Box Plots

Students are introduced to the notion of spread in a dataset. They learn about quartiles, box plots, and how to use them to talk about spread.

When we explored measures of center, we tried to answer a question about "typical" values. We considered a fact - that the Animal Shelter Bureau says the average pet weighs almost 40 pounds.

How useful is this fact, really? Maybe all the pets weigh between 35 and 45 pounds, with every pet close to the mean. But maybe all the pets are super small or huge, and no one is even near to the mean!

So once we have our summary for a "normal value", it’s likely we’ll ask another question: If the average pet is 40 pounds, just how typical is that?

There are differences in every class of students. Not everyone likes the same music, not everyone dresses the same, etc. So we’d expect some deviation - or spread - in any class of students! Some classes are more different than others. How do we measure the spread of a population?

Suppose we lined up all animals' weights from smallest to largest, and then split them in half by taking the median. We can learn something about the spread of the dataset by taking the median of each half, splitting the population into four equal-sized quarters. The boundary points between these quarters are called quartiles.

Besides looking at the median as center, and the spread between Q1 and Q3, we also gain valuable information from the spread of the entire dataset — that is, the distance between minimum and maximum. This is called the range of a dataset. (Note: the term “Range” means something different in statistics than it does in algebra and programming!)

Splitting a dataset into quarters gives us five numbers that we can use to measure spread: the minimum, the maximum and the three quartiles that split the dataset into quarters.

Taken together these are called the 5 Number Summary of a dataset, and this summary is one tool for calculating spread. We can use these numbers to calculate two new values:

We can use box plots to visualize the 5 number summary, the Range, and the Interquartile Range.

Below is the Contract for box-plot, along with an example that will make a box plot for the pounds column in the animals-table.

If students are struggling to write conclusions, go over the following five number summary from the box plot they made.

It is extremely common for students to forget that every quartile always includes 25% of the dataset. This will need to be heavily reinforced.

Students learn how to read a box plot, and consider spread and variability. They connect this visualization of spread to what they learned about histograms.

Just as pieand bar charts are ways of visualizing categorical data, box plots and histograms are both ways of visualizing the shape of quantitative data.

Box plots make it easy to see the 5-number summary, and compare the Range and Interquartile Range. Histograms make it easier to see skewness and more details of the shape, offering more granularity when using smaller bins.

Left-skewness is seen as a long tail in a histogram. In a box plot, it’s seen as a longer left "whisker" or more spread in the left part of the box. Likewise, right skewness is shown as a longer right "whisker" or more spread in the right part of the box.

Histograms: fixed intervals (“bins”) with variable numbers of data points in each one. Points “pile up in bins”, so we can see how many are in each. Larger bars show where the clusters are.

Box plots: variable intervals (“quartiles”) with a fixed number of data points in each one. Treats data more like “pizza dough”, dividing it into four equal quarters showing where the data is tightly clumped or spread thin. Smaller intervals show where the clusters are.

Histograms, box plots, and measures of center and spread are all different ways to get at the shape of our data. It’s important to get comfortable using every tool in the toolbox when discussing shape!

We started talking about measures of center with a single question: is "average" the right measure to use when talking about animals' weights? Now that we’ve explored the spread of the dataset, do you agree or disagree that average is the right summary?

Students apply what they have learned about box plots to their chosen dataset. They will add three items to their Data Exploration Project Slide Template: (1) at least two box plots, (2) the corresponding five-number summaries, and (3) any interesting questions they develop. To learn more about the sequence and scope of the Exploration Project, visit Project: Dataset Exploration. For teachers with time and interest, Project: Create a Research Project is an extension of the Dataset Exploration, where students select a single question to investigate via data analysis.

Let’s review what we have learned about making and interpreting box plots.

Let’s connect what we know about box plots to your chosen dataset.

Confirm that all students have created and understand how to interpret their box plots. Once you are confident that all students have made adequate progress, invite them to access their Data Exploration Project Slide Template from Google Drive.

Share your findings!

What shape did you notice in your box plots?

Did you discover anything surprising or interesting about your dataset?

What, if any, outliers did you discover when making box plots?

When your compared your findings with others, did they make any interesting discoveries? (For instance: Did everyone find outliers? Was there more or less similarity than expected?)