AP Statistics 5.1-5.2: Scatterplots and Correlation

Scatter plot

A graph of paired quantitative data $(x,y)$ in which each observation is represented by a point. It is used to examine the direction, form, strength, and unusual features of a relationship.

What does a scatter plot show about two quantitative variables?

It can reveal the overall direction and form of their association, how closely points follow a pattern, and deviations such as outliers or clusters.

How is a positive association identified in a scatter plot?

Higher values of $x$ tend to occur with higher values of $y$, and lower values of $x$ tend to occur with lower values of $y$. The pattern generally rises from left to right.

How is a negative association identified in a scatter plot?

Higher values of $x$ tend to occur with lower values of $y$, and lower values of $x$ tend to occur with higher values of $y$. The pattern generally falls from left to right.

How can the strength of a relationship be judged from a scatter plot?

For a linear relationship, points close to a straight line indicate a stronger association, while widely scattered points indicate a weaker association.

Why does a perfect horizontal pattern not represent a useful relationship between variables?

Although the points fit a horizontal line perfectly, changes in $x$ are not associated with changes in $y$. Thus, the variables have no linear association.

Before fitting a linear regression model, what should be examined and why?

Examine a scatter plot first to determine whether a roughly linear pattern is reasonable and whether unusual points or curvature could make a linear model inappropriate.

Correlation coefficient $r$

A numerical measure of the direction and strength of a linear association between two quantitative variables. It satisfies $-1\le r\le 1$.

What does the magnitude of the correlation coefficient indicate?

Values of $r$ close to $1$ or $-1$ indicate a strong linear association, while values near $0$ indicate little or no linear association.

What does the sign of the correlation coefficient indicate?

A positive $r$ indicates that $y$ tends to increase as $x$ increases; a negative $r$ indicates that $y$ tends to decrease as $x$ increases.

What do $r=1$ and $r=-1$ mean?

$r=1$ indicates perfect positive linear association, and $r=-1$ indicates perfect negative linear association. In either case, all data points lie exactly on a straight line.

Why is $r=0$ not sufficient evidence that two variables have no relationship at all?

$r=0$ indicates no linear association, but the variables may have a curved, nonlinear relationship. A scatter plot must be examined.

What type of relationship does the correlation coefficient measure?

The correlation coefficient measures only the strength and direction of a linear association. It does not measure the strength of a curved or other nonlinear association.

What are important properties of the correlation coefficient $r$?

It is unitless, has no units, is always between $-1$ and $1$, and is unchanged if the roles of $x$ and $y$ are switched. It measures linear association, not causation.

Does correlation depend on which variable is labeled explanatory and which is labeled response?

No. Switching $x$ and $y$ does not change the value of $r$, because correlation describes the association symmetrically rather than predicting one variable from the other.

Is the correlation coefficient affected by the units used to measure the variables?

No. Changing the measurement units or standardizing either variable does not change $r$, provided the transformation uses a positive scale factor.

Why can an outlier substantially change the correlation coefficient?

Correlation is not resistant to outliers. An unusual point can either strengthen or weaken the apparent linear association, so the scatter plot should always be inspected.

Does a strong correlation establish that one variable causes another?

No. Correlation measures association, not causation; a strong relationship may result from confounding variables, coincidence, or another causal structure.