O-Level E-Math: Misleading Statistical Diagrams (Misinterpretation of Data)
In the new O-level E-math syllabus (4048), an additional segment has been introduced to the statistics Data analysis section. This segment is about explaining why a given statistical diagram leads to misinterpretation of data.
Statistics can be easily twisted in favor of the presenter to distort or prove a point. As such, it is important for us to equip ourselves with knowledge to recognize the common statistical deception so as to avoid being misled.
The following are some of the important points to consider when looking at a graph:
1. Title
- Is it biased?
2. Pie Charts
- Are the section sizes proportional to their numbers?
- Does the percentages add up to 100%?)
3. Axes
- Are both axes labelled properly?
- Is the scale uniform and equally spaced?
- does the axes start from 0?
4. Data
- Are there any data that is missing or purposely left out?
5. Scale
- Are there any 3D effect that distorts actual size
- For pictorial representation of sizes, is it clear whether the length, area or volume of the figure represents the actual data size?
Worked Example 1:
The graph shows the results of voting for a Student Council President election in a school.
Explain one way in which the graph is misleading.
Solution:
1) The size of each sector is not proportional to the percentage of votes. This gives the impression that candidate B achieve more than twice the number of votes of A and C.
2) The percentages do not add up to 100%. A significant group of students (people who did not vote or has voided votes) are purposely left out and not represented in the graph.
3) Title of the bar chart is biased. It does not allow reader to make own judgement .
[AMKSS Prelim]
Worked Example 2:
The following chart showing the average number of viewers watching the evening news broadcast was published in a newspaper.
List two ways in which the graph is misleading.
Solution:
1) Vertical axis does not start from zero. It exaggerates the differences between the number of viewers.
2) Title is biased. It does not allow readers to make own judgement and misguides readers who are not discerning.
3) Missing data from 2012. Misrepresents the trend and there could have been a drop in viewership in 2012 or 2013 that is not reflected.
[TKGS Prelim]
Worked Example 3:
Doug’s Dog Food Company wanted to impress the public with the magnitude of the company’s growth. Sales of Doug’s Dog Food had doubled from 2002 to 2003, so the company displayed the following graph, in which the radius of the base and the height of the 2003 can are double those of the 2002 can. What does the graph really show with respect to the growth of the company?
Solution:
The three-dimensional drawing distorts the graph and exaggerates the actual increase in sales. A double in sales figures would appear like an increment by a factor of 8. This is because V=πhr^2, the volume would increase by a factor of 8 when doubling the radius and the height of the can.
[Arkansas Tech University 2006]
Practice Problems:
Practice makes perfect. Please attempt the following questions on misleading by yourself before referring to the answers below! These questions will be on explaining why a given statistical diagram leads to misinterpretation of data.
Practice Problem 1:
1) The pie chart shows the sales for 4 different brands of hand phones.
Explain one way in which the pie chart is misleading.
[HIHS Prelim]
Practice Problem 2:
2) The graph shows the number of prize winners at the Mathematics Olympiad over a number of years.
Explain one way in which the graph is misleading.
[CCHS(Main) Prelim]
Practice Problem 3:
The line graph shows the late coming occurrence of students in ABC School over 5 months.
Write down two statistical misrepresentations of this line graph.
Practice Problem 4:
The following statistical diagrams are used to illustrate that the number of elementary teaching majors in a college has doubled from 1993 to 2003, while the ratio of male to female teaching majors did not change. What is misleading about the way the graphs are constructed?
Practice Problem 5:
After looking at the pictograph above, Alvin claims that Men are worse drivers than women. Explain whether his claim is justified?
Solutions to Practice Problems:
Practice Problem 1
Misleading feature(s) :
1) Brand A and Brand C have equal percentage but the proportion on the pie chart are not equal
2) Title is biased
Effect of this feature on the graph :
1) It mislead people into believing that Brand A is selling better than Brand C
2) It does not allow reader to make own judgement.
Practice Problem 2
1) The y-axis (does not start at 0) starts at 15. This exaggerates the increase over the years. The actual increase from 2015 to 2016 is only 50% but the graph exaggerates it to look like a 300% increase.
2) The scale of the y-axis is not uniform so the length of each bar is not representative of the number of prize winners.
Practice Problem 3
1) It is not clear whether the vertical axis represents the number of students or the percentage of the students.
2) The scale of the vertical axis is not consistent and the intervals between the values on the vertical axis are not equal. This distorts the graph and cause changes at higher values to seem less significant than changes at lower values.
3) The vertical axis is truncated and does not start from zero. This exaggerates the differences in students between each month.
Practice Problem 4
When the radius of a circle is doubled, the area is quadrupled, which is misleading since the population has only doubled. Recall that the area of a circle of radius r is A=πr^2. Thus, this statistical diagram exaggerates the increment in number of elementary teaching majors.
Practice Problem 5
His claim is not justified because it is unclear whether the diagram shows the number of fatal accidents or the percentage of fatal accidents. In addition, it is unclear whether the diagram takes into account the total number of male to female drivers that is sampled. If there are more male than female drivers on the road, there would naturally be a larger number of male fatal accidents to female fatal accidents.
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