The school track team needs to raise some money to go to regionals. The teams decided to work on selling chocolate bars. They figure that this will be an excellent choice for a bunch of students. At the end of the fundraiser, the team began to tally up their totals. Mary sold 72 bars and Clint sold 65 bars. Mary wants to make it look like she has sold a lot more than Clint. Is this possible? How can he do it?

In this concept, you will learn how to understand misleading statistics.

### Misleading Statistics

When you **mislead** someone, you want them to think something other than what is the truth. Sometimes, there can be data displays that are misleading.

When you construct graphs and plots, you take 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. Sometimes people simply make mistakes. Other times, however, people may try to convince you of something by manipulating a graph that may otherwise not truly represent the data. You must keep a critical eye when you read graphs and compare the data.

Consider the graph below of the Yula High School Graduation Rate.

You use bar graphs because they are visual devices that allow you to compare the height 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.

Let’s look at another example.

A sales manager at Bank X prepared a report of the number of new clients they have in this quarter compared to their competitors. He prepared a graph and declared that, although they had fewer new clients, they’re not far far behind. What do you think?

The graph is misleading. The number of new clients looks similar. If you look at the data represented, you see the following:

Bank X: 325 new accounts

Bank Y: 475 new accounts

Bank Z: 517 new accounts

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, you have to look at how the graph is constructed to see if the data is misleading or not.

As you can see, it is important to keep an eye on the data and the way that it is presented to you. 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.

Let’s look at an 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?

### Examples

#### Example 1

Earlier, you were given a problem about the chocolate bar sales. For the team fundraiser, they are going to sell chocolate bars. Mary wants to make it look like she has sold a lot more than Clint. She sold 72 bars and Clint sold 65 bars.

First, 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 Mary skips 0-60 on her scale, the bars will look extremely different.

In this graph, it appears that Mary sold far more bars than Clint when in reality it was only 7 more bars. You can see how changing the \begin{align*}y\end{align*}-axis intervals can influence how the data is viewed.

#### Example 2

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.

Did flu cases decrease? Are they way down?

The answer is 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, you have to examine the actual data to see if the data display is accurate or misleading.

#### Use this graph to answer the following questions.

#### Example 3

Is there anything misleading about the data on the \begin{align*}x\end{align*}-axis?

The answer is no, the dates are evenly spaced.

#### Example 4

Is there anything misleading about the data on the \begin{align*}y\end{align*}-axis?

The answer is yes. The \begin{align*}y\end{align*}-axis shows the amount of money won. The money amounts are not evenly represented.

#### Example 5

How could you fix this graph?

You could fix the graph by making the intervals on the \begin{align*}y\end{align*}-axis are even. Then re-draw the line to accurately represent the data.

### Review

Answer each of the following questions true or false.

1. To sell more a product, a company may create a display that misleads consumers

2. You can create a graph to make it look like you have sold more of a product that you actually have.

3. Misleading statistics aren’t that relevant in sales.

4. Graphs aren’t actually misleading at all.

5. You can create a misleading graph only if your intervals are too small.

6. You can create a misleading graph whether your intervals are too small or too big.

7. The height of the bars in a bar graph can be misleading.

8. If the bars of a graph are too wide this can be misleading too.

9. You must be careful whenever you read a data display to be sure that the data is accurate.

Why are the following graphs misleading? What is the error in the conclusions drawn based on the graphs?

10. Conclusion: “The population in Dagwood is exploding!”

11. Conclusion: “George’s Café is far more successful than Rita’s Restaurant.”

12. Draw a graph to intentionally exaggerate 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.”

13. Explain the faulty conclusion that could be drawn from this graph.

14. Draw a graph to intentionally minimize this data: “Mike had 110 customers on his paper route in March and in June only 75.”

15. Explain the faulty conclusion that could be drawn from this graph.

16. Revise your graph in number 3 to represent the data more accurately.

17. Revise your graph in number 5 to represent the data more accurately.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 10.9.