10.5: Misleading Statistics
Introduction
Fundraising
Because the track team is going to regionals, they need to raise some money for the bus and the hotel. The boys team has decided to work on selling chocolate bars. They figure that this will be an excellent choice for a bunch of students.
“It is easy to sell and not expensive,” Chris says as he sells candy bars at lunch time.
At the end of the fundraiser, the team began to tally up their totals. Mark sold 72 bars and Clint sold 65 bars. Mark wants to make it look like he has sold more than Clint. Is this possible? How can he do it?
To accomplish this goal, while a dishonest, Mark will need to create a graph with misleading statistics. This happens all the time in the real – world. We will use Mark and Clint as an example of how easily people can be lead astray by misleading statistics.
What You Will Learn
In this lesson, you will learn how to complete the following skills.
- Identify and analyze misleading data displays.
- Identify and explain misleading comparisons of data.
- Design data displays to intentionally exaggerate or minimize possible comparisons.
- Revise data displays to eliminate misleading representations or conclusions.
Teaching Time
I. Identify and Analyze Misleading Data Displays
What does it mean to mislead? When we mislead someone, we want them to think something other than what is the truth. Sometimes, there can be data displays that are misleading. Let’s think about this.
When we have built graphs and plots, we have taken care to show the data correctly by choosing appropriate intervals and scales. There are a variety of ways that graphs can be displayed in a way that is misleading. At times, we simply make mistakes. Other times, however, people may try to convince us of something by manipulating a graph that may otherwise not truly support the data. We must keep a critical eye when we read graphs and compare the data.
Example
Consider the graph below of the Yula High School Graduation Rate.
We use bar graphs because they are visual devices which allow us to compare the size of the bars to interpret data.
What do the bars show you about the graduation rate at Yula High School?
It appears that the graduation rate skyrocketed. However, if you look at the scale on the \begin{align*}y\end{align*}-axis, you can see that the entire graph is not represented. The scale only represents a range from 81 to 88 percent. This exaggerates the difference in the data. Was there an improvement from 1991 to 1992? Yes. But the improvement was only 4% where the bars can mislead into thinking that the improvement was much greater.
So you can see that the size of the intervals and bars can impact the way we interpret the data.
Example
A sales manager at Bank X prepared a report of the number of new clients they have in the quarter compared to their competitors. He prepared a graph and declared that, although they had fewer new clients, they’re not far behind. What do you think?
The graph is misleading. The number of new clients looks similar. But look at the data table:
Bank X | Bank Y | Bank Z |
---|---|---|
325 | 475 | 517 |
Bank Y has 46% more new clients and Bank Z has 59% more new clients than Bank X. Because the scale on the \begin{align*}y\end{align*}-axis goes up to 1200, it decreases the relative size of the bars on the graph. An appropriate scale would extend to no further than 600.
Once again, we have to look at how the graph is constructed to see if the data is misleading or not.
II. Identify and Explain Misleading Comparisons of Data
We can also look at what it means when we have data that compares. In the last example, we compared different banks with new accounts. We can also look at comparisons within other arenas. Once again, look for misleading representations of the data.
Example
A hospital director was up for his evaluation. He wanted to show that he has helped the community in his years on the job. He prepared this graph for the evaluation committee to show how his work has helped to reduce cases of the flu.
True or false? Did flu cases decrease? Yes. Are they way down?
No. What trick did he use to mislead the committee? He did not change the scale but changed the size of the bar. The 1990 bar is very wide; it gives the appearance of being much bigger than 2000. Of course, the width of the bar does not make a difference for what the graph means. The flu cases dropped from about 875 to about 775. It is about a 12% decrease.
You can see that even if the comparison looks like it is true, we have to examine the actual data to see if the data display is accurate or misleading.
III. Design Data Displays to Intentionally Exaggerate or Minimize Possible Comparisons
As you can see, it is important to keep an eye on the data and the way that it is presented to us. In order to improve your skills in spotting misleading data, let’s see how you can design data displays to intentionally exaggerate or minimize comparisons. Look at this example.
Example
Mei Ling knows that her grade point average has dropped but doesn’t want her parents to notice. It went from 3.75 to 3.25 in one semester. How can she change manipulate a graph to mislead them? She decides to change the width of the bars to make the lower score look bigger.
In this graph, the second semester bar looks even bigger than the first. Of course, it’s not as tall. Do you think her parents will notice? Why or why not? Discuss your opinion with a neighbor.
IV. Revise Data Displays to Eliminate Misleading Representations or Conclusions
Now that you know how to look for misleading data representations and how to create data displays that are intentionally misleading, we can move on to correcting misleading displays. Once you have discovered that a display is misleading, the next step is to help to revise the data display so that the data is accurately shown.
Example
Four team members on the swim team recorded the following data for the 400 meter relay.
Team Member | David | Manual | Andre | Luke |
---|---|---|---|---|
Seconds | 32.5 | 36 | 34 | 33.5 |
David made a graph of the data. Here is his graph.
They all told Manuel that he needs to work harder because he’s bringing the team down. Manuel has been working hard and didn’t believe the graph. He noticed that the scale was designed to show a major difference between their times.
Manuel made the graph this way.
Notice that because of the intervals that Manuel used that his time wasn’t that far from the rest of the group. Manuel’s data display was much more accurate.
Now let’s go and apply what we have learned to the problem from the introduction.
Real-Life Example Completed
Fundraising
Here is the problem from the introduction. Reread it and then create a graph to mislead. When finished, correct your work with a friend to show an accurate graph.
Because the track team is going to regionals, they need to raise some money for the bus and the hotel. The boys team has decided to work on selling chocolate bars. They figure that this will be an excellent choice for a bunch of students.
“It is easy to sell and not expensive,” Chris says as he sells candy bars at lunch time.
At the end of the fundraiser, the team began to tally up their totals. Mark sold 72 bars and Clint sold 65 bars. Mark wants to make it look like he has sold more than Clint. Is this possible? How can he do it?
Remember, there are several parts to your answer.
Solution to Real – Life Example
Here is how we can create a misleading graph.
Create a bar graph and skip numbers in the scale. In an appropriate graph, a scale may reach 75 or 80, beginning from 0. However, if Mark skips 0-60 on her scale, the bars will look extremely different.
In this graph, it appears that Mark sold far more bars than Clint when in reality it was only 7 more bars. You can see how changing the size of the bars can influence how the data is viewed.
Now work with a partner and create a graph that is accurate and shows that Clint sold more bars than Mark did.
Time to Practice
Directions: Answer each of the following questions true or false.
- To sell more a product, a company may create a display that misleads consumers.
- You can create a graph to make it look like you have sold more of a product that you actually have.
- Misleading statistics aren’t that relevant in sales.
- Graphs aren’t actually misleading at all.
- You can create a misleading graph only if your intervals are too small.
- You can create a misleading graph whether your intervals are too small or too big.
- The height of the bars in a bar graph can be misleading.
- If the bars of a graph are too wide this can be misleading too.
- You must be careful whenever you read a data display to be sure that the data is accurate.
Directions: Why are the following graphs misleading? What is the error in the conclusions drawn based on the graphs?
- Conclusion: “The population in Dagwood is exploding!”
- Conclusion: “George’s Café is far more successful than Rita’s Restaurant.”
- Draw a graph to intentionally exaggerates this data: “From 2002 to 2004, the average number of semesters that students studied in order to complete their Bachelor’s Degree increased from 4.1 to 4.5.”
- Explain the faulty conclusion that could be drawn from this graph.
- Draw a graph to intentionally minimizes this data: “Mike had 110 customers on his paper route in March and in June only 75.”
- Explain the faulty conclusion that could be drawn from this graph.
- Revise your graph in number 3 to represent the data more accurately.
- Revise your graph in number 5 to represent the data more accurately.
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