Scatterplots, Association, and Correlation: Understanding Relationships Between Quantitative Variables

Scatterplots, Association, and Correlation

Introduction to Scatterplots and Association

Scatterplots are graphical displays that show the relationship between two quantitative variables. Each point on the plot represents an individual data pair. By examining the pattern of points, we can describe the association between the variables in terms of direction, form, and strength.

• Direction: Indicates whether the variables tend to increase together (positive association) or one increases as the other decreases (negative association).

• Form: Describes the shape of the relationship (e.g., linear, curved, or no pattern).

• Strength: Refers to how closely the points follow a clear form.

Example: The scatterplot above shows a positive association between students' heights and weights, with a fairly straight form and a possible outlier.

Correlation: Measuring Linear Association

The correlation coefficient, denoted as r, is a numerical measure of the strength and direction of the linear relationship between two quantitative variables. The value of r ranges from -1 to +1:

• r = +1: Perfect positive linear relationship

• r = -1: Perfect negative linear relationship

• r = 0: No linear relationship

Correlation does not depend on the units of measurement and is unaffected by changing the scale of either variable.

Properties of the Correlation Coefficient (r)

• Range: $-1 \leq r \leq 1$

• Direction: The sign of r indicates the direction of association (positive or negative).

• Strength: The closer |r| is to 1, the stronger the linear relationship.

• Symmetry: The correlation between x and y is the same as between y and x.

• No Units: Correlation is a unitless measure.

• Sensitivity to Outliers: Outliers can greatly affect the value of r.

Interpreting r in Context

For example, if the correlation between speed limit and average number of accidents per week is $r = 0.996$, this indicates a very strong, positive, linear relationship between the two variables.

Calculating the Correlation Coefficient

The formula for the sample correlation coefficient is:

$r = \frac{1}{n-1} \sum_{i=1}^{n} \left( \frac{x_i - \bar{x}}{s_x} \right) \left( \frac{y_i - \bar{y}}{s_y} \right)$

• $x_i, y_i$ are the individual data values

• $\bar{x}, \bar{y}$ are the sample means

• $s_x, s_y$ are the sample standard deviations

• $n$ is the number of data pairs

Correlation Conditions

Before using correlation, check these conditions:

• Quantitative Variables Condition: Both variables must be quantitative (not categorical).

• Straight Enough Condition: The relationship should be linear. If not, correlation may be misleading.

• Outlier Condition: Outliers can distort correlation. Consider reporting correlation with and without outliers.

Examples of Correlation Strength

• Strong Correlation: Points closely follow a straight line (e.g., $r = 0.961$ for fat and calories in burgers).

• Moderate Correlation: Points show a general linear trend but with more scatter.

• Weak Correlation: Points are loosely scattered with little linear trend.

• No Correlation: No discernible linear pattern.

Using Technology to Calculate Correlation

Statistical software and calculators can compute the correlation coefficient efficiently. The following images show steps for entering data and performing linear regression to obtain r:

Examples

Example 1: Heights and Weights There is a positive association between students' heights and weights, with a correlation of $r = 0.644$ indicating a moderate linear relationship.

Example 2: Fat and Calories in Burgers Correlation of $r = 0.961$ shows a strong, positive, linear relationship between fat content and calories in fast food hamburgers.

Straightening Scatterplots

Correlation measures only linear relationships. If a scatterplot shows a curved (nonlinear) pattern, re-expressing one or both variables (e.g., using logarithms or square roots) can sometimes straighten the relationship, making correlation an appropriate measure.

Common Pitfalls and Misinterpretations

• Correlation vs. Association: Use "correlation" only for linear relationships between quantitative variables. "Association" is a broader term that includes nonlinear and categorical relationships.

• Do Not Correlate Categorical Variables: Correlation is not appropriate for categorical data.

• Check for Linearity: A strong association may exist even if the relationship is not linear, but correlation will not capture this.

• Beware of Outliers: Outliers can dramatically change the value of r.

Correlation Does Not Imply Causation

It is crucial to remember that a strong correlation between two variables does not mean that one causes the other. There may be other factors (lurking variables) influencing both variables, or the relationship may be coincidental.

• Example: The number of ice cream cones sold and the number of shark attacks may be correlated, but eating ice cream does not cause shark attacks. Both may be related to a lurking variable such as temperature or season.

• Example: The number of cans of diet soda consumed and the number of accidents may be correlated, but this does not imply causation.

Lurking Variables

A lurking variable is a hidden variable that affects both variables under study, potentially creating a spurious association. Always consider the possibility of lurking variables when interpreting correlation.

Summary Table: Correlation Properties and Conditions

Property/Condition	Description
Range of r	-1 to +1
Direction	Sign of r (+ or -)
Strength	Closer to |1| is stronger
Symmetry	r(x, y) = r(y, x)
No Units	Unitless measure
Quantitative Variables	Both variables must be quantitative
Straight Enough Condition	Relationship should be linear
Outlier Condition	Outliers can distort r
Lurking Variables	May create spurious correlation

Additional info: This guide covers the essentials of scatterplots, association, and correlation, including calculation, interpretation, and common pitfalls, as relevant to introductory statistics.