Fun With NUMBERS
CATHERINE JOHNSON
One of the most common ways that people encounter mathematics in real life is through statistics.
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| A man looks at a financial chart labelled ‘Trouble in Thailand’. The graph depicts the course of the strengthening baht in 2006. WWW.ECONOMIST.COM |
Even before you finish reading this newspaper, you will encounter all sorts of data about the economic crisis and the popularity of various politicians on the world stage.
These data are used to try to model complex situations and make predictions about the future behaviour of stock markets or voters, but the information is all too often used in misleading ways.
Averages
Perhaps the most commonly misused statistical word is "average". There are different types of average and they can be used to deceive. Suppose that I am the owner of a company and I want to advertise the fact that my company offers a good salary.
Although most of my workers earn a small wage, I could include the huge salaries of my directors and calculate the mean by adding up all the wages and dividing the total by the number of workers. Because I have included some very big salaries, the average is scaled up.
Now, suppose that I want to use a small value as the average, perhaps to show that the bonus my company offers is a big percentage of the average wage. I would choose the median, which is the middle value (so 50 per cent of my workers earn less than the average, and 50 per cent earn more). Because the big salaries of the few people at the top do not affect the median, the average wage is smaller.
Sample size
My mythical company manufactures and sells weight-loss products. I want to collect some data to use in advertisements to show that my product really does help people to shed a few kilos, so I decide to test the product on a sample of people.
Now, large samples do not show much variation, so I choose to measure the weight loss of several small groups instead. Small samples show more varied results, so eventually I will get one group that records an unusually large weight loss.
This situation is like throwing a coin. If you throw hundreds of times, the number of heads and tails will be close to 50-50, but if you only throw 10 times and keep repeating the test, eventually you will get one sample that shows a large number of instances of "heads". This is the unusual sample that I want to use in my advertising, and I will have the evidence to back up my claim that people using my product will lose weight.
Correlation
I like to think that I am quite an honest person, so maybe I would not deliberately cheat. I decide to use a large sample of people and start them off on their regime, which includes my diet product.
After two months, I am delighted to find that they have lost weight! I have found an example of positive correlation, where there seems to be a genuine connection between using my product and weight loss. Therefore, my product caused the weight loss - didn't it?
Not necessarily, no. Many people confuse correlation with cause and effect. The people in my sample all knew they were taking part in an experiment to investigate weight loss and, presumably, most of them wanted to lose a little. The new regime could have encouraged them to make other lifestyle changes, such as avoiding fatty foods and taking more exercise. This could be what caused the weight loss, rather than my product.
Graphs
Presenting statistics in graphical form helps the reader to see any trends or patterns very easily, but pictures can also mislead. If my company showed an increase in profits from 10 million baht to 11 million, I could represent this on a simple line graph. If my scale goes from zero to 11 million, the rise in profits will look like a slight increase. However, suppose I selected a scale going from nine to 11 million. The gradient of the line looks much greater, as though profits have doubled.
These methods are only very simple examples of how statistics can be used to mislead and misdirect. When combined with persuasive advertising language, such statistical manipulation can be very convincing. Try to look behind the figures and question their validity.
If you aren't told which average has been used, why not? If samples have shown a positive result, how were the samples collected? If there really is a mathematical connection between two sets of data, does it mean that one factor caused the other to happen? Would the dramatic graph you are looking at look quite so impressive with a different scale?
Statistics is a very useful practical application of mathematics, but always bear in mind that it is often more of an art than a science.
Catherine Johnson is the head of mathematics at Shrewsbury International School. For more information, or if you have comments, you may email her at
catherine.mathematics@gmail.com .
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