Section author: Danielle J. Navarro and David R. Foxcroft

Box plots

Another alternative to histograms is a box plot, sometimes called a “box and whiskers” plot. Like histograms they’re most suited to interval or ratio scale data. The idea behind a box plot is to provide a simple visual depiction of the median, the interquartile range, and the range of the data. And because they do so in a fairly compact way box plots have become a very popular statistical graphic, especially during the exploratory stage of data analysis when you’re trying to understand the data yourself. Let’s have a look at how they work, again using the afl.margins variable from the aflsmall_margins data set as our example.

Fig. 23 Box plot of the afl.margins variable from the aflsmall_margins data set plotted in jamovi

The easiest way to describe what a box plot looks like is just to draw one. Click on the Box plot check box and you will get the plot shown on the lower right of Fig. 23. jamovi has drawn the most basic box plot possible. When you look at this plot this is how you should interpret it: the thick line in the middle of the box is the median; the box itself spans the range from the 25th percentile to the 75th percentile; and the “whiskers” go out to the most extreme data point that doesn’t exceed a certain bound. By default, this value is 1.5 times the interquartile range (IQR), calculated as 25th percentile - (1.5 * IQR) for the lower boundary, and 75th percentile + (1.5 * IQR) for the upper boundary. Any observation whose value falls outside this range is plotted as a circle or dot instead of being covered by the whiskers, and is commonly referred to as an outlier. For the afl.margins variable there are two observations that fall outside this range, and these observations are plotted as dots (the upper boundary is 107, and looking over the data column in the spreadsheet there are two observations with values higher than this, 108 and 116, so these are the dots).

Violin plots

A variation to the traditional box plot is the violin plot. Violin plots are similar to box plots except that they also show the kernel probability density of the data at different values. Typically, violin plots will include a marker for the median of the data and a box indicating the interquartile range, as in standard box plots. In jamovi you can achieve this sort of functionality by checking both the Violin and the Box plot check boxes. See Fig. 24, which also has the Data check box turned on to show the actual data points on the plot. This does tend to make the graph a bit too busy though, in my opinion. Clarity is simplicity, so in practice it might be better to just use a simple box plot.

Fig. 24 Violin plot of the afl.margins variable from the aflsmall_margins file plotted in jamovi, alsow showing a box plot and data points

Drawing multiple box plots

One last thing. What if you want to draw multiple box plots at once? Suppose, for instance, I wanted separate box plots showing the AFL margins not just for 2010 but for every year between 1987 and 2010. To do that the first thing we’ll have to do is find the data. These are stored in the aflmarginbyyear data set. So let’s load it into jamovi and see what is in it. You will see that it is a pretty big data set. It contains 4296 games and the variables that we’re interested in. What we want to do is have jamovi draw box plots for the margin variable, but plotted separately for each year. The way to do this is to move the year variable across into the Split by box, as in Fig. 25.

Fig. 25 jamovi screen shot showing the Split by box

The result is shown in Fig. 26. This version of the box plot, split by year, gives a sense of why it’s sometimes useful to choose box plots instead of histograms. It’s possible to get a good sense of what the data look like from year to year without getting overwhelmed with too much detail. Now imagine what would have happened if I’d tried to cram 24 histograms into this space: no chance at all that the reader is going to learn anything useful.

Fig. 26 Multiple box plots created in jamovi, for the variables margin split by year in the aflmarginbyyear data set

Using box plots to detect outliers

Because the box plot automatically separates out those observations that lie outside a certain range, depicting them with a dot in jamovi, people often use them as an informal method for detecting outliers: observations that are “suspiciously” distant from the rest of the data. Here’s an example. Suppose that I’d drawn the box plot for the afl.margins variable and it came up looking like Fig. 27.

Fig. 27 Box plot of the afl.margins variable showing two very suspicious outliers

It’s pretty clear that something funny is going on with two of the observations. Apparently, there were two games in which the margin was over 300 points! That doesn’t sound right to me. Now that I’ve become suspicious it’s time to look a bit more closely at the data. In jamovi you can quickly find out which of these observations are suspicious and then you can go back to the raw data to see if there has been a mistake in data entry. To do this you need to set up a filter so that only those observations with values over a certain threshold are included. In our example, the threshold is over 300, so that is the filter we will create. First, click on the Filters button at the top of the jamovi window, and then type margin > 300 into the filter field, as in Fig. 28.

Fig. 28 The jamovi filter screen

This filter creates a new column in the spreadsheet view where only those observations that pass the filter are included. One neat way to quickly identify which observations these are is to tell jamovi to produce a Frequency table (in the Exploration → Descriptives window) for the ID variable (which must be a nominal variable otherwise the Frequency table is not produced). In Fig. 29 you can see that the ID values for the observations where the margin was over 300 are 14 and 134. These are suspicious cases, or observations, where you should go back to the original data source to find out what is going on.

Fig. 29 Frequency table for ID showing the ID numbers for the two suspicious outliers: 14 and 134

Usually you find that someone has just typed in the wrong number. Whilst this might seem like a silly example, I should stress that this kind of thing actually happens a lot. Real world data sets are often riddled with stupid errors, especially when someone had to type something into a computer at some point. In fact, there’s actually a name for this phase of data analysis and in practice it can take up a huge chunk of our time: data cleaning. It involves searching for typing mistakes (“typos”), missing data and all sorts of other obnoxious errors in raw data files.

For less extreme values, even if they are flagged in a a box plot as outliers, the decision about whether to include outliers or exclude them in any analysis depends heavily on why you think the data look they way they do and what you want to use the data for. You really need to exercise good judgement here. If the outlier looks legitimate to you, then keep it. In any case, I’ll return to the topic again in section Model checking.