# Multiple mental models with median

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- Multiple mental models with median

## Multiple mental models with median

25 April 2019

Picture a graph of any data you like with a median of 25.

What do you see?

What type of graph is it? A bar chart? A dot plot, a cumulative frequency curve? Or do you see something else? What does it look like? Is there a peak? Is it stretched out? Is it smooth, or spiky? Or perhaps you aren’t picturing a traditional ‘graph’ at all.

I have taken to asking variations of this question when I am talking about statistics to teachers – and the answers are illuminating, prodding (albeit in a limited way) at people’s mental models of statistical ideas.

Understanding variation and distribution have long been identified as key elements of statistical thinking, and research argues that the aim of statistics instruction is to develop these concepts. So what does it mean if a teacher draws a blank when asked this question? It may be highly suggestive that these ideas of variation and distribution have not been connected to the individual skills that students are expected to learn, with median conceived as a statistic that represents the data in isolation, the result of a learned calculation or process.

In fact, the median could be considered as just one element of the aggregate distribution created by the underlying data; one small part of the shadow on the wall of Plato’s cave, through which we seek a pattern in the variation. In some sense the median ‘exists’ as a parameter of this aggregate alongside others such as spread, skewness, density and outliers.

So what could you have sketched in answer to the question? The graphs below give some possibilities:

It is a notable quirk that we often encourage students studying algebra to sketch graphs of functions designed to give an indication of the overall pattern and any key features, and yet we rarely do so for statistics, even though this tactic is equally legitimate here. The graphs above fulfil this role: they give an impression of the distribution without being time consuming or requiring high standards of accuracy or artistry.

The data represented by each sketch graph has a median of 25 but they all look very different. Why might this be? Can you imagine a data set that might have produced each one? Can you imagine more graphs that fit this parameter?

It is in this way that we can begin to develop a sense of the difference between novice and expert statistical thinking. The novice may have no picture of data in mind when being told that the median of a data set is 25 – they may simply see the number as an average value that has been calculated. Possibly they may have just a single image in mind; more than likely a recognisably normal-shaped distribution – the famous bell-shaped curve – with 25 at its centre. An expert statistician, however, may have all the images and more in mind fairly simultaneously; a superposition of possibilities that collapse as more parameters are uncovered until there can be only one.

This then is the goal of statistics teaching and learning: to encourage a rich experience of data that allows learners to develop a model of distribution that is robust enough to contain multiple images, and flexible enough to allow learners to move between them and make important decisions based on available (and perhaps changing) information.

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