Review: Comparing Variables

In previous chapters, we looked for relationships (associations) between variables by:

  • Comparing categorical variables with contingency tables and stacked barplots
  • Comparing numeric variables across groups with side-by-side boxplots
  • Looked at how variables change over time with timeplots

In this chapter, we will:

  • Look for relationships between two numeric variables

The Data

Recall the Motor Trend Cars data from previous chapters:

##                    mpg cyl disp  hp    wt  qsec vs     am
## Mazda RX4         21.0   6  160 110 2.620 16.46  V   auto
## Mazda RX4 Wag     21.0   6  160 110 2.875 17.02  V   auto
## Datsun 710        22.8   4  108  93 2.320 18.61  S   auto
## Hornet 4 Drive    21.4   6  258 110 3.215 19.44  S manual
## Hornet Sportabout 18.7   8  360 175 3.440 17.02  V manual
## Valiant           18.1   6  225 105 3.460 20.22  S manual

We might want to know:

  • Is there a relationship between engine displacement (size) and horsepower?
  • Is weight related to fuel efficiency?


How to we find relationships between numeric (quantitative) variables?

  • Visually: using scatterplots
  • Numerically: using the correlation coefficient
  • Usually, we do both
  • In this course, we will only focus on linear relationships



How to make scatterplots:

  • Define one variable as the \(X\) variable, and one as \(Y\)
  • Draw a point for each observation, using the values of the \(X\) and \(Y\) variables as coordinates
  • Typically, the \(X\) variable is on the horizontal axis and the \(Y\) variable on the vertical axis

What we look for:

  • Is there are trend or pattern?
  • Are there any outliers or unusual points?

Horsepower vs. Displacement

Horsepower vs. Displacement

Is there a relationship?

  • As engines get bigger, they tend to have more horsepower
  • We call this a positive association

Are there any unusual points?

  • There is a point well above the rest
  • Notice that it's engine size is right in the middle (about 300 cu. in.), but its horsepower is larger than any other car

Weight vs. Fuel Efficiency

Weight vs. Fuel Efficiency

Is there a relationship?

  • As cars get heavier, they tend to have lower fuel efficiency
  • We call this a negative association

Are there any outliers?

  • No points fall far away from the rest

Types of Relationships

There are many types of trends that can come up when we make scatterplots. In this class, we will focus on the most common:

  • Linear: The trend can be described fairly well by a straight line
  • Non-linear: Any other type of trend

Directions of Relationships

  • Positive: As one variable goes up, so does the other one
  • Negative: As one variable goes up, the other goes down

Why lines?

  • In statistics, we often try to find the simplest adequate method. Lines are simple.

Strong Positive Linear Trend

Moderate Negative Linear Trend

Weak Positive Linear Trend

No Trend

Non-Linear Trend

Non-Linear Trend

Roles of Variables

How do we decide which is \(X\) and which is \(Y\)?

The \(X\) Variable is:

  • The explanatory or independent variable.
  • We want to know if changes in this variable explains changes in \(Y\)

The \(Y\) Variable is:

  • The response or dependent variable
  • We want to see if this variable responds when we change \(X\)

Which is which depends on what question we're asking.

Variable Role Examples

Horsepower vs. Engine Displacement

  • It makes sense that giving a car a bigger engine gives it more power.
  • We can't just give a car more horsepower, horsepower responds to changes we make to the car.
  • Horsepower should be \(Y\), and Engine Displacement should be \(X\).

Fuel Efficiency vs. Weight

  • When we make a car heavier, it should mean that it takes more fuel to move it.
  • Fuel efficiency responds to changes in the properties of the car.
  • Fuel Efficiency should be \(Y\), and Weight should be \(X\).

Measuring the Strength

How do we measure how strong the relationship is?

  • We use the correlation coefficient
  • \(r = \frac{\sum z_y \times z_x}{n-1}\)
  • StatCrunch will find this for us

What is the correlation coefficient?

  • \(r\) is the strength of the linear relationship between two numeric variables
  • It tells us how well a straight line explains the relationship

Interpreting Correlation

  • \(-1 \le r \le 1\)
  • The value of \(r\) tells us the strength
  • The sign of \(r\) tells us the direction
  • \(r = 1\): the points make a perfect straight line with a positive slope
  • \(r = -1\): the points make a perfect straight line with a negative slope
  • \(r = 0\): there is no linear relationship at all


  • You can sometimes get high correlations even if the relationship isn't linear
  • You should always see a scatterplot along with a correlation coefficient to know whether or not it's meaningful

Strong Positive Linear Trend

Moderate Negative Linear Trend

Weak Positive Linear Trend

No Trend

Non-Linear Trend

Non-Linear Trend

Using the Correlation Coefficient

So how do we use \(r\)?

  • First, make a scatterplot
  • There needs to be a linear association, or \(r\) is meaningless
  • Check the sign: is the relationship positive or negative?
  • Check the value: how strong is the relationship?
  • Are there outliers? The correlation is very sensitive to them.


  • We often use the terms weak, moderate, and strong to describe the relationship, but these are up to interpretation.

Outliers in Correlation

  • \(r\) is the strength of the overall linear relationship in the data
  • If we have a point that is far away from the rest, it will decrease the strength of the relationship
  • If we remove an outlier, it will drive \(r\) away from 0 and towards -1 or 1
  • If we add an outlier, it will drive \(r\) towards 1

These relationships also hold if we alter a point's values (e.g., correct a typo in the data set)

  • Moving a point towards the rest improves \(r\)
  • Moving it away from the rest punishes \(r\)

Horsepower vs. Engine Displacement

Horsepower vs. Engine Displacement

More Properties of Correlation

  • \(r\) is unitless
  • \(r\) is not affected by changes of center or scale
  • If we change units, the correlation will not change (e.g., \(lbs \to kg\))
  • The correlation of \(X\) and \(Y\) is the same as the correlation between \(Z_x\) and \(Z_y\) (their z-scores)
  • The correlation stays the same if we flip \(X\) and \(Y\)
  • Correlation only applies to relationships between numeric variables. If there is an association involving categorical variables, it is not correlation.

Weight (lbs) vs. Fuel Efficiency

Weight (kg) vs. Fuel Efficiency

Weight vs. Fuel Efficiency (Z-Scores)

Fuel Efficiency vs. Weight

In StatCrunch


  1. Graph \(\to\) Scatter Plot
  2. X Column \(\to\) Select your explanatory \((X)\) variable
  3. Y Column \(\to\) Selected your response \((Y)\) variable
  4. Compute!


  1. Stat \(\to\) Summary Stats \(\to\) Correlation
  2. Select Column(s) \(\to\) Hold Shift/Ctrl/Command to select multiple variables (note: if you select more than two variables, it will find all pair-wise correlations)
  3. Compute!

Correlation \(\ne\) Causation

Must people are familiar with the phrase "correlation does not equal causation," but what does that really mean?

  • Even if we find a correlation between two variables, it does not mean that one causes the other.
  • This is especially common when two things both increase or decrease over time.
  • Both may be caused by other, unknown variables.
  • We call these unknown variables lurking variables or confounding variables.

For example:

  • What if we looked at the correlation between national ice cream sales and the number of forest fires, recorded for each month of the year?

Ice Cream Sales and Forest Fires