Percentage Perception
'What do these errors in perception really tell us?'
Think about the country you live in. For every 100 people that live in your country, how many of them do you think are Muslim? How many are Christian? The questions on this theme could continue...... If we took the average guess of a sample of people in that country, how close do you think it would be to the actual number? Close? Over estimated? Under estimated? If there is an error, how should we measure it? This activity is about all of these questions and works with in article that appeared in the news in late 2014 with the results of just such a survey. The results, as ever, are really interesting, but how is the reader supposed to interpret them? What conclusions are we supposed to draw? This is the bit that depends entirely on your understanding of mathematical ideas. This is another clear example of how difficult the media can be to understand if without mathematical thinking!
Aims
Students should practice working our percentage error and then consider carefully the differences between actual error and percentage error and when one or the other might be a better measure. Students will also grapple with some real world issues and some deep Theory of Knowledge ideas about ranking these results and the different ways it can be done.
Resources
The activity can be run in a number of ways - the easiest of which may be to simply hand out the task as it appears on this 
Percentage Perception activity sheet.
The answers to the first part of the activity can be found on this article on which the activity is based
Teacher Notes
Overview
- Students are told about the survey and then given the table with blanks in.
- Some work on calculation of percentage error may be necessary before or during.
- Students fill in the blanks by working backwards and forwards through percentage error calculations. These get more difficult as less information is given.
- Some thinking and reasoning is required.
- When enough students have successfully filled in the table, the answers can be shown.
- Then a discussion should ensue about which might be the best way to rank the results.
- Students might then explore the scatter graph.
- If time allows, students might explore or or more of the other surveys
- Students should make a record of the experience with some discussion of the key points.
- The activity provides a great opportunity to practice writing about mathematics.
Solutions
Click on the hidden box icon below to show the completed table
If you get to the point where you consider the scattergraph then you can show it using this hidden box.
ToK Moment
This task is also loaded with ToK references that it warrants this little section on the page. Often, students will make these links for themselves, but just in case, here are some key points.
Patterns in estimation - Do we have a tendency to overestimate things in general or just the number of Muslims in our country? Do we underestimate stuff as well? How and why does this happen?
Influence of media on opinions - To what extent is media coverage to blame for inaccurate perception of these numbers? The temptation is to say 100% straight away, but answers need to be tempered by discussions to the point above.
How to present the data? - Well, the exercise invites us to think about what is the best way to present the information....
- Ranked by the size of the difference between the guess and the reality
- Ranked by the percentage error
- As a scatter graph
Each method involves the correct application of a mathematical technique to the same information, but all three tell a different story! There are good arguments for all three methods, so where does that leave us? How are we supposed to interpret the information?
The activity might invite us to say that the article would have been more accurate if it ranked by percentage error, but this could be countered by the 'Jedi knights are the worlds fastest growing religion' argument where the population grew from 2 to 4. That is clearly a case where percentage change is not the most appropriate.... Check out this activity as a fun follow up Percentage Error
The best conclusion we can reach is that fro every headline and data set there is a good deal of work to be done before any useful conclusions can be drawn...
Have a great discussion!
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I did it my way!
As a practising mathematics teacher I know that most us like to give activities our own little twist and do them 'our way'. It would be great to add a little collection of 'twists' from users. You can either add your twist to the comments section below or e-mail them directly to me at [email protected] In time some of these twists may appear here....