Chapter 6: Scatterplots, Association and Correlation

Scatterplot

• Showing relationship between 2 quantitive variables

• Ideal way to picture associations between two quantitative variables

Example where a scatterplot could be used:

Average daily temperature and daily ice cream sales over 12 days

Scatterplot direction of the association - Negative

Runs from the upper left to the lower righ

Scatterplot direction of the association - Positive

Running from the lower left to the upper righ

Positive linear relationship

Nonlinear relationship

No relationship

There's no correlation between X and Y 

Scatterplot - Form

Is it straight, curved, something exotic, or no pattern?

• If there is a straight-line relationship, it will appear as a

  cloud or swarm of points stretched out in a generally

Linear form

Consistent, straight form

Scatterplots. -Strength

• How much scatter?

The more scattered = the weaker the relationship between X and Y 

The less scattered = the stronger the relationship between X and Y 

Scatterplots - The unexpected

• Are there unusual observations or subgroups?

• An outlier is an unusual observation, standing away from the overall pattern of the scatterplot

Correlation Coefficient (r)

Measure that describes the direction and strength of a linear assocation

What is the unit of correlation coefficient?

Unit-less

What are possible values of the correlation coefficient and what do they indicate

-1: Strongest negative linear association

0: No linear association

1: Strongest positive linear association

Sample correlation coefficient (𝑟) is computed as

N = number of data points  

Two of the more common alternative formulas for

correlation are

N = number of data points  

Covariance

• An alternative to the correlation coefficien

  • Depends on the unit of measurement

• Good for direction but not strength

• Covariance is NOT unit-less

Associations

Change in the value of one variable associated with change in the value of the other variable

Bivariate analysis

Investigation of 2 variables

• To make a scatterplot of two quantitative variables, assign

  one to the y-axis and the other to the x-axis

  • Coordinates (x,y)

Explanatory or Predictor variable

X-axis (independent)

Response variable

Y-axis (dependent)

Correlation

Measures the strength of the linear association between two quantitative variables

Quantitative variables condition

Correlation applies only to quantitative variables

Linearity Condition

Correlation measures the strength only of the linear association

Outlier Condition

Unusual observations can distort the correlation

Correlation Properties

• Correlation is always between −1 and +1

• Correlation treats x and y symmetrically

• Correlation has no units

• Not affected by changes in the center or scale of either variable.

Example of a change in scale

Change in currency (ie converting from CAD to USD) 

CAD --> USD = You can either multiply or divide  1

Correlation Table

• Compact and give a lot of summary information at a glance.

• Example of a correlation table for Amazon books

Correlation ≠ Causation

Two variables may be correlated but that does not mean there is a causal effect between them

Example that Correlation ≠ Causation

Poverty rates causes crimes to go up  

• It might be correlation not causation

Lurking variables

It is not included in the original analysis but affects the outcome.

Example of a lurking variable

Increased sales of ice cream and the number of deaths by drowning 

• Lurking variable (that feeds into both): Temperature  

• Positive correlation