Comparing Statistical Diagrams

What each diagram shows and what it hides: choosing between dot diagrams, stem-and-leaf diagrams, histograms and the Secondary 1 charts, and giving a reason.

What You Should Already Know

From Secondary 1: a bar graph compares the counts of categories; a pie chart shows each category's share of a whole; a pictogram shows counts with symbols; a line graph shows how a quantity changes over time.
From lessons 12.1 to 12.4: dot diagrams, stem-and-leaf diagrams and histograms (ungrouped and grouped) show numerical data.

One Set of Data, Three Diagrams

Each diagram is a different picture of the data. Some keep every value; some give up the single values to show the overall shape more clearly. The best way to see this is to draw the same data in different ways.

Example 1: The same marks, three ways

The marks, out of 50, of 24 pupils are shown below as a dot diagram, a stem-and-leaf diagram and a grouped histogram. For each question, say which diagrams can answer it, and answer it. (a) What was the highest mark? (b) What was the most common mark? (c) How many pupils scored from 35 to 39? (d) Which diagram shows the shape of the marks most simply?

Solution

(a) The dot diagram (the last dot, at 48) and the stem-and-leaf diagram (4 | 8) both show 48. The histogram shows only that the highest mark is in the class $45 \le x < 50$ .

(b) The dot diagram shows it at once: the tallest column, at 35. The stem-and-leaf diagram shows it as the leaf 5 written three times on stem 3. The histogram cannot: it shows the modal class, $35 \le x < 40$ , not a single mark.

(d) The histogram: six bars rising to $35 \le x < 40$ and falling after it. The stem-and-leaf diagram shows the same shape on its side, in rows of 10 marks, and keeps the marks too.

(a) Dot diagram and stem-and-leaf: 48. (b) Dot diagram and stem-and-leaf: 35. (c) All three: 8 pupils. (d) The grouped histogram (the stem-and-leaf diagram also shows it).

Diagram	Keeps every value?
Dot diagram	Yes: one dot for each value.
Stem-and-leaf diagram	Yes: one leaf for each value.
Histogram for ungrouped data	Yes: each bar is one value and its frequency.
Histogram for grouped data	No: only how many values are in each class.
Bar graph, pie chart, pictogram	They show counts or shares of categories, not single values.

What Each Diagram Is For

Choosing a diagram

Bar graph: categorical data; compares the counts of categories. Bars apart, in any order.

Pie chart: categorical data; shows each category's share of a whole. Exact counts are hard to read, and similar sectors are hard to compare.

Pictogram: categorical data with small counts; eye-catching, but part-symbols are hard to read exactly.

Line graph: a quantity that changes over time; shows rises, falls and trends.

Dot diagram: numerical data with few different values; shows every value, clusters, gaps and the most common value. It becomes long and thin when the values are many and spread out.

Stem-and-leaf diagram: numerical data of two or three digits; keeps every value and shows the shape; back-to-back, it compares two sets. It becomes unwieldy for a very large set of data.

Histogram for ungrouped data: numerical data with few different values, each repeated often; shows the frequency of each value and the mode.

Histogram for grouped data: numerical data with many different values, or a large set of data; shows the overall shape and the modal class, but not the single values.

Example 2: Choosing a diagram and giving a reason

Choose a suitable diagram for each purpose, and give a reason. (a) The favourite fruit of 40 pupils. (b) The temperature at noon each day for two weeks. (c) How a family's monthly spending is shared among food, transport, housing and other items. (d) The masses of 200 durians sold at a stall. (e) Comparing the test marks of Class 2A and Class 2B while keeping every mark. (f) The number of goals scored in each of 12 matches.

Solution

(a) A bar graph (or a pie chart): fruit are categories, not numbers.

(b) A line graph: the temperature changes over time, and a line shows the rises and falls from day to day.

(d) A grouped histogram: the masses are numbers with many different values, and with 200 of them a diagram of every value would be crowded. Grouping shows the shape.

(e) A back-to-back stem-and-leaf diagram: the two classes share the stems, so they can be compared row by row, and every mark is kept.

(f) A dot diagram (or a histogram for ungrouped data): goals are whole numbers with only a few different values.

(a) bar graph (b) line graph (c) pie chart (d) grouped histogram (e) back-to-back stem-and-leaf diagram (f) dot diagram or ungrouped histogram, each for the reason given.

Poor Choices of Diagram

A diagram can be drawn correctly and still be the wrong choice: it may show something that means nothing, or suggest something that is not true.

Example 3: Why is each a poor choice?

Explain why each diagram is a poor choice, and say what would be better. (a) A pie chart of the heights of 5 pupils: Aisha 152 cm, Ben 160 cm, Chen 148 cm, Dev 165 cm, Ella 155 cm. (b) A line graph of the number of pupils in each CCA: Choir 7, Football 9, Robotics 6, Scouts 5, Art Club 3. (c) A dot diagram of the daily step counts of 200 pupils, from 2000 to 15 000.

(a)

(b)

Solution

(a) A pie chart shows shares of a whole, but the total of five heights (780 cm) is not a whole of anything. The five sectors look almost the same, and no height can be read from them. A bar graph with one bar for each pupil shows and compares the heights.

(b) A line joins one CCA to the next as if the number of pupils changed from Choir to Football to Robotics over time. The CCAs are separate categories in no particular order, so a bar graph is right.

(c) A line from 2000 to 15 000 would need a place for each of 13 001 step counts, and almost every pupil has a different count, so the 200 dots would stand one by one, far apart, and show no shape. A grouped histogram (classes $2000 \le x < 4000$ , $4000 \le x < 6000$ , …) shows the shape clearly.

(a) Heights are not shares of a whole: use a bar graph. (b) CCAs are not times in order: use a bar graph. (c) Too many values over too wide a range: use a grouped histogram.

Common Mistakes

Choosing a diagram by habit rather than by the kind of data: categories need a bar graph, pie chart or pictogram; numbers need a dot diagram, stem-and-leaf diagram or histogram; change over time needs a line graph.
Reading single values from a grouped histogram. It shows only how many values are in each class.
Using a pie chart for numbers that are not parts of one whole.
Joining categories with a line, as if they followed one another in time.
Giving a choice without a reason. Say what the diagram shows that the purpose needs.

Practice Questions

Try each question first, then open the solution to check your working.

Question 1

Say whether each set of data is categorical or numerical, and name a suitable diagram. (a) The blood types of 50 blood donors. (b) The number of spelling mistakes made by each of 30 pupils in a dictation. (c) Singapore's population each year from 2015 to 2024. (d) The heights of 300 pupils.

ShowHide solution

(a) Categorical (A, B, AB, O): a bar graph or a pie chart.

(b) Numerical, few different values: a dot diagram or a histogram for ungrouped data.

(d) Numerical, many different values, a large set: a histogram for grouped data.

(a) categorical: bar graph or pie chart (b) numerical: dot diagram or ungrouped histogram (c) numerical over time: line graph (d) numerical: grouped histogram

Question 2

Choose a diagram for each purpose and give a reason. (a) A teacher wants to compare the marks of two classes and still see every mark. (b) A school canteen wants to show what fraction of its sales came from each stall. (c) A polyclinic wants to show the waiting times of 500 patients.

ShowHide solution

(a) A back-to-back stem-and-leaf diagram: both classes share the stems, so their marks can be compared row by row, and every mark is kept.

(b) A pie chart: the stalls' sales are parts of one whole, the canteen's total sales.

(a) back-to-back stem-and-leaf diagram (b) pie chart (c) grouped histogram, for the reasons given.

Question 3

The stem-and-leaf diagram and the histogram show the same data: the ages of the 20 members of a club. Which diagram, or both, can answer each question? Answer it if you can. (a) How old is the oldest member? (b) How many members are in their thirties (30 to 39)? (c) How many members are under 35?

ShowHide solution

(a) Only the stem-and-leaf diagram: 6 | 1, so 61 years. The histogram shows only that one member is in $60 \le a < 70$ .

(b) Both: 7 leaves on stem 3; the bar $30 \le a < 40$ reaches 7.

(c) Only the stem-and-leaf diagram: 23, 25, 28, 31, 32, 34, 34, which is 7 members. The histogram cannot tell how many of the 7 in $30 \le a < 40$ are under 35.

(a) stem-and-leaf only: 61 years (b) both: 7 (c) stem-and-leaf only: 7

Question 4

The dot diagram and the histogram show the scores, out of 20, of 20 pupils in a quiz. In the histogram the last class, $15 \le x \le 20$ , includes 20. (a) What is the most common score? Which diagram did you use, and why can the other not tell you? (b) How many pupils scored at least 15? (c) Which diagram keeps every score?

ShowHide solution

(a) From the dot diagram: the tallest column is at 12 (3 pupils). The histogram shows only the modal class, $10 \le x < 15$ ; it does not show which score in that class is the most common.

(b) Either diagram: the dots at 15 to 20 number $2 + 1 + 1 + 2 + 1 = 7$ , and the bar for 15 to 20 reaches 7.

(a) 12, from the dot diagram; the histogram shows classes, not single scores. (b) 7 (c) the dot diagram

Question 5

Mei drew this pie chart of the masses of the school bags of 4 friends: 4 kg, 5 kg, 6 kg and 5 kg. Why is it a poor choice? Suggest a better diagram.

ShowHide solution

The pie shows each bag's share of the total mass of the four bags, 20 kg, but that total is not a whole that means anything. No mass can be read from the sectors, and they look nearly the same.

A bar graph with one bar for each friend shows each mass and compares them.

The masses are not parts of a whole, and the pie hides the actual masses. A bar graph is better.

Question 6

For each set of data, would you choose a dot diagram, a stem-and-leaf diagram or a grouped histogram? Give a reason. (a) The scores out of 10 of 30 pupils in a quiz. (b) The marks out of 100 of 25 pupils, keeping every mark. (c) The times of 400 runners in a school cross-country race.

ShowHide solution

(a) A dot diagram: the scores are whole numbers with only 11 possible values, so 30 dots stack into clear columns and show every score and the most common one.

(b) A stem-and-leaf diagram: the marks have two digits and are spread from 0 to 100, so a dot diagram would need a place for every mark, with most columns one dot high. The stem-and-leaf diagram keeps every mark and still shows the shape.

(c) A grouped histogram: 400 times, almost all different, are too many to draw one by one as dots or leaves. Grouped into classes of equal width, they show the shape.

(a) dot diagram (b) stem-and-leaf diagram (c) grouped histogram, each for the reason given

Question 7

For each diagram, answer "Yes" or "No" to two questions: (i) Does it keep every value? (ii) Does it show the shape of numerical data? (a) A dot diagram. (b) A stem-and-leaf diagram. (c) A grouped histogram. (d) A pie chart.

ShowHide solution

(a) Dot diagram: (i) Yes, one dot for each value. (ii) Yes.

(b) Stem-and-leaf diagram: (i) Yes, one leaf for each value. (ii) Yes, on its side.

(d) Pie chart: (i) No. (ii) No: it shows the shares of categories in a whole.

(a) Yes, Yes (b) Yes, Yes (c) No, Yes (d) No, No

Question 8

A CCA teacher drew this line graph of the number of pupils who chose each of four CCAs. What is wrong with the choice of diagram, and what should be drawn instead?

ShowHide solution

The line suggests that the number of pupils falls from Basketball to Chess and then rises to Drama, as if one led to the next in time. The CCAs are separate categories; their order means nothing.

A bar graph (or a pie chart, to show shares of the whole group) is right.

A line graph is for change over time; the CCAs are categories, so a bar graph should be drawn.

Question 9

A property agent has the resale prices of 300 HDB flats sold in one town last year. (a) Which diagram should she use to show how the prices are spread? (b) Which diagram should she use to show how the average price in the town changed each year from 2015 to 2024? Give reasons.

ShowHide solution

(a) A grouped histogram: 300 prices with many different values are grouped into classes (for example, $50 000 wide) to show the shape.

(b) A line graph: one price for each year, changing over time.

(a) a grouped histogram (b) a line graph

Key Points

Choose a diagram by the kind of data and by the purpose, and give the reason.
Categorical data: bar graph, pie chart (shares of a whole) or pictogram. Change over time: line graph.
Numerical data: dot diagram or ungrouped histogram (few different values), stem-and-leaf diagram (keeps two- or three-digit values; back-to-back compares two sets), grouped histogram (many different values or a large set).
Dot diagrams, stem-and-leaf diagrams and ungrouped histograms keep every value; a grouped histogram shows only how many values are in each class.
A grouped histogram cannot give the highest value, the mode or a count that cuts a class.
A pie chart of numbers that are not parts of a whole, or a line joining categories, is a poor choice.