Choosing a Statistical Diagram

What each form of statistical representation is for, its advantages and disadvantages, how to choose the best one for given data and purpose, and when a diagram cannot show the data at all.

Before you start

You have met five ways of presenting data: a table, a pictogram, a bar graph, a pie chart and a line graph. In this lesson they are compared, not taught again.
Categorical data are names or groups (favourite flavour, CCA, way of travelling to school). Numerical data are amounts that are counted or measured: discrete if they are counted (number of books; amounts of money are discrete too), continuous if they are measured (height, temperature, time).
A pie chart shows parts of one whole: each sector of $a^\circ$ stands for $\frac{a}{360}$ of the total.

One Set of Data, Four Forms

The same data can be presented in more than one form. Each form makes some things easy to see and hides others, so the best form depends on what the data are and what the reader needs to find out.

Here is one set of data, the favourite bubble tea flavour of the 40 students of class 1C, in a table, a pictogram, a bar graph and a pie chart.

Favourite bubble tea flavour of the 40 students of class 1C
Flavour	Number of students
Brown sugar	14
Taro	8
Matcha	6
Mango	12
Total	40

All four hold the same numbers, but each answers a different question best. The table gives each number exactly. The pictogram and the bar graph show at once which flavour is the most popular and which the least. The pie chart shows what share of the class chose each flavour.

Example 1: Which form answers the question?

Use the four forms above. For each question, say which form answers it most directly, and answer it.

(a) Exactly how many students chose Matcha?

(b) Which flavour is the most popular, and which is the least popular?

(d) The class wants to show the results to a group of Primary 2 children. Which form should it use?

Solution

(a) The table gives every number exactly: Matcha, 6 students. (The bar graph gives it too, since the Matcha bar ends on the gridline at 6.)

(b) The bar graph (or the pictogram): the longest bar is Brown sugar and the shortest is Matcha. No numbers need to be read or compared one by one.

(c) The pie chart: Brown sugar is $35\%$ and Mango $30\%$ , so together they are $35\% + 30\% = 65\%$ of the class. The two sectors together can be seen to be nearly two thirds of the circle.

Check from the table: $\frac{14 + 12}{40} \times 100\% = \frac{26}{40} \times 100\% = 65\%$ .

(d) The pictogram: young children can count the symbols and need no scale and no percentages.

(a) The table: 6 students. (b) The bar graph: Brown sugar most, Matcha least. (c) The pie chart: $65\%$ . (d) The pictogram.

The Purpose of Each Form

Each form of statistical representation has its own purpose, the job it does best. A table is a form too: it is usually the first place data are written down after they are collected, and every diagram is drawn from it.

What each form is for

A table records the data as exact numbers, so that values can be looked up and calculations made.

A pictogram shows a few counts with symbols, so that they can be compared at a glance.

A bar graph compares amounts in different categories by the lengths of the bars.

A pie chart shows how one whole is divided into parts, and how big each part is compared with the whole.

A line graph shows how a quantity changes over time, so that rises, falls and the trend can be seen.

The kind of data narrows the choice. Categorical data (flavours, CCAs, ways of travelling) are shown in a table, a pictogram, a bar graph or, if they are parts of one whole, a pie chart. Data that follow one another in time (the temperature every two hours, the loans each month) are shown in a line graph, or in a bar graph with the times in order.

Advantages and Disadvantages

The table below brings together the advantages and disadvantages of the five forms. When you are asked for an advantage or a disadvantage, give one that belongs to that form, and if you can, say what it is compared with ("the table gives exact values, but the bar graph shows the largest at a glance").

Advantages and disadvantages of the forms of statistical representation
Form	Advantages	Disadvantages
Table	Every value is exact, and totals, differences and percentages can be worked out from it. It can hold many categories, and two things about each person at once, such as gender and CCA (a two-way table).	The greatest and least values, and any pattern or trend, are not seen at a glance: the numbers must be read and compared one by one.
Pictogram	It is attractive and easy to understand, even for young children. A value is found by counting symbols and using the key; no scale is needed. The rows can be compared at a glance.	Part symbols are hard to draw and to read exactly, and some numbers cannot be shown exactly. With large numbers each symbol stands for many, so small differences are lost. It takes longer to draw than a bar graph, and it is used for counts, not for measurements such as heights or temperatures.
Bar graph	The lengths of the bars make it easy to compare the categories and to see the greatest and the least. Values are read from a scale, more exactly than from a pictogram. A dual bar graph compares two sets of data on one graph.	A value that ends between gridlines can only be estimated. It does not show at once what fraction of the whole each category is. With many categories it becomes crowded. Its value axis must start at 0, so values that are large and close together (such as temperatures from 25 °C to 32 °C) give bars of nearly the same length.
Pie chart	It shows at a glance how one whole is shared out, and how big each part is compared with the whole and with the other parts.	It can only show parts of one whole. It does not show the amounts: each one must be worked out from its angle and the total. With more than about 7 parts, or very small parts, the sectors are hard to compare and to label. It does not show change over time.
Line graph	It shows how a quantity changes over time: rises, falls and the trend are seen at a glance. Two lines on one graph compare two sets of data over the same times. For a quantity that changes continuously, such as temperature, a value between two readings can be estimated.	It can only show data in time order; it cannot show categories. A value between gridlines can only be estimated. For counts, such as sales each month, the line between two points shows the trend only: there is no value between two months to read.

Example 2: An advantage and a disadvantage

Mei Ling puts the pie chart of class 1C's favourite flavours (above) on the class noticeboard. Give one advantage and one disadvantage of the pie chart, compared with the bar graph of the same data.

Solution

Advantage. The pie chart shows at once each flavour's share of the whole class. For example, the Brown sugar and Mango sectors together fill nearly two thirds of the circle. From the bar graph, the total of 40 must first be found and each percentage worked out.

Disadvantage. The pie chart does not show how many students chose each flavour. To find the number who chose Taro, the percentage and the total are needed: $\frac{20}{100} \times 40 = 8$ students. On the bar graph the Taro bar is read straight off the scale as 8.

Advantage: it shows each flavour's share of the class at a glance. Disadvantage: the number of students for each flavour is not shown and must be calculated from the percentage and the total.

Choosing the Best Diagram

To choose a form, look first at the data and then at the purpose. A reason for a choice must say what this form shows that the others do not. "It is clear" or "it is easy to read" is true of every good diagram, so it is not a reason.

How to choose

1. Are exact values needed, for looking up or for calculating? Use a table.

2. Do the values follow in time, and is the change what matters? Use a line graph.

3. Are the values parts of one whole, and does each part's share of the whole matter? Use a pie chart.

4. Are amounts in different categories to be compared? Use a bar graph; for a few counts shown to young children, a pictogram.

Example 3: Choosing the best diagram

For each set of data, choose the best form and give a reason.

(a) How Mr Lee's household uses its income of $5000 a month: on housing, food, transport, other things and savings.

(b) The number of cans collected for recycling by each of the eight Secondary 1 classes in a recycling drive, to see which classes collected the most.

(d) The number of visitors to the school library on each school day of the year, kept for the librarian's records.

(e) The favourite fruit of a Primary 1 class, for a poster on their classroom wall.

Solution

(a) Pie chart. The five amounts make up one whole, the $5000, and the purpose is to see what share of the income goes on each item.

(b) Bar graph. The classes are categories and the purpose is to compare their amounts. Eight bars side by side show at once which are the longest. (A pie chart would have eight sectors, several of nearly the same size, which are hard to compare.)

(c) Line graph. The readings follow one another in time and the purpose is to see how the temperature changes from day to day. The line shows its rises, falls and trend.

(d) Table. The librarian needs the exact number for each day, and about 200 school days would make any diagram far too crowded.

(e) Pictogram. There are only a few fruits, and young children can count the symbols without reading a scale.

(a) Pie chart (b) Bar graph (c) Line graph (d) Table (e) Pictogram, each for the reason given.

When a Diagram Cannot Show the Data

Some forms do not just suit the data less well: they cannot show them at all, because what the diagram means is not true of the data. Two cases come up most often: a pie chart for values that are not parts of one whole, and a line graph for categories.

When a form cannot be used

A pie chart needs the values to be parts of one whole: the whole is shared out so that each item (each student, each litre, each hour) is in exactly one part, and the parts add up to the total. Measurements can be shown when they are parts of one whole, such as a household's water use shared among its uses, or the 24 hours of a day. A pie chart cannot show values that are not parts of one whole, such as marks in different subjects, the heights of several students, the temperature on each day, percentages of different groups, or counts in which one person is counted more than once.

A line graph needs the data to be in time order, so that the line from one point to the next means something. It cannot show categories: they do not follow one another in time, and most of them (flavours, CCAs, subjects) have no order at all.

A pictogram shows counts. It is not used for measurements such as heights, masses or temperatures.

Example 4: A line graph of categories

Ravi drew class 1C's favourite flavours as a line graph (the first graph below). His friend drew the same data with the flavours in a different order (the second graph). Explain why a line graph cannot show these data.

Solution

A line from one point to the next says that the quantity changes from one value to the next, in order. The flavours have no order: Taro does not come "after" Brown sugar, and there is nothing halfway between Taro and Matcha.

Ravi's graph seems to say that the number "falls and then rises". His friend's graph, made from exactly the same numbers, seems to say it "rises, falls and rises". The shape depends only on the order in which the flavours were written, so it tells us nothing about the data.

The flavours are categories, so a bar graph (or a pictogram, or a pie chart, since the 40 students make one whole) should be used, as in the first section.

The flavours are categories with no order, so the lines between them mean nothing and their shape changes with the order chosen. A bar graph should be used.

Example 5: Can a pie chart show it?

For each set of data, say whether a pie chart can show it, and give a reason.

(a) The percentage of students in each of four Secondary 1 classes who passed a swimming test: 1A $80\%$ , 1B $75\%$ , 1C $90\%$ , 1D $85\%$ .

(b) The heights of five students: 148 cm, 152 cm, 155 cm, 160 cm and 149 cm.

Solution

(a) $80\% + 75\% + 90\% + 85\% = 330\%$ , not $100\%$ . Each percentage is a part of a different class, so they are not parts of one whole. A pie chart cannot show them. A bar graph can compare the four classes.

(b) The heights are not parts of one whole: each belongs to a different student. Adding them gives 764 cm, which is not a whole that has been shared out among the students. A pie chart cannot show them (nor can a pictogram, which shows counts). A bar graph can.

(c) $16 + 12 + 8 + 4 = 40$ : every student is counted once, and the parts make up the whole class. A pie chart can show it, with $360^\circ \div 40 = 9^\circ$ for each student: bus $16 \times 9^\circ = 144^\circ$ , MRT $12 \times 9^\circ = 108^\circ$ , walk $8 \times 9^\circ = 72^\circ$ , car $4 \times 9^\circ = 36^\circ$ .

Check: $144^\circ + 108^\circ + 72^\circ + 36^\circ = 360^\circ$ .

(a) No: the percentages are of different classes and add up to $330\%$ . (b) No: the heights are not parts of one whole. (c) Yes: the parts make up the whole class of 40.

The Same Data Drawn Two Ways

Often two forms can both show the data. Then the better one is the one that suits the purpose: the question the reader wants answered. Drawing the same data both ways shows the difference clearly.

Example 6: A bar graph or a pie chart?

Siti's household has an income of $3600 a month. It spends $1200 on housing, $1000 on food, $400 on transport and $400 on other things, and saves $600. The data are drawn below as a bar graph and as a pie chart.

Which diagram is better for each purpose? Answer each question from the better diagram.

(a) To see what fraction of the income goes on housing.

(b) To find how much more is spent on food than is saved.

Solution

(a) The pie chart, because the question is about a part of the whole. The housing sector is $120^\circ$ , and $\frac{120^\circ}{360^\circ} = \frac{1}{3}$ . One third of the income goes on housing. The bar graph shows $1200, but to find the fraction we must also know the total $3600.

(b) The bar graph, because the question is about amounts. The food bar reads 1000 and the savings bar 600 on the scale, so the difference is $1000 - 600 = 400$ .

From the pie chart the same answer needs the angles and the total: $\frac{100^\circ - 60^\circ}{360^\circ} \times 3600 = \frac{40}{360} \times 3600 = 400$ .

(a) The pie chart: $\frac{1}{3}$ of the income goes on housing. (b) The bar graph: $400 more is spent on food than is saved.

Example 7: A line graph or a bar graph?

Hui Min recorded the temperature in the school garden every two hours on one day.

Temperature in the school garden
Time	7 a.m.	9 a.m.	11 a.m.	1 p.m.	3 p.m.	5 p.m.
Temperature (°C)	25	27	30	32	31	28

She drew the data as a line graph and as a bar graph.

Which diagram is better for describing how the temperature changed during the day and estimating the temperature at 10 a.m.? Give the description and the estimate.

Solution

The readings follow one another in time, and the temperature changes continuously between them. So the line graph is better: the line shows the rise and fall in one glance, and a point on the line between two readings gives an estimate of the temperature at a time in between.

The change: the temperature rose from $25^\circ$ C at 7 a.m. to its highest, $32^\circ$ C, at 1 p.m., and then fell to $28^\circ$ C by 5 p.m.

The estimate: 10 a.m. is halfway between 9 a.m. and 11 a.m., so on the line it is halfway between $27^\circ$ C and $30^\circ$ C: $\frac{27 + 30}{2} = 28.5$ , so about $28.5^\circ$ C. This is an estimate, since the temperature need not rise steadily between two readings.

The bar graph shows each reading separately, and there is nothing between two bars to read at 10 a.m. Its axis must start at 0, so the bars, from 25 to 32, differ little in height and the rise and fall are harder to see. All six bars are tall and differ only near their tops, so the rise and fall are harder to see than on the line graph.

The line graph. The temperature rose from $25^\circ$ C at 7 a.m. to $32^\circ$ C at 1 p.m., then fell to $28^\circ$ C at 5 p.m.; at 10 a.m. it was about $28.5^\circ$ C.

Common Mistakes

A pie chart for data that are not parts of one whole, such as test marks in different subjects, the heights of several students, or the percentages of different classes. First check that the values add up to a total that has been shared out. (Whether the data are counted or measured does not decide it: a household's water use, shared among its uses, is measured and can be shown.)
A line graph for categories. The points of a line graph are joined only when the data are in time order. Flavours, CCAs or subjects have no order, so they are shown in a bar graph.
A reason that fits every diagram. "It is neat" or "it is easy to understand" does not explain the choice. Say what the chosen form shows that the others do not: exact values, a comparison, a share of the whole, or a change over time.
Choosing by looks. A pictogram is attractive, but with large or awkward numbers it cannot show the values exactly; choose for what the reader needs to find out.
Two pie charts to compare two groups of different sizes. A pie chart shows shares, not numbers: equal angles need not mean equal numbers. To compare the numbers in two groups, use a dual bar graph.

Practice Questions

Give a reason with every choice. Work out each answer before you open its solution.

Practice 1: The purpose of each form

Name the form of statistical representation that is used for each purpose.

(a) To show how a whole is divided into parts.

(b) To keep the exact values, so that calculations can be made from them.

(d) To compare amounts in several categories at a glance, with values read from a scale.

(e) To show a few counts with symbols, in a way young children can understand.

ShowHide solution

(a) A pie chart: the whole circle is the whole, and each sector is one part.

(b) A table: it gives every value exactly.

(d) A bar graph: the lengths of the bars compare the categories at a glance, and each value is read from the scale.

(e) A pictogram: the values are counted in symbols, using the key, with no scale to read.

(a) Pie chart (b) Table (c) Line graph (d) Bar graph (e) Pictogram

Practice 2: A table or a bar graph?

The table shows the number of library books borrowed last month by five Secondary 1 classes.

Library books borrowed last month
Class	1A	1B	1C	1D	1E
Number of books	46	38	52	41	35

The librarian could also draw the data as a bar graph with a scale marked 0, 10, 20, 30, 40, 50, 60.

(a) Give one advantage of the table over the bar graph.

(b) Give one advantage of the bar graph over the table.

ShowHide solution

(a) The table gives every number exactly. On the bar graph, 1D's bar would end between the gridlines at 40 and 50, and 41 could only be estimated.

(b) On the bar graph the longest bar (1C) and the shortest (1E) are seen at a glance. In the table the five numbers must be read and compared one by one.

(c) $46 + 38 + 52 + 41 + 35 = 212$ , so the five classes borrowed 212 books. A calculation like this is made from the exact values of the table.

(a) For example: it gives exact values (41 cannot be read exactly from the bar graph). (b) For example: the greatest and least are seen at a glance. (c) 212 books

Practice 3: A pictogram

The pictogram shows the number of Secondary 1 students who entered four events at a Sports Day.

(a) How many students entered the relay?

(b) Give one advantage of the pictogram.

(c) 15 students entered the shot put. Can a row for the shot put be drawn exactly with this key? What disadvantage of a pictogram does this show?

ShowHide solution

(a) The relay has $3\frac{1}{2}$ symbols, and each represents 8 students: $3 \times 8 + 4 = 28$ students.

(b) It is attractive and easy to understand: the number of students is found by counting symbols and using the key, with no scale to read, and the rows can be compared at a glance.

(c) $15 \div 8 = 1\frac{7}{8}$ . With a key of 8, a quarter symbol stands for 2, a half for 4 and three quarters for 6, so a row can show 14 ( $1\frac{3}{4}$ symbols) or 16 (2 symbols) but not 15. The row cannot be drawn exactly.

This shows that a number that is not a whole number of halves or quarters of the key cannot be shown exactly on a pictogram. A bar graph can show 15 exactly, for example halfway between the gridlines at 10 and 20.

(a) 28 students (b) For example: it is easy to understand, by counting symbols. (c) No: 15 needs $1\frac{7}{8}$ symbols; a value that is not a whole number of halves or quarters of the key cannot be shown exactly.

Practice 4: Choosing the best diagram

For each set of data, choose the best form and give a reason.

(a) How a household's water use in a month is shared among showering, washing, cooking and other uses.

(b) The mass of a baby, measured on the first day of each month in her first year.

(d) The prices of the 20 items sold at a school canteen stall, for students to look up.

ShowHide solution

(a) Pie chart. The four uses together make up the whole of the month's water, and the purpose is to see each use's share.

(b) Line graph. The masses are measured one month after another, and the purpose is to see how the baby grows over time.

(d) Table. The students need the exact price of each item, and 20 items are listed most clearly in a table.

(a) Pie chart (b) Line graph (c) Bar graph (d) Table, each for the reason given.

Practice 5: Visitors to the book fair

The school held a book fair for one week. The table shows the number of visitors on each day.

Visitors to the book fair
Day	Mon	Tue	Wed	Thu	Fri
Number of visitors	120	150	90	180	210

(a) Siti wants to show how the number of visitors changed during the week. She can draw a line graph or a bar graph with the days in order. Which shows the change more clearly? Describe the change.

(b) On the line graph, Ravi reads 135 visitors halfway between Monday and Tuesday. Explain why this number means nothing.

ShowHide solution

(a) The days are in time order, so both forms can show the data. The line graph shows the change more clearly: the slope of each piece of line shows at once whether the number rose or fell, and by how much. The number rose from 120 on Monday to 150 on Tuesday, fell to 90 on Wednesday, then rose to 180 on Thursday and 210 on Friday, the most of the week.

(b) Each reading is the number of visitors on one whole day, a count. There is no day between Monday and Tuesday, so the point halfway along the line, $\frac{120 + 150}{2} = 135$ , is not the number of visitors on any day or in any part of a day. A value between two readings can be estimated only for a quantity that changes continuously, such as a temperature.

(c) Each bar stands alone for one day, so the bar graph does not suggest values between the days, and the days are easy to compare by the lengths of their bars (Friday longest, Wednesday shortest).

(a) The line graph: the number rose from Monday to Tuesday, fell on Wednesday, then rose on Thursday and Friday to 210, the most of the week. (b) Each reading is a count for a whole day; there is no day between Monday and Tuesday, so 135 is not the number of visitors at any time. (c) For example: each bar is a separate day, so no values between days are suggested.

Practice 6: Ali's line graph

Ali asked 36 students to name their favourite hawker centre food and drew the line graph below.

(a) Ali says, "The graph shows that the number of students falls from chicken rice to laksa and then rises to nasi lemak." Explain why this means nothing.

(b) Name two forms that could show these data properly, and draw one of them.

ShowHide solution

(a) The foods are categories with no order. Ali could have written them in any order, and the line would then rise and fall in a different way. A line joining laksa to nasi lemak suggests a change from one to the next, but there is no change: they are just two different foods.

(b) A bar graph, since the foods are categories to be compared. A pie chart could also be used, since every student named one food and the counts make up the whole 36: $12 + 6 + 9 + 5 + 4 = 36$ .

(A pictogram, since these are counts, or a table would also be right: any two of these four forms.)

The bar graph, on a scale from 0 to 12 in steps of 2 (5 and 9 lie halfway between gridlines):

For a pie chart each student is $360^\circ \div 36 = 10^\circ$ : the angles would be $120^\circ$ , $60^\circ$ , $90^\circ$ , $50^\circ$ and $40^\circ$ , which add up to $360^\circ$ .

(a) The foods have no order, so the line between them means nothing. (b) Any two of a bar graph, a pie chart, a pictogram and a table; the bar graph is drawn above.

Practice 7: Which forms can be used?

Ali ran 2.4 km at the end of each term and recorded his times: Term 1 16 minutes, Term 2 15 minutes, Term 3 14.5 minutes, Term 4 13.5 minutes. He wants to show whether he has improved.

Which of these forms can show the data, and which cannot? Give a reason for each: a table, a bar graph, a pie chart, a line graph.

ShowHide solution

Table: can. It lists the four times exactly.

Bar graph: can. A bar for each term, with the terms in order, compares the four times; the shorter bars show the faster runs.

Pie chart: cannot. The four times are not parts of one whole: each is the time for a whole 2.4 km run. Their sum, $16 + 15 + 14.5 + 13.5 = 59$ minutes, is not a time that has been shared out among the terms.

Line graph: can. The terms are in time order, and the line shows at once that his time fell every term: he improved. It is the best of the four for his purpose.

A table, a bar graph and a line graph can (the line graph shows the improvement best); a pie chart cannot, because the four times are not parts of one whole.

Practice 8: Bubble tea sold on Saturday

A bubble tea stall sold 60 cups on Saturday.

Cups of bubble tea sold on Saturday
Flavour	Brown sugar	Taro	Matcha	Mango	Total
Number of cups	24	9	12	15	60

(a) Draw a bar graph of the data.

(b) Calculate the angle of each sector and draw a pie chart of the data.

(c) Which of the two diagrams is better for showing that brown sugar was $40\%$ of the cups sold? Which is better for finding how many more cups of mango than of taro were sold? Explain, and find the difference.

ShowHide solution

(a) The greatest value is 24. A scale from 0 to 24 in steps of 2 (12 steps) puts every value on a gridline or halfway between two; steps of 5 would leave 24, 9 and 12 between gridlines:

(b) $360^\circ \div 60 = 6^\circ$ for each cup.

Flavour	Number of cups	Angle of sector
Brown sugar	24	$24 \times 6^\circ = 144^\circ$
Taro	9	$9 \times 6^\circ = 54^\circ$
Matcha	12	$12 \times 6^\circ = 72^\circ$
Mango	15	$15 \times 6^\circ = 90^\circ$
Total	60	$360^\circ$

(c) Brown sugar's share: the pie chart, because a share of the whole is a sector of the circle: $\frac{144^\circ}{360^\circ} \times 100\% = 40\%$ . The brown sugar sector is clearly the largest, a little less than half the circle.

Mango against taro: the bar graph, because the amounts are read straight from the scale: mango 15 cups, taro 9 cups, so $15 - 9 = 6$ more cups of mango. (From the pie chart: $\frac{90^\circ - 54^\circ}{360^\circ} \times 60 = 6$ , which needs the total and a calculation.)

(a) and (b) As drawn above; the angles are $144^\circ$ , $54^\circ$ , $72^\circ$ , $90^\circ$ . (c) The pie chart for the $40\%$ share; the bar graph for the difference, 6 cups.

Practice 9: Two pie charts or a dual bar graph?

In a survey, each of the 150 boys and 120 girls in Secondary 1 at Ravi's school chose the one of three CCAs he or she would most like to try: basketball, band or scouts. Ravi drew a pie chart for the boys and one for the girls.

(a) Ravi says, "The two Scouts sectors are the same size, so the same number of boys and girls chose Scouts." Is he right? Show your working.

(b) Which form is better for comparing the numbers of boys and girls who chose each CCA? Draw it.

ShowHide solution

(a) Each Scouts sector is $120^\circ$ , which is $\frac{120}{360} = \frac{1}{3}$ of its own pie chart.

Boys who chose Scouts: $\frac{1}{3} \times 150 = 50$ . Girls who chose Scouts: $\frac{1}{3} \times 120 = 40$ .

Ravi is wrong: 50 boys and 40 girls. Equal angles mean equal shares of two different totals, not equal numbers.

(b) The numbers for each CCA. Boys: Basketball $\frac{144}{360} \times 150 = 60$ , Band $\frac{96}{360} \times 150 = 40$ , Scouts 50; check $60 + 40 + 50 = 150$ . Girls: Basketball $\frac{60}{360} \times 120 = 20$ , Band $\frac{180}{360} \times 120 = 60$ , Scouts 40; check $20 + 60 + 40 = 120$ .

A dual bar graph draws the boys and the girls to the same scale, with the two bars for each CCA side by side:

(c) The pie charts show each CCA's share of its own group: for example, half of the girls chose band, and a third of each group chose scouts. From the dual bar graph these fractions need the totals and a calculation.

(a) No: 50 boys and 40 girls chose Scouts; equal angles are equal shares of different totals. (b) A dual bar graph, drawn above. (c) Each CCA's share of the boys and of the girls, such as half of the girls choosing band.

Practice 10: Hui Min's marks

Hui Min scored these marks, each out of 100, in her end-of-year examinations: English 72, Mathematics 85, Science 78, Chinese 64, History 70. She wants to draw a pie chart of her marks.

(a) Explain why a pie chart is not suitable, and name a form that is.

(b) Hui Min also wants to show her Mathematics marks in the four tests she took during the year, from Term 1 to Term 4, to see whether she has improved. Which form should she use? Why?

ShowHide solution

(a) The five marks are not parts of one whole: each is out of its own 100, and their sum, $72 + 85 + 78 + 64 + 70 = 369$ , is not a total that has been shared out among the subjects. A pie chart would show Mathematics as $\frac{85}{369}$ of a circle, which means nothing.

The subjects are categories and the purpose is to compare her marks, so a bar graph is suitable (a table too, if the exact marks are wanted).

(b) A line graph. The four tests come one after another in time, and the purpose is to see how her marks change from term to term: the line shows at once whether they rise.

(a) The marks are not parts of one whole; a bar graph. (b) A line graph, since the tests follow in time and she wants to see the change.

Practice 11: Mrs Goh's library report

Mrs Goh, the school librarian, is writing a report on the library. Choose the best form for each part, with a reason.

(a) The number of books borrowed in each month from January to October, to see whether borrowing goes up before the examinations.

(b) The books borrowed last month, by section, to show what share of the loans each section was. The library shelves every book in exactly one of four sections: fiction, non-fiction, comics and poetry.

(d) The exact numbers behind all of these, for the school's records.

(e) Give one disadvantage of the form you chose in (b).

ShowHide solution

(a) Line graph. The months follow in time and the purpose is to see how borrowing rises and falls. (A bar graph with the months in order can also show it.)

(b) Pie chart. Every book is in exactly one section, so the four sections make up the whole of last month's loans, and the purpose is each section's share.

(d) Table. It keeps every number exactly, and each diagram is drawn from it.

(e) The pie chart does not show how many books from each section were borrowed: each number must be worked out from its angle and the total. (Or: a small section, such as poetry, would have a thin sector that is hard to see and to label.)

(a) Line graph (b) Pie chart (c) Bar graph (d) Table (e) For example: the pie chart does not show the numbers of books.

Practice 12: Mrs Lim's household

Mrs Lim's household had an income of $4000 in March. It spent $1400 on housing, $1200 on food, $600 on transport and $300 on other things, and saved $500.

Siti says, "A line graph is best, because it shows the trend from housing to other things."

(a) Explain why Siti is wrong.

(b) Which form best shows what fraction of the money goes on each item? Calculate the angle for each item.

(d) Mrs Lim wants to compare the amount for each item in March this year with March last year, when the household's income was $3600. Which form should she use, and why not two pie charts?

ShowHide solution

(a) Housing, food, transport, savings and other things are categories with no order in time, so there is no trend "from housing to other things". A line joining them means nothing, and its shape would change if the items were listed in another order.

(b) A pie chart, since the five amounts make up the whole $4000. Each angle is $\frac{\text{amount}}{4000} \times 360^\circ$ :

Item	Amount ($)	Angle of sector
Housing	1400	$\frac{1400}{4000} \times 360^\circ = 126^\circ$
Food	1200	$\frac{1200}{4000} \times 360^\circ = 108^\circ$
Transport	600	$\frac{600}{4000} \times 360^\circ = 54^\circ$
Savings	500	$\frac{500}{4000} \times 360^\circ = 45^\circ$
Others	300	$\frac{300}{4000} \times 360^\circ = 27^\circ$
Total	4000	$360^\circ$

Check: $126^\circ + 108^\circ + 54^\circ + 45^\circ + 27^\circ = 360^\circ$ .

(c) Advantage: it shows at a glance what share of the $4000 goes on each item; for example, housing and food together take more than half the circle ( $126^\circ + 108^\circ = 234^\circ$ ). Disadvantage: the amounts are not shown; to find the amount saved, for example, we need $\frac{45}{360} \times 4000 = 500$ . (Or: the small sector for other things, $27^\circ$ , is hard to label.)

(d) A dual bar graph, with this year's and last year's amount for each item side by side on the same scale. Two pie charts would show each item's share of a different total, $4000 and $3600, so equal sectors would not mean equal amounts, and the amounts could not be compared directly.

(a) The items are categories, so there is no trend and a line means nothing. (b) A pie chart: $126^\circ$ , $108^\circ$ , $54^\circ$ , $45^\circ$ , $27^\circ$ . (c) For example: it shows each share at a glance; it does not show the amounts. (d) A dual bar graph, because the two totals are different.

Key Points

A table records exact values; a pictogram shows a few counts with symbols; a bar graph compares amounts in categories; a pie chart shows how one whole is divided; a line graph shows change over time.
Choose a form by the kind of data and the purpose: what the reader needs to find out.
A reason for a choice says what that form shows that the others do not.
A pie chart can only show parts of one whole: each item in exactly one part, the parts adding up to the total.
A line graph can only show data in time order; never categories.
A pictogram shows counts, not measurements.
When two forms can both show the data, the better one is the one that answers the stated purpose more directly.
To compare the numbers in two groups of different sizes, use a dual bar graph, not two pie charts; two pie charts compare their shares.