle of its class: 5, 15, 25 and 35.

Points: $(5, 14)$, $(15, 18)$, $(25, 26)$, $(35, 12)$

Join them in order with straight lines. Do not join the last point back to the first.

(a) The highest frequency is 10, in the class $160 < h \le 170$

(b) The points are plotted at the ends of the classes instead of the middles (135, 145, ...).

The last point is joined back to the first, which a frequency polygon should not do.

(a) The largest frequency is 10, in the class $160 < h \le 170$.

(b) The points are plotted at the start of each class instead of at the midpoints (135, 145, …).

The last point has been joined back to the first, but a frequency polygon is not a closed shape.

School A: $x = (360 - 60) \div 2 = 150°$

Tigers at A $= \frac{150}{360} \times 480 = 200$

School B: tigers angle $= 360 - 160 - 110 = 90°$

Tigers at B $= \frac{90}{360} \times 760 = 190$

$200 \ne 190$, so Henry is not correct.