Measures of Center

Students learn about mean (or "average"), and how it is one way (among others!) to summarize a quantitative column.

According to the Animal Shelter Bureau, the average pet weighs almost 40 pounds.

Some medicines are dosed by weight: heavier animals need a larger dose that could be dangerous for smaller animals. If someone from the shelter needs to give a dose of medicine to the animals, is the “average” the best estimate we can use?

“The average pet weighs almost 40 pounds” is a statement about the entire dataset, which summarizes a whole column of values with a single number. Summarizing a big dataset means that some information gets lost, so it’s important to pick an appropriate summary. Picking the wrong summary can have serious implications! Here are just a few examples of summary data being used for important things. Do you think these summaries are appropriate or not?

Every kind of summary has situations in which it does a good job of reporting what’s typical, and others where it doesn’t really do justice to the data. In fact, the shape of the data can play a huge role in whether or not one kind of summary is appropriate!

One of the ways that Data Scientists summarize quantitative data is by talking about its center - literally asking "what is a typical value in this sample?", in the hopes of inferring something about a larger population. But there are many different ways to define "center", and each method has strengths and weaknesses. Let’s check the “40 pounds” claim and see if it’s an appropriate measure of center. Later on, you’ll have a chance to apply what you’ve learned to your own dataset, to find the best way to provide an overall summary of the data.

Before digging into a discussion of mean, let’s look at a visual.

If we plotted all the animals' weights as points on a number line, it would look something like this:

Each of these are different ways of “measuring center”.

The Animal Shelter Bureau used one method of summary, called the mean, or "average". The mean of a dataset is the sum of values divided by the number of values. To take the average of a column, we add all the numbers in that column and divide by the number of rows.

CODAP has a way for us to compute the mean of any quantitative column in a Table. It consumes a Table and the name of the column you want to measure, and produces the mean — or average — of the numbers in that column.

Notice that calculating the mean requires being able to add and divide, so the mean only makes sense for quantitative data. For example, the mean of a list of Presidents doesn’t make sense. Same thing for a list of zip codes: even though we can divide a sum of zip codes, the output doesn’t correspond to some “center” zip code.

Students learn a second measure of center: the median. They learn the algorithm and the code to find the median, as well as situations where taking the median is more appropriate than the mean.

You computed the mean of that column to be almost exactly 40 pounds. That IS the average, but if we scan the dataset we’ll quickly see that most of the animals weigh less than 40 pounds! In fact, more than half of the animals weigh less than just 15 pounds. What is throwing off the average so much?

Kujo and Mr. Peanutbutter!

In this case, the mean is being thrown off by a few extreme data points. These extreme points are called outliers, because they fall far outside of the rest of the dataset. Calculating the mean is great when all the points are fairly balanced on either side of the middle, but it distorts things for datasets with extreme outliers. The mean may also be thrown off by the presence of skewness: a lopsided shape due to values trailing off to the left or right.

A different way to measure center is to line up all of the data points — in order — and find a point in the center where half of the values are smaller and the other half are larger. This is the median, or “middle” value of a list.

As an example, consider this list of ACT scores:

25, 26, 28, 28, 28, 29, 29, 30, 30, 31, 32

Here 29 is the median, because it separates the "bottom half” (5 values below it) from the top half” (5 values above it).

The algorithm for finding the median of a quantitative column is:

3, 7, 9, 21

The median of this list is 8, because 8 is the mean of the two middle numbers, 7 and 9. To find their mean, we added 7 and 9 to get 16 and split 16 in half.

Looking at the shape of the data (via a histogram, for example), helps us determine whether it’s probably better to use the mean or median.

Strong left skewness and/or low outliers can pull the mean down below the median, while right skewness and/or high outliers can pull it up above the median.

Mean is generally the best measure of center, because it includes information from every single point. But it’s misleading for highly-skewed datasets, so statisticians fall back to the median.

Students learn about the mode(s) of a dataset, how to compute them, and when it is appropriate to use them as a measure of center.

The third measure of center is called the modes of a dataset. The modes of a dataset are the values that appear most often.

Median and Mean always produce one number and many datasets are what we call “unimodal”, having just one mode. But sometimes there are exceptions!

Consider the following three datasets:

Modes are rarely used to summarize quantitative data. It is very common as a summary of categorical data, telling us which category occurs most often.

The easiest way to determine modes in CODAP is to sort a column. Do this by clicking on the column name and then selecting from the drop-down menu either Sort option. Scan the column to see which values are the most common.

The most common animal weights are 0.1 and 6.5! That’s well below our mean and even our median, which is further evidence of outliers or skewness.

At this point, we have a lot of evidence that suggests the Bureau’s use of “mean” to summarize animal weights isn’t ideal. We have three reasons to suspect that mean isn’t the best value to use:

Consider how many policies or laws are informed by statistics like this! Knowing about measures of center helps us see through misleading statements.

You now have three different ways to measure center in a dataset. But how do you know which one to use? Depending on the shape of the dataset, a measure could be really useful or totally misleading! Here are some guidelines for when to use one measurement over the other:

Optional: We strongly recommend having students practice the Data Cycle with measures of center, using Data Cycle: Measures of Center. Sometimes what’s created isn’t a table or a display, and this activity demonstrates that. It also drives home an important difference between Arithmetic and Statistical Questions.

Students apply what they have learned about measures of center to their chosen dataset. In their Data Exploration Project Slide Template, they will complete the first four rows of the "Measures of Center and Spread" table. They will also interpret those measures of center, and record any interesting questions that emerge. 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 computing and interpreting three measures of center - mean, median, and modes.

Let’s connect what we know about measures of center to your chosen dataset.

Invite students to discuss their results and consider how to interpret them.

Share your findings!

Did you discover anything surprising or interesting about your dataset?

Which measures of center do you think were the most useful for the quantitative columns you chose?

What questions did the measures of center inspire you to ask about your dataset?

When you compared your findings with other students, did you make any interesting discoveries? (For instance: Did everyone find mode(s)? Did anyone have a measure of center that was dramatically influenced by an outlier?)