Correlations

Students identify and make use of patterns in scatter plots, learning to characterize them as being linear, curved, or showing no clear pattern. This builds intuition for determining if the form is linear, in which case we can proceed to correlation and linear regression.

By now we have learned ways to summarize a single quantitative variable, like the age of an animal in our dataset: report the center, spread, and shape of the distribution. Together, those numbers tell us what age is typical, how much the ages vary, and what kind of age values are usual or unusual. We could do the same for for animals' weights (or any other quantitative column).

But those individual summaries tell us nothing about the relationship between animals' ages and weights. In order to understand such relationships, we have to expand our view from a single dimension (along one axis) to two dimensions. This goes hand in hand with expanding our display from a one-dimensional histogram to a two-dimensional scatter plot.

Rather than summarizing each distribution in one dimension, we can summarize a linear relationship between two quantitative variables. But this only makes sense if the scatter plot follows a straight-line pattern, as opposed to being curved. So the very first assessment we have to make is to identify the form of the relationship as being linear or not.

The relationship between two quantitative variables can take many forms - some patterns are linear, and appear as a straight line sloping up or down. Some patterns are non-linear, and may look like a curve or an arc. And sometimes there is no pattern or relationship at all!

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 only linear relationships make sense if we want to summarize with a correlation.

Once students have learned to identify a possible linear relationship, they can turn their attention to other qualities of that relationship: its direction and strength. Each of these is expressed in the r$\displaystyle r$-value, which students learn to read.

Assuming a relationship is linear, data scientists calculate a single number called "correlation" - or r$\displaystyle r$-value - that reports both the direction and strength.

A linear relationship between two quantitative variables is positive if, in general, the scatter plot points are sloping up: smaller x values tend to go with smaller y values, and larger x values tend to go with larger y values. The relationship is negative if points slope down: smaller x values tend to go with larger y values, and larger x values tend to go with smaller y values.

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

Have students turn to Identifying Form, Direction and Strength (Page 81) in their student workbooks. For each scatter plot, identify whether the relationship is positive or negative, and whether it is strong or weak.

The correlation r is a number (falling anywhere from -1 to +1) that tells us the direction and strength of a linear relationship between two variables. r$\displaystyle r$ 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$\displaystyle r$-value of zero means there is no correlation at all, and stronger correlations will be closer to −1 or 1.

An r$\displaystyle r$-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$\displaystyle r$-value of ±0.50 might be considered impressively strong!

Calculating r$\displaystyle r$ from a data set 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.

Have students turn to Identifying Form and r-Values (Page 82) in their student workbooks. For each scatter plot, identify whether the relationship is linear, and use r$\displaystyle r$ to summarize direction and strength. You could also have them complete a card sort activity on identifying strength (Desmos) and a card sort activity on identifying direction (Desmos).

(Note: An excellent resource to build intuition for r-values is Guess the Correlation!)

It is useful to ask students probing questions, to help address the misconceptions listed above. Some examples:

It’s easy to be seduced by large r$\displaystyle r$-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…​

Here are some possible correlations that have absolutely no causal relationship; they come about either by chance or because both of them are related to another variable that’s (often) lurking in the background.

Here are a few real correlations, drawn from the Spurious Correlations website. If time allows, have your students explore the site to see more! - “Number of people who drowned after falling out of a fishing boat” v. “Marriage rate in Kentucky” (r$\displaystyle r$ = 0.98) - “Average per-person consumption of chicken” v. “U.S. crude oil imports” (r$\displaystyle r$ = 0.95) - “Marriage rate in Wyoming” v. “Domestic production of cars” (r$\displaystyle r$ = 0.99) - “Number of people who get tangled in their own bedsheets” v. “Amount of cheese consumed that year” (r$\displaystyle r$ = 0.95)

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

What correlations do you think there are in your dataset? Would you like to investigate a subset of your data to find those correlations?

What correlations did you find? Did you need to filter out certain rows in order to get those correlations?

After looking at the scatter plot for our animal shelter, do you still agree with the claim on (Dis)Proving a Claim (Page 79)? (Perhaps they need more information, or to see the analysis broken down separately by animal!)