Stem and leaf plots, box plots, quartiles and interquartile ranges

Graphs and calculations involving stem and leaf plots, box plots, quartiles and interquartile ranges are relatively new in the context of mathematical concepts and skills.

Box and whisker box plots

Box and whisker or box plots are a method of showing the distribution of a set of data using the data’s quartiles, that is, “breaking” the data into quarters (or 25% percentiles) and showing each of these quarters as a part of the box plot.

The terminology used and function of these types of graphs can be confusing for students, and it may be a challenge to understand they are used to show the distribution or spread of the data.

Box plots can be generated using an Excel spreadsheet. An example plot is illustrated below showing how the size of US states are distributed:

• The image above shows that the extremes of US state land area can be very small or very large, with the “whiskers” extending to just over zero on the scale and to over 1500 square metres. These extremes actually correspond to Rhode Island (3000 sq. km in area) and Alaska (1.718 million sq. km in area) which are the smallest and largest of the 50 states of the USA.
• The dotted line shows the range of the data in which all 50 states of the US fit.
• The box shows the interquartile range of US state size with the left edge of the box the Quartile 1 (Q1), and the right edge of the box Quartile 3 (Q3), and the width of the box being the interquartile range. Half of all the data sits in the interquartile range, and therefore 25 US states data is in the box (remembering there are 50 states in the US). Therefore we know that half of all US states sit in the 100 to 200 thousand sq. km range.
• The thick black line in the box is the median state size, which is roughly 140 thousand sq. km according to the plot.

This video further describes how boxplots demonstrate the distribution of data including through minimum, maximum, median, and quartiles. If you cannot access YouTube, there is a transcript in the downloads section.

Stem and leaf plots

Stem and leaf plots are used to show the frequency of values and are often interchangeable with frequency distribution tables. Stem and leaf plots can also be used to easily find the mean or mode of a set of data.

Australian maths teacher Eddie Woo unpacks stem and leaf plots for his students by using a large amount of data and talking through how a stem and leaf plot is created. He starts by making categories for the numbers (100-109, 110-119, etc.), and then starts sorting out the numbers into these categories. Once it becomes obvious that this is tiresome, he then explains that instead of writing the whole number of say, 108 into the category of 100-109, he can just write 8 into a category of 10. The 10 is the stem, and the 8 is the leaf.

Once he has written out all the numbers, Woo then flips the page and reveals to the students that what this has created is essential a dot plot that shows the clusters and outliers, but can also show the numerical values. By explaining it like this, Woo has connected the student’s previous knowledge to a new concept, which allows them to have a deeper understanding of what stem and leaf plots are.

Watch the full video below if you are interested in seeing how Eddie Woo taught the lesson. If you cannot access YouTube, there is a transcript in the downloads section with screenshots.