Correlations

Students identify and make use of patterns in scatter plots, learning to characterize them as appearing to be linear, curved, or showing no clear pattern. Determining that a form appears to be linear is a prerequisite for proceeding to correlation and linear regression.

We can analyze a single quantitative variable, such as age or pounds to identify a value that is typical, how much the values vary, and what kind of values are usual or unusual.

(The bolded words above all deal with notions of what it means for a value to be "normal" or "abnormal". These words have loaded meaning in the context of variability, and should be used carefully!)

But those analyses tell us nothing about the relationship between animals' ages and weights. In order to understand such relationships, we have to expand our view from one column to two. This goes hand-in-hand with expanding our display from a 1-dimensional histogram or box plot to a 2-dimensional scatter plot.

Rather than summarizing each distribution in one dimension, we can search for a linear relationship between two quantitative variables. Linear relationships only make sense if the scatter plot follows a straight-line pattern, so the first thing we need to ask is whether the form of the relationship appears to be linear or not.

Form indicates whether a relationship is linear, nonlinear or undefined:

Some patterns are linear, and cluster around a straight line sloping up or down.

Some patterns are nonlinear, and may look like a curve, a hockey-stick, or some other shape!

And sometimes there is no relationship or pattern at all! That means there’s no predictable change in the y-axis as we go from one side of the x-axis to the other.

Data Scientists use their eyes all the time! It doesn’t make sense to search for correlations when there’s no pattern at all, and summarizing with a correlation only makes sense for linear relationships!

Once students have learned to identify a possible linear relationship, they can turn their attention to other qualities of that relationship, like its direction.

We can also examine the direction of a linear relationship.

A positive direction means that the line slopes up as we look from left-to-right. Positive relationships are by far most common because of natural tendencies for variables to increase in tandem. For example, “the older the animal, the more it tends to weigh”. This is usually true for human animals, too!

A negative direction means that the line slopes down as we look from left-to-right. Negative relationships can also occur. For example, “the older a child gets, the fewer new words he or she learns each day.”

If the form is nonlinear or non-existent, "direction" doesn’t apply: A parabola might look like it has both a positive and negative correlation, and if there’s no form at all then there certainly can’t be a direction!

We’ll explore another quality of a possible linear relationship: its strength.

Strength indicates how closely the two variables are correlated.

How well does knowing the x-value allow us to predict what the y-value will be?

A relationship is strong if knowing the x-value of a data point gives us a very good idea of what its y-value will be (knowing a student’s age gives us a very good idea of what grade they’re in). A strong linear relationship means that the points in the scatter plot are all clustered tightly around an invisible line.

A relationship is weak if x tells us little about y (a student’s age doesn’t tell us much about their number of siblings). A weak linear relationship means that the cloud of points is scattered very loosely around the line.

If the form is non-existent, "strength" doesn’t apply: without any form at all, there’s nothing for data points to be tightly or loosely clustered around!

This page includes a series of probing questions that get at the common misconceptions listed above. Discuss the answers as a class.

Optional: If time permits, have students complete Identifying Form, Direction and Strength (Matching).

Now that students know how to identify direction and strength for linear relationships, they’ll learn to read how these are expressed in the 𝑟-value.

Students have learned that a correlation can be described by three pieces of information: Form, Direction, and Strength. Statisticians and Data Scientists have a shorter way of describing all three, called r-value.

𝑟 is positive or negative depending on whether the correlation is positive or negative. The strength of a correlation is the distance from zero: an 𝑟-value of zero means there is no correlation at all, and stronger correlations will be closer to −1 or 1.

An 𝑟-value of about ±0.65 or ±0.70 or more is typically considered a strong correlation, and anything between ±0.35 and ±0.65 is “moderately correlated”. Anything less than about ±0.25 or ±0.35 may be considered weak. However, these cutoffs are not an exact science! In some contexts an 𝑟-value of ±0.50 might be considered impressively strong!

If it works for you, give students five minutes to play a few rounds of the online game Guess the Correlation to develop intuition with r-values. (This will require creating an account.)

Calculating 𝑟 from a dataset only tells us the direction and strength of the relationship in that particular sample. If the correlation between adoption time and age for a representative sample of about 30 shelter animals turns out to be +0.44, the correlation for the larger population of animals will probably be close to that, but certainly not the same.

It’s easy to be seduced by large 𝑟-values, and believe that we’re really onto something that will help us claim that one variable really impacts another! But Data Scientists know better than that…​

If time allows, you may want to emphasize the point that correlation does not imply causation by having students look at the nonsense claims that could be made from the graphs of real world data on the Spurious Correlations website.

Students often giggle at some of the Spurious Correlations examples, but fail to internalize the point when it comes to the Animals dataset or their own analysis. Pay close attention to students' language when describing their correlations, and make sure they are not using causative wording!

Which corresponded more strongly with time to adoption, "age" or "pounds"? What does this mean?

The correlation with "pounds" is higher, meaning that an animal’s weight is a better predictor of the number of weeks an animal will live at the shelter before being adopted than its age.

Students apply what they have learned about correlations to their chosen dataset. They will add two or more items to their Data Exploration Project Slide Template: (1) a correlation they think they see in the data set, and (2) the form, direction and strength of that correlation. 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 correlations.

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

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

Have students share their findings.

Did you discover anything surprising or interesting about their dataset?

Were any of the correlations especially strong? Were any of them surprising?

When students compared their your findings with those of their classmates, did they make any interesting discoveries? (For instance: Did everyone find a strong correlation? A linear one?)