2.1 Scatter plots

Scatter plots with quantitative variables

A scatter plot depicts the relationship between two variables on a rectangular coordinate system, where each axis corresponds to one variable. Scatter plots are used for both quantitative and categorical data.

Correlation and causation

A scatter plot is often used to visually determine whether two variables are correlated. Two variables are correlated when a variable behaves with a predictable trend or pattern with respect to changes in another variable. Ex: Ice cream sales tend to increase as temperatures increase.

Correlation is not the same as causation. Some variables that appear correlated in a scatterplot may not actually have an underlying relationship. In the figure below, divorces and margarine consumption appear to follow the same pattern from 2000 to 2009. However, no rational person would actually believe that margarine consumption causes divorce.

Source: Spurious Correlations TylerVigen

Example 2.1.1: Number of engineering faculty versus campus rank.

The data below shows the number of engineering faculty and engineering campus rank for eight University of California campuses, with being the highest possible rank. In the table below, the campus with engineering faculty is ranked number in the country, while the campus with only engineering faculty is ranked number .

Engineering faculty	USNWR rank

Below is a scatter plot showing engineering faculty size vs. engineering campus rank in 2014 for the eight campuses in the University of California system. Each instance in the table above becomes a coordinate in the following scatter plot.

The scatter plot clearly shows the relationship between number of faculty and rank. This relationship suggests that an increasing number of engineering faculty is correlated to a decreasing (better) USNWR rank. However, this relationship does not necessarily imply that additional faculty causes an improvement in rank. An underlying relationship between the number of engineering faculty and USNWR rank may not actually exist or the existence of an unknown variable may make the relationship appear stronger.

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Example 2.1.2: Head coach salary versus college football rankings.

The scatter plot below shows college football team rankings (where a team with rank is best) and the total salary for each team's head coach¹². The scatter plot shows that half of coaches earn over one million dollars, and that many poorly-ranked teams' coaches earn just a few hundred thousand dollars. However, several teams in the lower left are well-ranked despite having coaches earning less than a million dollars.

The scatterplot shows a correlation between the salary of a head coach and overall ranking. However, many other variables may exist to explain this relationship. Care should be taken not to infer a causal relationship between two correlated variables.

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participation activity

2.1.1: Scatter plot.

Consider the scatter plot above showing the number of engineering faculty versus rank for eight UC campuses.

1)

What is the approximate rank for the campus having around engineering faculty?

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2)

Looking at the overall trend, would another campus in the UC system with engineering faculty likely have a better or worse rank than the campus with around engineering faculty?

better

worse