## 6 August 2014

### Teaching Box and Whisker Plots

 Image: ruthmaas.com
When I wrote Long Live Stem and Leaf, I'd been challenged to find a practical application of stem and leaf diagrams outside the maths classroom. After extensive googling - and helpful input from twitter - the only examples I could find were bus and train timetables. But I argued that real life application isn't the be all and end all.

Stem and leaf diagrams were invented in the 1970s by John Tukey, who also invented box and whisker plots. As a matter of interest, I've also investigated real life applications of box plots. I found that box plots are widely used in research papers and analyses. Unlike stem and leaf diagrams, they are a statistical tool genuinely used by statisticians. Because box plots are commonly used, much has been written about their effectiveness. As this blogger says, “a box plot is a simple yet powerful tool, with a truly great design - a universally beautiful thing that stands the test of time”.

In this post, I’m going to focus on teaching ideas and resources.

Practical activities
I’ve read a lot of blog posts about human box plots. For example, 11 students line up at the front of the classroom in height order. Their classmates determine who is the median, lower hinge and upper hinge (yes, I’m using the word hinge instead of quartile. Read my blog post on Consistent Mathematics if you want to know why). This is a lovely activity in a lesson introducing the idea of median, but not so effective in a box plot lesson. Because even though you can use a bedsheet and string to get the students to look like a box plot, if they’re not appropriately spaced out (ie if there’s no scale) then it’s not a box plot. If you want a proper box plot you’ll have to actually measure the heights and draw a scale on the floor.

An alternative idea (inspired by this post) involves paper aeroplanes. You’ll need a big space for this - the school hall or playground. Tape (or draw in chalk) a long scale on the floor. Have each of your pupils make a paper aeroplane, write their name on it, and throw it from the start of the scale. Leave the aeroplanes where they land. Once all pupils have thrown their aeroplane, you can draw or tape a box plot on the floor around the planes to represent the distance flown. Pupils will be able to see which quartile their plane landed in. Hopefully one future engineer will throw an amazing outlier which will make a good discussion point! You could even split the class in half and do two box plots on the floor side by side, then the class could discuss which team had better aeroplanes and which team’s planes were more consistent.

If you like practical activities, here’s two more:
• This simple paper folding activity is a nice introduction to median and quartiles.
 Source: bigideasmath.com

Resources
There’s surprisingly few good box plot resources online. But Don Steward never lets us down. His website has a number of box plot activities - my favourite is this true or false activity.

I also like the activity below from bigideasmath.com:
If you’re looking to generate box plots on your interactive whiteboard for class discussion, this online tool is nice – it has pre-populated data or you can input your own.

I've listed some more good box plot resources below. I’ve found that many resources ask questions about skewness, which we don’t cover until S1 (but perhaps we should cover earlier). Treatment of outliers is also not covered at GCSE but makes for a nice discussion.

If you want to show your class some ‘real world’ box plots then these box plots showing the ages of World Cup players could prompt some good discussions about comparisons. Read @loumeracy’s blog post for more on this.

If you want to ‘wow’ your class with an example of how box plots can be used to compare huge amounts of data in a small space, show them this example which shows the age distribution of Olympics athletes (more of this here).

Final thoughts
In French the box plot is called boîte à moustaches (box with a moustache). The writer of this article attempts to create a box and beard plot!

There are many variations to the box and whisker plot, such as the bean plot and violin plot. Look these up on google images for lots of examples.

I read an interesting idea here that I’d never thought of before: “One way to help you interpret box plots is to imagine that the way a data set looks as a histogram is something like a mountain viewed from ground level and a box and whisker diagram is something like a contour map of that mountain as viewed from above”.

Well, there you are – a 'box plot blog post'. Try saying that five times out loud. I've invented a new tongue twister.

1. Another interesting read thanks. It's always useful to know who invented/discovered various concepts and I like seeing where box and whisker plots are used (or have been used).

I knew that there would be a formula for working out which data would be classed as outliers, but couldn't remember it (not something I use every week!). This is quite a reasonable explanation: www.purplemath.com/modules/boxwhisk3.htm Some more: http://web.pdx.edu/~stipakb/download/PA551/boxplot.html

1. Thanks! I like your second link which describes the 'yellow card zone' and 'red card zone'. Not seen that before, will use it in S1.

2. nice sharing

3. Nice post, thanks for sharing. Recommend also following
this web page .

4. I do human box plots -- but not the way you describe them. Instead, every student comes up to the front of the room and we mark in a vertical line on the white board where the top of that student's head is. Once all the marks are on the board, we find the five number summary. This also prevents lots of tittering about the unforunate shortest and tallest students.

In my opinion the 1st and 3rd quartiles should not be called hinges. Their proper name is quartiles.

5. The paper airplane idea is a huge time waste. If you don't want to line students up at the board and measure their heights to quickly create box plots, then do a simple heart rate experiment -- take sitting pulse and record, then take pulse after 30 seconds ofd jumping jacks and record. You'll get two box plots with outliers, you'll have the fun of deciding what to do with the student who claims they have no pulse at all or a pulse of 250, and you then have two comparable box plots so that you can do what we actually do in the real world with box plots -- use them to understand and compare large data sets