Spread of a Data Set

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.

A teacher may report that her students averaged a 75 on a test, but it’s important to know how those scores were spread out: did all of them get exactly 75, or did half score 100 and the other half 50? When Data Scientists use the mean of a sample to estimate the mean of a whole population, it’s important to know the spread in order to report how good or bad a job that estimate does.

Suppose we lined up all of the values in the pounds column of the animals data set from smallest to largest, and then split the line up into two equal groups by taking the median. We can learn something about the spread of the data set by taking things further: The middle of the lighter half of animals is called the first quartile - or "Q1" - and the middle of the heavier half of animals is the third quartile (also called "Q3"). Once we find these numbers, we can say that the middle half of the animals’ weights are spread between Q1 and Q3.

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 data set—that is, the distance between minimum and maximum. This is called the range of a data set. (Note: the term “Range” means something different in statistics than it does in algebra and programming!)

We can use box plots to visualize all of this information. These plots are constructed using just five numbers, which makes them convenient ways to display both center and spread of a data set in a clear and simple way. 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.

🖼Show image

This plot shows us the center and spread in our dataset according to those five numbers.

Data Scientists subtract the 1st quartile from the 3rd quartile to compute the range of the “middle half” of the dataset, also called the interquartile range.

Now that you’re comfortable creating box plots and looking at measures of spread on the computer, it’s time to put your skills to the test!

Just as pie and 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, and offer 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.

Box plots and Histograms can both tell us a lot about the shape of a dataset, but they do so by grouping data quite differently. A box plot is always divided into four parts, which may fall on differently-sized intervals but all contain the same number of points. A histogram, on the other hand, has identically-sized intervals which can contain very different numbers of points.

Challenge Questions: - Compare the for the pounds column of both cats and dogs in the dataset. Are their shapes different? How much overlap is there? - Compare histograms for the age column of both cats and dogs in the dataset. Are their shapes different? How much overlap is there? - Can you explain why the amount of overlap between these two distributions is different?

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

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!

Students assess the degree of visual overlap of two numerical distributions.

"Do dogs take longer to get adopted than cats?"

This is asking us about the interaction between a categorical variable (species) and a quantitative one (weeks). Instead of creating a whole new display, all we have to do is make separate box plots for the distribution of weeks for both cats and dogs. Note: this works fine as long as we’re sure to use a common scale! Both box plots (see below) share the same axis for adoption times, which ranges from about 1 to 10 weeks.

Box plots make it easy to decide if values of a quantitative variable seem to be fairly similar or quite different, depending on which group an individual is in. The trick is to train your eyes to look for whether there’s a lot of overlap in the two box plots, or if one is noticeably higher than the other.

Have students break into groups of 3-4, and compare the box plot of weeks-to-adoption for cats with the one for dogs. Note: they can generate the pair of box plots themselves, but we recommend simply giving them this image: cats v. dogs 🖼Show image

Next, each group examines the pair of box plots that compare weeks to adoption for fixed versus unfixed animals: fixed v. unfixed 🖼Show image. Once again, consider how similar or different the two plots seem.

Students should confirm that the box plots for adoption times of unfixed versus fixed animals have more overlap than the box plots for adoption times of cats versus dogs.

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.

Referring to our Dogs v. Cats box plots, the dogs’ adoption times were much higher than the cats’; the top half of the dogs’ box plot doesn’t overlap at all with the cats’ box plot. Does this suggest that species does or does not play a role in how long it takes for an animal to be adopted?

Referring to our Fixed v. Unfixed box plots, we saw that adoption times for unfixed and fixed animals overlapped a lot, and the medians were pretty close. Does this suggest that being fixed does or does not play a role in how long it takes for an animal to be adopted?

Which variable seems to have more of an effect on adoption time: species (cat or dog) or whether an animal is fixed or not? Have students share back their findings.

Students repeat the previous activity, this time applying it to their own dataset and interpreting their own results. Note: this activity can be done briefly as a homework assignment, but we recommend giving students an additional class period to work on this.

Have students share their findings with one another.