Measures of Center

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

Some medicines are dosed by weight: heavier animals need a larger dose. 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 41 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 “41 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.

If we plotted all the pounds values as points on a number line, what could we say about the average of those values? Is there a midpoint? Is there a point that shows up most often? Each of these are different ways of “measuring center”.

The Animal Shelter Bureau used one method of summary, called the mean, or "average". In general, the mean of a data set 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.

Pyret 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.

Type mean(animals-table, "pounds"). What does this give us? Does this support the Bureau’s claims?

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 41 pounds. That IS the average, but if we scan the dataset we’ll quickly see that most of the animals weigh less than 41 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 left or right of center.

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:

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:

By looking at the histogram, we can develop an intuition for whether it’s probably better to use the mean or median. Pronounced 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. Either way, such shapes distort the mean as a measure of what’s typical for the data set. Data scientists generally prefer to use the mean as their measure of center, because it contains information from every single data value. However, if a data set has substantial skewness or outliers, they use median to report the center .

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

The third measure of center is called the mode of a dataset. The mode of a data set is the value that appears most often. Median and Mean always produce one number, but if two or more values are equally common, there can be more than one mode. If all values are equally common, then there is no mode at all! Often there will be just one mode in the list of most common values: many data sets are what we call “unimodal”. But sometimes there are exceptions! Consider the following three datasets:

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

In Pyret, the mode(s) are calculated by the modes function, which consumes a Table and the name of the column you want to measure, and produces a List of Numbers.

The most common number of pounds an animal weighs is 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. Our mean weight agrees with their findings, but we have three reasons to suspect that mean isn’t the best value to use:

“In 2003, the average American family earned $43,000 a year — well above the poverty line! Therefore very few Americans were living in poverty."

Do you trust this statement? Why or why not? 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: