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# Wrong, right, misleading?

Saturday 8 June 2013

### What is wrong with this picture?

This post is a great example of the type of distraction that crosses my path all too often these days. My life was so much simpler before twitter and facebook. This one started when I picked up this article from the bbc about football transfer fee records. Please don't let a disinterest in football stop you reading, this is not about football! I was enjoying the graph at the bottom of the page and set about trying to fit a model to it. It was then that I realised there was no scale on the x-axis and started scoffing at the use of misleading graphs in popular media. I dutifully posted this example in the Mathematics Teacher Exchange facebook group, where a couple of teachers quickly shared my sentiments and I bookmarked the page for future reference. It was later that day when Adil Jaffer commented that he could not see what was wrong with the graph, that I started to think more about it and the many ways this sort of thing can be used in mathematics teaching. Below is a bigger version of the graph in question...

The problem I have is that there is no scale on the x-axis and as such, the interpetation of the graph as a time series is dubious. Adil pointed out that it was only the fact that numbers were used on the x-axis that made me feel this way (or words to that effect) and that if the football players names were used then I would not have had the same reaction. See the graph below as an example of this....

I have to agree. Still though, I have some conflict. So often with this type of graph students will be confused about the significance (if any) of the order the items in the x-axis appear. Here, the order is highly significant because we are looking at the progression of record transfer fees over time. For example, the graph below shows the same data with the players in alphabetical order by first name....

By the same token, this graph is not incorrect, but clearly tells us very little indeed. At this point the focus shifts to 'what are you trying to tell us with this graph?' and it seems obvious to me that the main message is to show how the transfer record fee appears to be growing exponentially against time - or perhaps to simply show how it is changing so that the reader can draw their own conclusion. At this point I wanted to see what the data would look like if there was a scale on the axis. As such, the following is a scattergraph.....

Granted that this is less visually appealing! That said, it strikes me that this shows something much more akin to exponential growth than the original graph. As such, I think it tells a better story. Anyway, the whole exercise has left me inspired to do more of this alternative representation of the same data with students to help them grasp the subtelties of different types of representation that have been occupying the minds of those of us discussing this article. In doing so I am thinking about the following questions that can lead to activity about this and perhaps taking some examples and representing the data differently and asking students to judge them regarding correctness, relevance and usefulness.

Given the following data, what would do with it? What story does it tell? How would you tell it?

I have put the data in a google spreadsheet too

So, on balance, I am agreeing with Adil that there is nothing wrong with the graph, but at the same time arguing that the reader has to be aware of a number of mathematical subtelties to properly interpet the graph and that it is not the best way to present it and even that the article would have delivered an even more startling message had they used a scale and posisbly even used it to predict future fees!

For interest, I have also been reminded of how much great data there is available out there to play with when I found the following two articles on wikipedia about transfer fee records.

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