Misleading Statistical Diagrams

Spot why a statistical diagram gives a wrong picture of its data, explain it with numbers measured off the drawing, and redraw the diagram honestly.

Before you start

In a bar graph the length of each bar stands for its value, so the value axis starts at 0. A line graph may start above 0 only with a zigzag break drawn at the bottom of the axis.
In a pie chart the angles at the centre add up to $360^\circ$ and the percentages add up to $100\%$ ; a sector of $p\%$ has an angle of $\frac{p}{100} \times 360^\circ$ .
Percentage increase $= \frac{\text{increase}}{\text{original value}} \times 100\%$ . From 50 to 60 is an increase of $\frac{10}{50} \times 100\% = 20\%$ .

How a Diagram Can Mislead

We read a statistical diagram with our eyes before we read its numbers: we compare the heights of bars, the sizes of symbols and the sizes of sectors. A diagram is fair when what the eye sees is in the same proportion as the numbers. A bar for 40 is drawn twice as tall as a bar for 20; a sector for $30\%$ is twice the size of a sector for $15\%$ .

A diagram misleads when its drawing gives a picture that the numbers do not support. This may happen by accident, or on purpose, for example in an advertisement or a poster that wants to make a result look better. Every number printed on a diagram can be correct and the diagram still mislead.

In this lesson each misleading diagram is drawn beside an honest version of the same data, so that you can see the difference.

A fair statistical diagram

Bar graph: the value axis starts at 0; the numbers on the axis go up in equal steps; every bar has the same width.

Line graph: the value axis starts at 0, or a zigzag break is drawn where it starts above 0; the times along the horizontal axis go up in equal steps.

Pictogram: every symbol is the same size and stands for the same amount; a larger amount is shown by more symbols, not bigger ones.

Pie chart: it is drawn flat; its percentages add up to $100\%$ and its angles to $360^\circ$ .

Every diagram: its title and any claim made with it say only what the numbers show.

Explaining why a diagram misleads

1. Read the true values from the numbers on the diagram.

2. Measure what the drawing shows: how many grid squares tall each bar is, how many times as wide or as tall a symbol is, what angle a sector is drawn with.

3. Compare the two in a sentence with numbers, for example: "B's bar looks 3 times as tall as A's, but B's value is only $4\%$ more than A's."

A Value Axis That Does Not Start at 0

The height of a bar stands for its value only when the bar is measured from 0. If the value axis starts at 196, every bar loses its bottom 196, and only the tops of the bars are left to compare. A small difference then looks large.

A line graph may start above 0 to show small changes in detail, but a zigzag break must be drawn at the bottom of the axis to warn the reader, and the scale must still be read with care. An axis that is cut off with no break is misleading.

Example 1: Two bubble tea shops

Shop A and Shop B sell bubble tea in the same mall. Last Saturday Shop A sold 200 cups and Shop B sold 208 cups. Shop B put up Graph 1 in its window. Graph 2 shows the same numbers on an axis that starts at 0.

(a) In Graph 1, how many times as tall as Shop A's bar is Shop B's bar?

(b) What percentage more cups did Shop B sell than Shop A?

Solution

(a) The value axis of Graph 1 starts at 196, and each grid square is 2 cups.

Shop A's bar goes from 196 up to 200: $\frac{200 - 196}{2} = 2$ squares tall.

Shop B's bar goes from 196 up to 208: $\frac{208 - 196}{2} = 6$ squares tall.

$6 \div 2 = 3$ , so Shop B's bar is 3 times as tall as Shop A's.

(b) Shop B sold $208 - 200 = 8$ more cups.

Percentage more $= \frac{8}{200} \times 100\% = 4\%$

(c) Graph 1 starts its value axis at 196, not at 0. Only the parts of the bars above 196 are drawn, so Shop B's bar looks 3 times as tall as Shop A's, as if Shop B sold 3 times as many cups. In fact it sold only $4\%$ more.

In Graph 2, drawn from 0, the bars are almost the same height, $\frac{208}{200} = 1.04$ times, which is the true picture.

Shop B's bar looks 3 times as tall as Shop A's, but Shop B sold only 8 more cups, $4\%$ more. Graph 1 misleads because its value axis starts at 196, not 0.

Example 2: A line graph cut off with no break

Each year the school canteen asks its students whether they are satisfied with the food. The canteen's newsletter printed Graph 1. Graph 2 shows the same data.

(a) By how much did the percentage of satisfied students rise from 2021 to 2025?

(b) In Graph 1, how many grid squares above the bottom of the graph are the points for 2021 and 2025? How much of the height of the graph does the line climb?

Solution

(a) From the graph, $87\%$ were satisfied in 2021 and $89\%$ in 2025.

The rise is $89\% - 87\% = 2$ percentage points (the difference between two percentages is measured in percentage points).

As a percentage of the starting value, the rise is $\frac{2}{87} \times 100\% \approx 2.3\%$ (to 1 decimal place).

(b) Graph 1's axis starts at 86 and each square is $1\%$ . The 2021 point is $87 - 86 = 1$ square up; the 2025 point is $89 - 86 = 3$ squares up.

The graph is $90 - 86 = 4$ squares tall, and the line climbs $3 - 1 = 2$ of them: half the height of the graph.

(c) The axis starts at 86 with no break, so the reader is not warned that the bottom 86 has been cut off. The 2025 point is drawn 3 times as high as the 2021 point and the line climbs half the height of the graph, which looks like a large rise.

In Graph 2, drawn from 0, the line climbs only $\frac{2}{100}$ of the height of the graph and looks almost flat. The heading "Huge rise" is not supported: the rise was 2 percentage points in four years.

If the axis is to start at 86 to show the change in detail, the graph needs a zigzag break at the bottom of the axis and a heading that states the data, such as "Students satisfied with the food, 2021 to 2025".

The rise is only 2 percentage points (about $2.3\%$ ), but Graph 1 cuts its axis at 86 with no break, so the line climbs half the graph's height and the heading calls it huge.

Unequal Steps on an Axis

The numbers on an axis must go up in equal steps, so that equal distances stand for equal amounts. If the numbers go up in growing steps, such as 0, 10, 20, 50, 100, while the gridlines stay the same distance apart, the large values are squeezed together and the small ones are stretched. The differences between the bars then look much smaller than they are.

Example 3: Members of four CCAs

Mr Tan counted the members of four CCAs: Chess 10, Choir 20, Football 50 and Band 100. He drew Graph 1. Graph 2 shows the same data.

(a) Write down the steps between the numbers on the value axis of Graph 1.

(b) In Graph 1, how many times as tall as the Chess bar is the Band bar? How many times as many members does Band have as Chess?

Solution

(a) The numbers are 0, 10, 20, 50, 100. The steps are $10 - 0 = 10$ , $20 - 10 = 10$ , $50 - 20 = 30$ and $100 - 50 = 50$ . They are not equal, yet the gridlines are equally spaced.

(b) In Graph 1 the Chess bar reaches the first gridline (1 square) and the Band bar the fourth (4 squares). The Band bar is 4 times as tall as the Chess bar.

Band has $\frac{100}{10} = 10$ times as many members as Chess.

In the same way, the Football bar is 3 squares and the Choir bar 2, so Football looks $\frac{3}{2} = 1.5$ times Choir; in fact Football has $\frac{50}{20} = 2.5$ times as many members.

(c) Each square of Graph 1 stands for a different amount (10, 10, 30, then 50), so the heights of the bars are not in proportion to the numbers. Band looks only 4 times as big as Chess when it is 10 times as big. Graph 2 uses equal steps of 10, and its Band bar is 10 times as tall as its Chess bar.

The steps are 10, 10, 30, 50. The Band bar looks 4 times as tall as the Chess bar, but Band has 10 times as many members; Graph 1 misleads because its axis is not in equal steps.

Bars of Unequal Width

The eye judges a bar by how much space it covers, its area, and not only by its height. When every bar has the same width, the area of each bar is in proportion to its height, which is its value. A bar drawn wider than the others covers more space and looks bigger than its value.

Example 4: Voting for a class outing

The 40 students of Class 1E voted for a class outing: Zoo 14, Beach 10, Science Centre 9, Bowling 7. Hui Min, who wanted the beach, drew Graph 1. Graph 2 shows the same data.

(a) Which outing received the most votes?

(b) In Graph 1 the Beach bar is twice as wide as the others. Taking the width of an ordinary bar as 1 unit and each vote as 1 unit of height, find the area of the Zoo bar and of the Beach bar. How many times the area of the Zoo bar is the Beach bar?

Solution

(a) The Zoo has the tallest bar, 14 votes.

(b) Zoo bar: $1 \times 14 = 14$ square units.

Beach bar: $2 \times 10 = 20$ square units.

$\frac{20}{14} \approx 1.4$ , so the Beach bar covers about 1.4 times the area of the Zoo bar.

(c) The Beach bar is twice as wide as the others, so it covers more of the graph than any other bar, about 1.4 times the Zoo bar. It looks like the most popular choice, but the Beach had only 10 votes against the Zoo's 14, that is $\frac{10}{14} \approx 0.7$ of the Zoo's votes. In Graph 2 every bar is the same width, and the Zoo bar is clearly the biggest.

The Zoo received the most votes, but the Beach bar, drawn twice as wide, covers about 1.4 times the area of the Zoo bar; Graph 1 misleads because its bars are not all the same width.

Pictograms with Enlarged Symbols

In a pictogram each symbol stands for the same amount, so every symbol must be the same size, and a larger amount is shown by more symbols.

Sometimes one symbol is drawn for each category and made bigger for a bigger amount. If a symbol is drawn twice as wide and twice as tall, it covers $2 \times 2 = 4$ times the area. It looks 4 times as big, not twice as big.

Enlarging a symbol

A symbol drawn $k$ times as wide and $k$ times as tall covers $k \times k = k^{2}$ times the area. Twice as wide and tall gives 4 times the area; 3 times as wide and tall gives 9 times the area.

Example 5: Library loans

A school library lent 3000 books in 2024 and 6000 books in 2025. Its poster showed Graph 1, in which the 2025 book is drawn twice as wide and twice as tall as the 2024 book. Graph 2 shows the same data.

(a) How many times as many books were lent in 2025 as in 2024?

(b) How many times the area of the 2024 symbol is the 2025 symbol in Graph 1?

Solution

(a) $\frac{6000}{3000} = 2$ , so twice as many books were lent in 2025.

(b) The 2025 symbol is 2 times as wide and 2 times as tall, so its area is $2 \times 2 = 4$ times the area of the 2024 symbol.

(c) The eye compares the space each symbol covers. The 2025 book covers 4 times the space of the 2024 book, so loans look as if they had grown 4 times, when they only doubled.

Graph 2 uses symbols of one size, each standing for 1000 books: 3 symbols for 2024 and 6 for 2025. Twice the loans are shown by twice as many symbols.

Loans doubled, but the 2025 symbol covers 4 times the area of the 2024 symbol, so the growth looks 4 times. A fair pictogram uses symbols of one size: 3 for 2024 and 6 for 2025.

Tilted (3-D) Pie Charts

A pie chart drawn as a tilted disc, to look solid, is squashed from top to bottom. Its angles at the centre are no longer the true angles: sectors at the front and back are drawn with wider angles, and sectors at the sides with narrower ones. The sectors at the front also have the thick side of the disc drawn below them, which makes them look bigger.

A pie chart is read by comparing its sectors, so it must be drawn flat, with every angle at the centre equal to its true angle.

Example 6: How students travel to school

Graph 1 and Graph 2 show how the students of a school travel to school: Bus $40\%$ , Walk $20\%$ , MRT $30\%$ , Car $10\%$ . Graph 1 is drawn as a tilted disc.

Siti looks at Graph 1 and says that more students walk than take the MRT.

(a) Is Siti right?

(b) On Graph 1, the angle at the centre of the Walk sector measures about $111^\circ$ and that of the MRT sector about $69^\circ$ . What should these two angles be?

Solution

(a) Walk is $20\%$ and MRT is $30\%$ , so more students take the MRT. Siti is wrong.

(b) Walk: $\frac{20}{100} \times 360^\circ = 72^\circ$

MRT: $\frac{30}{100} \times 360^\circ = 108^\circ$

(c) The Walk sector is at the front of the tilted disc, where the squashing widens it: it is drawn with about $111^\circ$ in place of $72^\circ$ , and the thick side of the disc is drawn under it. The MRT sector is at the side, where the squashing narrows it: about $69^\circ$ in place of $108^\circ$ .

So on the drawing the Walk sector's angle is $\frac{111}{69} \approx 1.6$ times the MRT sector's, when Walk is really only $\frac{20}{30} = \frac{2}{3}$ of MRT. In the flat Graph 2 the angles are $72^\circ$ and $108^\circ$ , and the MRT sector is clearly the bigger.

Siti is wrong: $30\%$ take the MRT and $20\%$ walk. The tilt draws Walk ( $72^\circ$ ) at about $111^\circ$ with the disc's side below it, and MRT ( $108^\circ$ ) at about $69^\circ$ .

Percentages That Do Not Add Up to 100%

The sectors of a pie chart are the parts of one whole, so their percentages must add up to $100\%$ and their angles to $360^\circ$ . Check this before reading any pie chart.

If the percentages add up to more than $100\%$ , the people asked could give more than one answer. The data are then not parts of one whole, and a pie chart cannot show them; a bar graph can. If they add up to less than $100\%$ , part of the whole has been left out. (A total of $99\%$ or $101\%$ can come from rounding each percentage to a whole number; a bigger difference cannot.)

Example 7: Favourite hawker foods

A survey asked 200 students which of three hawker foods they like. Each student could choose one or more. Graph 1 is a pie chart of the results; Graph 2 is a bar graph of the same results.

(a) Add up the percentages in Graph 1.

(b) What angle should a sector of $45\%$ have? A pie chart can only share out the whole circle, so Graph 1 has been drawn with its sectors in the ratio $45 : 35 : 30$ . What angle is the Chicken rice sector drawn with?

Solution

(a) $45\% + 35\% + 30\% = 110\%$ , which is more than $100\%$ . The pie chart is wrong.

(b) A sector of $45\%$ should have an angle of $\frac{45}{100} \times 360^\circ = 162^\circ$ .

Graph 1 shares $360^\circ$ out in the ratio $45 : 35 : 30$ , which has $45 + 35 + 30 = 110$ parts.

Chicken rice is drawn with $\frac{45}{110} \times 360^\circ \approx 147^\circ$ (to the nearest degree), which is only $\frac{147}{360} \times 100\% \approx 41\%$ of the circle, though it is labelled $45\%$ .

(c) Some students chose more than one food, so the three groups overlap: they are not parts of one whole of 200 students, and their percentages add up to more than $100\%$ . No pie chart can draw every sector its true size. A bar graph shows each percentage as a separate bar measured from 0, so it can show these data truthfully.

The percentages add up to $110\%$ . The $45\%$ sector should be $162^\circ$ but is drawn at about $147^\circ$ ; the answers overlap, so the data are not parts of one whole, and a bar graph should be used.

Unequal Time Steps on a Line Graph

Along the horizontal axis of a line graph, equal distances must stand for equal lengths of time. If some years or months are left out but the points are still drawn equally spaced, the slope of the line changes: a step that covers a long time looks just as long as a step that covers a short time. A steady change can then look as if it sped up or slowed down.

Example 8: The photography club

The table shows the number of members of a school's photography club, counted every two years.

Members of the photography club
Year	2014	2016	2018	2020	2022	2024
Members	30	40	50	60	70	80

Ravi drew Graph 1 from four of the readings. Graph 2 shows every reading.

(a) How many years does each step along the horizontal axis of Graph 1 stand for?

(b) In Graph 1, how many squares does the line rise in each step? How many members did the club gain per year in each step?

Solution

(a) The steps are 2014 to 2020, 2020 to 2022 and 2022 to 2024: $2020 - 2014 = 6$ years, then 2 years, then 2 years. They are drawn the same width.

(b) Each square is 10 members. From 2014 to 2020 the line rises from 30 to 60, $\frac{60 - 30}{10} = 3$ squares; in each of the next two steps it rises $\frac{70 - 60}{10} = 1$ square and $\frac{80 - 70}{10} = 1$ square.

Members gained per year: $\frac{30}{6} = 5$ from 2014 to 2020, $\frac{10}{2} = 5$ from 2020 to 2022, and $\frac{10}{2} = 5$ from 2022 to 2024.

(c) The heading is not true: the club gained 5 members a year all the way through. Graph 1 draws a 6-year step and a 2-year step the same width, so the first step rises 3 squares and the later ones 1 square, and the line looks 3 times as steep before 2020 as after. In Graph 2 every step is 2 years and the line is straight, which shows the steady growth.

The steps are 6, 2 and 2 years; the line rises 3, 1 and 1 squares, but the club grew by 5 members a year throughout. Growth did not slow: Graph 1 misleads by drawing unequal times equally spaced.

A Title or Claim the Numbers Do Not Support

A diagram can be drawn correctly and still mislead through its words. Check every title, heading and claim against the numbers. "Most" means more than half. "Doubled" means 2 times as many. A "huge rise" must be large compared with the starting value.

Check also what is left out: a category missing from a pie chart, or a graph that shows only the part of a time period that looks good.

Example 9: Favourite drinks

Mei Ling asked the 40 students of her class to choose their favourite drink. She drew Graph 1. Graph 2 is the same bar graph with a different heading.

(a) What percentage of the class chose bubble tea?

(b) Is the heading of Graph 1 true? Explain.

Solution

(a) $\frac{14}{40} \times 100\% = 35\%$

(b) "Most students" means more than half of them, more than 20 of the 40. Only 14 chose bubble tea, and $40 - 14 = 26$ students, $65\%$ , chose another drink. The heading is not true, even though the bars are drawn correctly.

(c) Bubble tea was chosen by more students than any other drink, so a true heading is "Bubble tea is the most popular drink, chosen by $35\%$ of the class", or simply "Favourite drinks of 40 students".

$35\%$ chose bubble tea, which is less than half, so "most students" is false. A true heading: "Bubble tea is the most popular drink, chosen by $35\%$ of the class".

Checking and Redrawing a Diagram

Questions to ask of any statistical diagram

1. Does the value axis start at 0? If not, is a break drawn, and is it a line graph (a bar graph must start at 0)?

2. Do the numbers on the value axis go up in equal steps? Are the times along the horizontal axis equally spaced?

3. Are all the bars the same width, and all the symbols the same size?

4. Is a pie chart drawn flat, and do its percentages add up to $100\%$ ?

5. Do the title and any claim agree with the numbers? Is anything left out?

To redraw a misleading diagram honestly, keep the data and change only the drawing: start a bar graph's value axis at 0 (a line graph's at 0, or above 0 with a zigzag break), go up in equal steps, give every bar the same width and every symbol the same size, draw a pie chart flat, space the times equally, and write a title that says what the data are.

Example 10: Fried rice sales

Ravi's father has a stall at a hawker centre. It sold 150 plates of fried rice in January and 180 in June. His poster shows this graph.

(a) Find the increase in sales as a number and as a percentage.

(b) Explain two ways in which the graph is misleading, with numbers.

Solution

(a) Increase $= 180 - 150 = 30$ plates.

Percentage increase $= \frac{30}{150} \times 100\% = 20\%$

(b) First, the value axis starts at 120, not at 0. Each square is 10 plates, so the January bar is $\frac{150 - 120}{10} = 3$ squares tall and the June bar $\frac{180 - 120}{10} = 6$ squares: June looks twice January.

Second, the heading says "Sales doubled!". Doubled would be $2 \times 150 = 300$ plates; June's sales were $\frac{180}{150} = 1.2$ times January's, a $20\%$ rise.

(c) Keep the data, 150 and 180. Start the value axis at 0 and choose equal steps of 20 up to 200, so 180 is on a gridline and 150 is halfway between 140 and 160. Draw two bars of the same width, and write a heading that states the data.

Redrawn, the June bar is 1.2 times as tall as the January bar, as it should be.

Sales rose by 30 plates, $20\%$ . The graph misleads by starting its axis at 120 (June's bar looks twice January's) and by claiming that sales doubled; the honest graph starts at 0 with the heading "Plates of fried rice sold".

Common mistakes

Comparing the heights of bars without first looking at where the value axis starts and what each step is. Read the numbers, then compare.
Thinking that every axis that starts above 0 is wrong. A line graph may start above 0 if a zigzag break is drawn; a bar graph must always start at 0.
Saying that a symbol drawn twice as wide and twice as tall is twice as big. It covers $2 \times 2 = 4$ times the area.
Mixing up percentage points and percentage. A rise from $87\%$ to $89\%$ is 2 percentage points, which is $\frac{2}{87} \times 100\% \approx 2.3\%$ of the starting value.
Reading a pie chart without adding up its percentages. If they come to more than $100\%$ (by more than rounding can explain), the answers overlap and a pie chart cannot show the data; if less, a part is missing.
Explaining a misleading diagram only in words, such as "it exaggerates". Say by how much, with numbers from the drawing and from the data.

Practice

The questions get harder as you go. Try each one before you open its solution.

Practice 1: Books borrowed

In March, Class 1A borrowed 48 library books and Class 1B borrowed 54. Class 1B's form teacher put up this bar graph.

(a) How many grid squares tall is each bar? How many times as tall as Class 1A's bar is Class 1B's?

(b) What percentage more books did Class 1B borrow than Class 1A?

ShowHide solution

(a) The axis starts at 45 and each square is 1 book. Class 1A: $48 - 45 = 3$ squares. Class 1B: $54 - 45 = 9$ squares.

$9 \div 3 = 3$ , so Class 1B's bar is 3 times as tall.

(b) Class 1B borrowed $54 - 48 = 6$ more books. $\frac{6}{48} \times 100\% = 12.5\%$

(c) The value axis starts at 45, not 0, so Class 1B's bar looks 3 times as tall as Class 1A's, as if the class borrowed 3 times as many books. It borrowed only $12.5\%$ more. Drawn from 0, Class 1B's bar is $\frac{54}{48} = 1.125$ times as tall as Class 1A's:

(a) 3 squares and 9 squares: 3 times as tall. (b) $12.5\%$ more. (c) The axis starts at 45, so a $12.5\%$ difference looks like 3 times as many.

Practice 2: Plates of chicken rice

A chicken rice stall sold 100 plates on Monday and 300 plates on Saturday. The stall owner drew this pictogram, with the Saturday plate 3 times as wide and 3 times as tall as the Monday plate.

(a) How many times as many plates were sold on Saturday as on Monday?

(b) How many times the area of the Monday symbol is the Saturday symbol?

ShowHide solution

(a) $\frac{300}{100} = 3$ times as many.

(b) The Saturday symbol is 3 times as wide and 3 times as tall, so its area is $3 \times 3 = 9$ times the Monday symbol's.

(c) The eye compares the space each symbol covers, so Saturday's sales look 9 times Monday's when they are only 3 times. An honest pictogram uses symbols of one size, each standing for 100 plates: 1 symbol for Monday and 3 for Saturday.

(a) 3 times. (b) 9 times. (c) The sales look 9 times as big; draw one symbol for every 100 plates, all the same size: 1 for Monday and 3 for Saturday.

Practice 3: A household's spending

Mrs Goh drew this bar graph of her household's spending last month: phone $50, transport $100, utilities $200, food $400.

(a) What is wrong with the value axis?

(b) On the graph, how many times as tall as the Phone bar is the Food bar? How many times as much was spent on food as on the phone?

(d) The axis is to be redrawn from 0 in equal steps. What is the largest equal step that puts every amount on a gridline?

ShowHide solution

(a) The numbers 0, 50, 100, 200, 400 go up in steps of 50, 50, 100 and 200, but they are drawn equally spaced.

(b) Phone reaches the first gridline (1 square) and Food the fourth (4 squares): the Food bar is 4 times as tall. In fact $\frac{400}{50} = 8$ , so 8 times as much was spent on food.

(c) Utilities is 3 squares and Transport 2 squares: it looks $\frac{3}{2} = 1.5$ times as much. In fact $\frac{200}{100} = 2$ times as much.

(d) Every amount (50, 100, 200, 400) is a multiple of 50, and 50 is the largest number that divides all four, so the largest step is 50: equal steps of 50 from 0 to 400. Redrawn, the Food bar is 8 times as tall as the Phone bar:

(a) The steps are unequal (50, 50, 100, 200). (b) 4 times as tall, but 8 times the money. (c) 1.5 times as tall, but 2 times the money. (d) 50.

Practice 4: Sports Day entries

The numbers of students of each house who entered events at Sports Day were Red 40, Blue 50, Green 30 and Yellow 45. The Green House captain drew this bar graph.

(a) Which house had the most entries, and which the fewest?

(b) Taking an ordinary bar's width as 1 unit and each student as 1 unit of height, find the area of the Green bar and of the Blue bar.

ShowHide solution

(a) Blue had the most entries, 50; Green had the fewest, 30.

(b) Green: $2 \times 30 = 60$ square units. Blue: $1 \times 50 = 50$ square units.

(c) The Green bar is twice as wide as the others, so it covers $\frac{60}{50} = 1.2$ times the area of the Blue bar, and more than any other bar, though Green had the fewest entries. With bars of equal width, Green's bar is the smallest:

(a) Blue had the most (50), Green the fewest (30). (b) Green 60, Blue 50 square units. (c) The wide Green bar covers 1.2 times Blue's area, though Green had the fewest entries.

Practice 5: A pie chart with a part missing

A school asked its students which language they speak most often at home, and made this pie chart of the answers.

(a) Add up the percentages. What has gone wrong?

(b) What angle should the English sector have? The drawing shares the circle out in the ratio $45 : 30 : 10$ ; what angle is the English sector drawn with, to the nearest degree?

(c) The students who are not shown speak Tamil or another language most often at home. Find the angle of every sector of an honest pie chart.

ShowHide solution

(a) $45\% + 30\% + 10\% = 85\%$ . The percentages should add up to $100\%$ , so $100\% - 85\% = 15\%$ of the students have been left out.

(b) The English sector should be $\frac{45}{100} \times 360^\circ = 162^\circ$ .

The ratio $45 : 30 : 10$ has $45 + 30 + 10 = 85$ parts. English is drawn with $\frac{45}{85} \times 360^\circ \approx 191^\circ$ , which is about $\frac{191}{360} \times 100\% \approx 53\%$ of the circle, though it is labelled $45\%$ . Every sector shown is drawn too big.

(c) Add a sector "Tamil or another language" of $15\%$ . English $\frac{45}{100} \times 360^\circ = 162^\circ$ ; Mandarin $\frac{30}{100} \times 360^\circ = 108^\circ$ ; Malay $\frac{10}{100} \times 360^\circ = 36^\circ$ ; Tamil or another language $\frac{15}{100} \times 360^\circ = 54^\circ$ .

Check: $162^\circ + 108^\circ + 36^\circ + 54^\circ = 360^\circ$

(a) They add up to $85\%$ : $15\%$ of the students are missing. (b) $162^\circ$ , but it is drawn at about $191^\circ$ . (c) English $162^\circ$ , Mandarin $108^\circ$ , Malay $36^\circ$ , Tamil or another language $54^\circ$ .

Practice 6: School numbers soar?

A school printed this line graph of its number of students.

(a) Find the rise in the number of students from 2021 to 2025, and the rise as a percentage of the 2021 number, to 1 decimal place.

(b) How many grid squares above the bottom of the graph are the 2021 and 2025 points? How many times as high is the 2025 point drawn?

ShowHide solution

(a) Rise $= 1280 - 1240 = 40$ students.

Percentage rise $= \frac{40}{1240} \times 100\% \approx 3.2\%$

(b) The axis starts at 1230 and each square is 10 students. 2021: $\frac{1240 - 1230}{10} = 1$ square up; 2025: $\frac{1280 - 1230}{10} = 5$ squares up. The 2025 point is drawn 5 times as high.

(c) First, the value axis starts at 1230 with no break, so the reader is not warned that it is cut off. The line climbs 4 of the graph's 6 squares, two thirds of its height, for a rise of only $3.2\%$ .

Second, the heading "School numbers soar!" is not supported: 40 more students in four years is a small rise.

An honest graph can keep the axis from 1230 to show the changes in detail, as long as a zigzag break is drawn at the bottom of the axis and the heading states the data. With a break the reader is warned; the rise of 40 is still only $3.2\%$ . (Drawn from 0, the line would be almost flat and the yearly changes could not be read.)

(a) 40 students, about $3.2\%$ . (b) 1 square and 5 squares: 5 times as high. (c) The axis is cut at 1230 with no break, and the heading exaggerates a $3.2\%$ rise.

Practice 7: A tilted pie chart

Ali asked a group of students for their favourite bubble tea flavour and drew this tilted pie chart: Brown sugar $35\%$ , Taro $30\%$ , Milk tea $20\%$ , Mango $15\%$ .

On the drawing, the Taro sector's angle at the centre measures about $140^\circ$ and the Brown sugar sector's about $110^\circ$ .

(a) Which sector looks the largest? Which flavour is in fact the most popular?

(b) Find the true angle of each sector.

ShowHide solution

(a) The Taro sector looks the largest. Brown sugar, $35\%$ , is in fact the most popular.

(b) Brown sugar $\frac{35}{100} \times 360^\circ = 126^\circ$ ; Taro $\frac{30}{100} \times 360^\circ = 108^\circ$ ; Milk tea $\frac{20}{100} \times 360^\circ = 72^\circ$ ; Mango $\frac{15}{100} \times 360^\circ = 54^\circ$ . Check: $126^\circ + 108^\circ + 72^\circ + 54^\circ = 360^\circ$ .

(c) Taro sits at the front of the tilted disc, so it is drawn wider than it is, about $140^\circ$ in place of $108^\circ$ , with the thick side of the disc below it. Brown sugar is at the side, drawn at about $110^\circ$ in place of $126^\circ$ . So the smaller share looks the bigger one. Drawn flat, every sector has its true angle:

(a) Taro looks largest; Brown sugar is the most popular. (b) $126^\circ$ , $108^\circ$ , $72^\circ$ , $54^\circ$ . (c) The tilt widens the front sector (Taro, $108^\circ$ drawn at about $140^\circ$ ) and narrows the side one (Brown sugar, $126^\circ$ at about $110^\circ$ ).

Practice 8: Hui Min's savings

Hui Min saves the same amount every month. The table shows her total savings at the end of each month.

Hui Min's savings
Month	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Savings ($)	20	40	60	80	100	120	140	160	180	200	220	240

She drew this line graph from five of the months.

(a) How many months does each step along the horizontal axis stand for?

(b) How many squares does the line rise in each step? How much did she save per month in each step?

ShowHide solution

(a) Jan to Feb 1 month, Feb to Mar 1 month, Mar to Jun 3 months, Jun to Dec 6 months.

(b) Each square is $20. The rises are $40 - 20 = 20$ , $60 - 40 = 20$ , $120 - 60 = 60$ and $240 - 120 = 120$ , which are 1, 1, 3 and 6 squares.

Per month: $\frac{20}{1} = 20$ , $\frac{20}{1} = 20$ , $\frac{60}{3} = 20$ and $\frac{120}{6} = 20$ , so she saved $20 a month every time.

(c) The heading is not true: her savings grew by the same $20 every month. The graph draws steps of 1, 1, 3 and 6 months the same width, so the last step rises 6 times as much as the first and the line looks as if it is getting steeper. With every month drawn, the line is straight:

(a) 1, 1, 3 and 6 months. (b) 1, 1, 3 and 6 squares; $20 a month in every step. (c) No: she saves $20 every month, but unequal time steps drawn equally spaced make the line look steeper and steeper.

Practice 9: A claim about library books

The pictogram shows the books borrowed from a class library in three months.

The class monitor says: "Loans in March were double those in January."

(a) Find the number of books borrowed in each month.

(b) Is the monitor right? Explain with numbers.

ShowHide solution

(a) Each symbol stands for 20 books. January: $3 \times 20 = 60$ . February: $3\frac{1}{2} \times 20 = 70$ . March: $4\frac{1}{2} \times 20 = 90$ .

(b) Double January's loans would be $2 \times 60 = 120$ books. March had 90, and $\frac{90}{60} = 1.5$ . The monitor is wrong; the pictogram is drawn fairly, but the claim is not supported by it.

(c) March had $90 - 60 = 30$ more loans, and $\frac{30}{60} \times 100\% = 50\%$ . "Loans in March were 1.5 times those in January, $50\%$ more."

(a) 60, 70 and 90 books. (b) No: double would be 120, but March had 90, 1.5 times January. (c) "Loans in March were $50\%$ more than in January."

Practice 10: Two stalls, two graphs

Two dessert stalls at a hawker centre sell ice kachang. On Sunday Stall A sold 90 bowls and Stall B sold 92 bowls. Each stall drew its own bar graph.

Siti puts the two graphs side by side and says that Stall A sold far more than Stall B.

(a) Which stall sold more, and by how many cups?

(b) What fraction of the height of its axis does each bar reach?

ShowHide solution

(a) Stall B sold more, by $92 - 90 = 2$ bowls.

(b) Stall A's axis runs from 0 to 100, and its bar reaches $\frac{90}{100} = \frac{9}{10}$ of the height.

Stall B's axis runs from 80 to 100, which is $100 - 80 = 20$ bowls tall, and its bar reaches $92 - 80 = 12$ of them: $\frac{12}{20} = \frac{3}{5}$ of the height.

(c) The two graphs use different scales, and Stall B's axis starts at 80, not at 0. So Stall A's bar is drawn $\frac{9}{10} \div \frac{3}{5} = 1.5$ times as tall as Stall B's, though Stall B sold more. To compare the stalls, draw both bars on one graph whose axis starts at 0:

(a) Stall B, by 2 bowls. (b) Stall A $\frac{9}{10}$ , Stall B $\frac{3}{5}$ . (c) The graphs have different scales and B's starts at 80, so A's bar looks 1.5 times as tall; compare them on one graph from 0.

Practice 11: How to make a bar look 3 times as tall

Shop A sold 40 phone covers and Shop B sold 50. Mr Tan wants to draw a bar graph in which Shop B's bar looks 3 times as tall as Shop A's.

(a) Let the value axis start at $m$ . Write an equation for $m$ and solve it.

(b) Draw what his graph would look like beside an honest one, and explain why his graph would mislead.

ShowHide solution

(a) With the axis starting at $m$ , Shop A's bar is $40 - m$ tall and Shop B's bar is $50 - m$ tall.

Shop B's bar is to be 3 times as tall: $50 - m = 3(40 - m)$

$50 - m = 120 - 3m$

$2m = 70$

$m = 35$

Check: $50 - 35 = 15$ and $40 - 35 = 5$ , and $15 = 3 \times 5$ .

(b) With the axis from 35 in steps of 5, Shop A's bar is 1 square tall and Shop B's is 3 squares. Graph 1 is Mr Tan's graph; Graph 2 is the honest one, drawn from 0.

Shop B sold $\frac{50}{40} = 1.25$ times as many as Shop A, that is $25\%$ more, but Mr Tan's graph makes it look 3 times as many, because only the parts of the bars above 35 are drawn. Drawn from 0, Shop B's bar is 1.25 times as tall as Shop A's.

(a) $50 - m = 3(40 - m)$ , so $m = 35$ . (b) Starting at 35, Shop B looks 3 times Shop A, though it sold only $25\%$ more.

Practice 12: Bigger symbols, fewer students

The pictogram shows the students who joined the school choir in 2024 and 2025. Each symbol in the 2025 row is drawn 1.5 times as wide and 1.5 times as tall as each symbol in the 2024 row.

The choir's teacher says: "More students joined the choir in 2025 than in 2024."

(a) Find the number of students who joined in each year.

(b) How many times the area of a 2024 symbol is a 2025 symbol? Taking the area of one 2024 symbol as 1 unit, how many units of area do the symbols of each row cover? Which row is the longer?

ShowHide solution

(a) 2024: $5 \times 10 = 50$ students. 2025: $4 \times 10 = 40$ students.

(b) Each 2025 symbol is 1.5 times as wide and 1.5 times as tall, so it covers $1.5 \times 1.5 = 2.25$ times the area of a 2024 symbol.

The 2024 row covers $5 \times 1 = 5$ units of area; the 2025 row covers $4 \times 2.25 = 9$ units.

The 2025 row is the longer: measured from its first symbol to its last, it is about 1.1 times as long as the 2024 row, although it has fewer symbols.

(c) The teacher is wrong: 40 joined in 2025 and 50 in 2024, so $50 - 40 = 10$ fewer joined, which is $\frac{10}{50} \times 100\% = 20\%$ fewer.

The bigger symbols make the 2025 row longer and cover $\frac{9}{5} = 1.8$ times the area of the 2024 row, so it looks bigger. With every symbol the same size, the 2025 row is the shorter one:

(a) 50 in 2024 and 40 in 2025. (b) 2.25 times; 9 units against 5; the 2025 row is the longer. (c) No: $20\%$ fewer joined in 2025; the bigger symbols make the smaller number look bigger.

Key Points

A diagram misleads when what the eye sees (heights, areas, angles) is not in proportion to the numbers, even if every number printed on it is correct.
A bar graph's value axis starts at 0. A line graph may start above 0 only with a zigzag break, and its scale must still be read with care.
The numbers on an axis go up in equal steps, and the times along a line graph's horizontal axis are equally spaced.
Every bar has the same width and every pictogram symbol the same size. A symbol $k$ times as wide and as tall covers $k^{2}$ times the area.
A pie chart is drawn flat, with percentages adding up to $100\%$ and angles to $360^\circ$ . Answers that overlap (more than $100\%$ , beyond rounding) need a bar graph.
Check titles and claims against the numbers: "most" means more than half, "doubled" means 2 times.
Explain with numbers: how many times as big it looks, measured from the drawing, against how many times as big it is.
To redraw honestly, keep the data and fix the drawing: a bar graph's axis from 0 (a line graph's from 0 or with a break), equal steps, equal widths and sizes, a flat pie, equal time steps, a title that states the data.