Variation and the Coefficient of Determination in Regression Analysis

Variation and the Coefficient of Determination

Understanding the Coefficient of Determination ($R^2$)

The coefficient of determination, denoted as $R^2$, is a key statistical measure in regression analysis. It quantifies how much of the variation in the dependent variable (y) is explained by the variation in the independent variable (x).

• Definition: $R^2$ measures the proportion of the total variation in y that is explained by the regression model using x.

• Interpretation: An $R^2$ value close to 1 indicates that almost all of the variation in y is explained by x. An $R^2$ value close to 0 means that none of the variation is explained by x; the data is nearly uncorrelated.

Formula:

$R^2 = 1 - \frac{\text{Unexplained Variation}}{\text{Total Variation}}$

Alternatively, $R^2$ can be calculated as the square of the correlation coefficient ($r$):

$R^2 = r^2$

Explained vs. Unexplained Variation

In regression analysis, the total variation in the dependent variable (y) can be split into two components:

• Explained Variation: The part of the variation in y that is accounted for by the regression model (i.e., by changes in x).

• Unexplained Variation: The part of the variation in y that is not accounted for by the regression model; often due to random error or other variables not included in the model.

Example: Suppose you have data on test scores (y) versus hours studied (x). If $R^2 = 0.555$, then 55.5% of the variation in test scores is explained by hours studied, and 44.5% is unexplained.

Application: Calculating $R^2$ from Data

Given a dataset, you can determine the value of the correlation coefficient ($r$) and then compute $R^2$ to assess the strength of the relationship between variables.

• Step 1: Enter the data into lists (e.g., L1 and L2) on a calculator.

• Step 2: Use the regression function to calculate the correlation coefficient ($r$).

• Step 3: Square the correlation coefficient to obtain $R^2$.

Calculator Instructions (TI-84):

• Enter data in L1 and L2.

• Press STAT → CALC → LinReg(ax+b).

• Set Xlist: L1, Ylist: L2.

• View output: $r$ = Correlation Coefficient, $R^2$ = Coefficient of Determination.

Worked Example Table

The following table illustrates how to compute the coefficient of determination from a set of data:

Hours Studied (x)	Test Score (y)
2	65
4	70
6	75
8	80
10	85

Suppose the correlation coefficient $r = 0.745$. Then:

$R^2 = (0.745)^2 = 0.555$

This means 55.5% of the variation in test scores is explained by hours studied.

Additional Example: Retail Analysis

A retail analyst studies the relationship between the number of in-store promotional displays (x) and weekly sales revenue (y) at 12 store locations. The data is entered into a calculator to find the coefficient of determination.

Displays (x)	Weekly Revenue (y)
5	1440
6	1560
7	1680
8	1800
9	1920
10	2040
11	2160
12	2280

By following the calculator steps above, the analyst can determine $R^2$ and interpret how much of the variation in weekly revenue is explained by the number of displays.

Summary Table: $R^2$ Interpretation

$R^2$ Value	Interpretation
Close to 1	Nearly all variation in y is explained by x
Close to 0	Almost none of the variation in y is explained by x

Additional info: The coefficient of determination is a central concept in regression analysis, helping to assess the goodness-of-fit of a model and the strength of the relationship between variables.