Correlation and Regression: Study Notes for Introductory Statistics

Correlation and Regression

Introduction to Correlation

Correlation is a fundamental concept in statistics that describes the relationship between two variables. In this context, data are often represented as ordered pairs (x, y), where x is the independent (explanatory) variable and y is the dependent (response) variable.

• Correlation: A relationship between two variables, which can be visualized using scatter plots.

• Independent variable (x): The variable that is manipulated or categorized.

• Dependent variable (y): The variable that is measured or observed.

• Scatter plot: A graphical representation used to determine whether a linear (straight line) correlation exists between two variables.

• Ordered pairs: Each data point is represented as (x, y).

Types of Correlation

There are several types of correlation, each describing a different kind of relationship between variables:

• Positive Linear Correlation: As x increases, y tends to increase.

• Negative Linear Correlation: As x increases, y tends to decrease.

• No Correlation: No discernible relationship between x and y.

• Nonlinear Correlation: The relationship between x and y is not linear.

Example: Constructing a Scatter Plot

An economist wants to determine whether there is a linear relationship between a country’s gross domestic product (GDP) and carbon dioxide (CO2) emissions. The data are shown in the table below. Displaying the data in a scatter plot helps determine whether there appears to be a positive or negative linear correlation or no linear correlation.

• GDP (trillions of $): 1.6, 3.6, 4.9, 1.1, 0.9, 2.9, 2.7, 2.3, 1.6, 1.5

• CO2 emission (millions of metric tons): 428.2, 828.8, 1214.2, 444.6, 264.0, 415.3, 571.8, 454.9, 358.7, 573.5

Conclusion: The scatter plot shows a positive linear correlation. As GDP increases, CO2 emissions tend to increase.

Correlation Coefficient

The correlation coefficient is a measure of the strength and direction of a linear relationship between two variables. The sample correlation coefficient is denoted by r, and the population correlation coefficient is denoted by \rho (rho).

• Range of r: -1 to 1

• r = -1: Perfect negative correlation

• r = 1: Perfect positive correlation

• r ≈ 0: No linear correlation

Formula for the sample correlation coefficient r:

$r = \frac{n \sum xy - \sum x \sum y}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}$

• n: Number of data pairs

Examples of Correlation Coefficient Values

• Strong negative correlation: r = -0.91

• Strong positive correlation: r = 0.88

• Weak positive correlation: r = 0.42

• Nonlinear correlation: r = 0.07

Using Technology to Find a Correlation Coefficient

Statistical software or calculators can be used to compute the correlation coefficient. For example, using technology to analyze the Old Faithful geyser data yields r ≈ 0.979, indicating a strong positive correlation.

• Procedure: Enter data, use the DiagnosticOn command, and calculate r.

• Interpretation: A value close to 1 or -1 suggests a strong linear relationship.

Correlation and Causation

It is important to distinguish between correlation and causation. A strong correlation does not necessarily imply a cause-and-effect relationship. Several possibilities should be considered:

1. Direct cause-and-effect: Does x cause y?

2. Reverse cause-and-effect: Does y cause x?

3. Third variable: Is the relationship caused by another variable or a combination of variables?

4. Coincidence: Is the relationship merely coincidental?

Summary of Section 9.1

• Introduced linear correlation, independent and dependent variables, and types of correlation

• Found a correlation coefficient

• Tested a population correlation coefficient \rho using a table

• Performed a hypothesis test for a population correlation coefficient \rho

• Distinguished between correlation and causation