5.4: Boxplots

Another alternative to histograms is a boxplot, sometimes called a “box and whisker” plot. Like histograms, they’re most suited to interval or ratio scale data. The idea behind a boxplot 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, boxplots have become a very popular statistical graphic, especially during the exploratory stage of data analysis when you’re trying to understand the data yourself. Figure *** shows a typical boxplot and what each element represents.

Figure 6.13: A basic boxplot (panel a), plus the same plot with annotations added to explain what aspect of the data set each part of the boxplot corresponds to (panel b).

Let’s have a look at how they work, again using the Margin of Victory variable in the MLBGL2021.sav data file as our example. Firstly, let’s actually calculate these numbers ourselves using the Analyze/Descriptive Statistics/Explore... menu:

As shown in the tables, the minimum value is 1, the maximum value is 21, the median is 3, the 25th percentile is 1, and the 75th percentile is 5, and thus the IQR is 4.

So how does a boxplot capture these numbers? The easiest way to describe what a boxplot looks like is just to draw one. The boxplot shown in Figure *** is the most basic boxplot 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” cover the full range from the minimum value to the maximum value.

In real life, this isn’t quite how boxplots usually look. In most applications, the “whiskers” don’t cover the full range from minimum to maximum. Instead, they actually 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. Any observation whose value falls outside Q1 - 1.5(IQR) or Q3 + 1.5(IQR) Is plotted as a circle instead of being covered by the whiskers, and is commonly referred to as an outlier. For our MLBGL2021 margin of victory data, there are several observations that fall outside this range. As a consequence, the upper whisker is pulled back to the observations that do not exceed the Q3 + 1.5(IQR) value, which is 11. This is illustrated in Figure below:

The numbers next to the outliers represent the case number of the outlier data points.

Using Box Plots to Detect Outliers

Because the boxplot automatically separates out those observations that lie within a certain range, 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 we draw boxplot for the MLB margins data, and it looks like this:

Note that I used the Transpose command when making the boxplot to make the graph horizontal to more easily see the outliers. You can see in the graph that there are some suspicious points way out on the right. Incidentally, you may notice that the very extreme outliers are marked with asterisks instead of circles. That is an indication by SPSS that those are extreme outliers, or 3*IQR above the 75th percentile. We might want to take a further look at those points to ensure they belong.