Coefficient of Determination: Videos & Practice Problems

The coefficient of determination, represented as $R^2$, measures how much variation in the dependent variable (y) is explained by the independent variable (x). It is calculated by squaring the linear correlation coefficient $r$. A higher $R^2$ value indicates a stronger linear relationship, with values closer to 1 suggesting that most variation is explained by the correlation, while values near 0 indicate minimal explanation. This concept is crucial in regression analysis and understanding data relationships.

Coefficient of Determination

In statistical analysis, understanding the relationship between two variables is crucial, and two important measures that help in this regard are the linear correlation coefficient and the coefficient of determination. The linear correlation coefficient, denoted as \( r \), quantifies the strength and direction of a linear relationship between two variables, while the coefficient of determination, represented as \( R^2 \), provides insight into how much of the variation in the dependent variable (y) can be explained by the independent variable (x).

The coefficient of determination is calculated by squaring the linear correlation coefficient: \( R^2 = r^2 \). This means that if you know the value of \( r \), you can easily find \( R^2 \) by performing this simple calculation. For instance, if \( r = 0.745 \), then \( R^2 = (0.745)^2 = 0.555 \). This indicates that approximately 55.5% of the variation in the dependent variable can be explained by the independent variable.

It is important to note that while \( r \) can range from -1 to 1, indicating both the strength and direction of the relationship, \( R^2 \) will always be a positive value between 0 and 1. A higher \( R^2 \) value suggests a stronger linear relationship, meaning that the data points are closer to the regression line, while a value closer to 0 indicates a weak relationship with more scattered data points.

Graphically, the coefficient of determination can be understood through the concepts of explained variation and total variation. The explained variation refers to the sum of the squared distances from the regression line to the mean of the data, while the total variation is the sum of the squared distances from each data point to the mean. The ratio of explained variation to total variation gives the \( R^2 \) value, which can also be expressed as a percentage. For example, if \( R^2 = 0.555 \), it can be interpreted as 55.5% of the variation in the dependent variable being explained by the independent variable, with the remaining 44.5% attributed to other factors or randomness.

In practice, to calculate \( R^2 \) using a graphing calculator, one would input the data into the calculator, access the statistical menu, and perform a linear regression analysis. The output will provide both \( r \) and \( R^2 \) values, allowing for a comprehensive understanding of the data's linear relationship.

In summary, both the linear correlation coefficient and the coefficient of determination are essential tools in statistics for analyzing relationships between variables. Understanding how to calculate and interpret these coefficients is vital for effective data analysis and drawing meaningful conclusions from datasets.

In a given dataset, you determine the value of the correlation coefficient to be $r=-0.957$. Find the coefficient of determination. What does this tell you about the explained variation of the data about the regression line? What about the unexplained variation?

• A. Explained = 91.6%; Unexplained = 8.4%
• B. Explained = 95.7%; Unexplained = 4.3%
• C. Explained = 8.4%; Unexplained = 91.6%
• D. Explained = 4.3%; Unexplained = 95.7%

Answer: (A)

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

• A. 0.0031
• B. 0.0016
• C. 0.9984
• D. 0.9969

Answer: (D)

Here’s what students ask on this topic:

What is the coefficient of determination (R²) and how is it calculated?

The coefficient of determination, denoted as R², measures the proportion of variation in the dependent variable (y) that is explained by the independent variable (x) in a regression model. It is calculated by squaring the linear correlation coefficient (r). Mathematically:

$R^2=r^2$

R² values range from 0 to 1. A value closer to 1 indicates a strong linear relationship, meaning most of the variation in y is explained by x. Conversely, a value near 0 suggests minimal correlation. For example, if r = 0.8, then R² = 0.64, meaning 64% of the variation in y is explained by x.

What is the difference between the linear correlation coefficient (r) and the coefficient of determination (R²)?

The linear correlation coefficient (r) measures the strength and direction of a linear relationship between two variables, ranging from -1 to 1. A positive r indicates a positive relationship, while a negative r indicates a negative relationship. The coefficient of determination (R²), on the other hand, measures the proportion of variation in the dependent variable (y) explained by the independent variable (x). It is derived by squaring r:

$R^2=r^2$

Unlike r, R² is always positive and ranges from 0 to 1. While r provides information about the direction and strength, R² focuses on the explanatory power of the model.

How do you interpret the value of R² in regression analysis?

In regression analysis, R² represents the proportion of the total variation in the dependent variable (y) that is explained by the independent variable (x). For example, if R² = 0.75, it means 75% of the variation in y is explained by the linear relationship with x, while the remaining 25% is due to other factors or randomness. Higher R² values indicate a better fit of the regression model to the data. However, an R² value close to 1 does not guarantee causation, and a low R² does not necessarily mean the model is useless, especially in fields with inherently high variability.

How can you calculate R² using a graphing calculator?

To calculate R² using a graphing calculator, follow these steps:

1. Enter the data into the calculator: Input the x-values into L1 and the y-values into L2.
2. Access the regression function: Open the STAT menu, go to CALC, and select LinReg(ax+b).
3. Specify the data lists: Assign L1 as the x-data and L2 as the y-data.
4. Calculate: Press ENTER to compute the regression. The calculator will display the values of r and R².

For example, if r = 0.745, then R² = 0.555, meaning 55.5% of the variation in y is explained by x.

What does it mean if R² is close to 0 or 1?

If R² is close to 1, it indicates that most of the variation in the dependent variable (y) is explained by the independent variable (x), suggesting a strong linear relationship. For example, an R² of 0.9 means 90% of the variation in y is explained by x. Conversely, if R² is close to 0, it means the independent variable explains very little of the variation in y, implying a weak or no linear relationship. For instance, an R² of 0.1 means only 10% of the variation in y is explained by x, with the rest due to other factors or randomness.

How is R² used to evaluate the effectiveness of a predictive model?

R² is a key metric for evaluating the effectiveness of a predictive model. It quantifies how well the independent variable(s) explain the variation in the dependent variable. A higher R² value indicates a better fit, meaning the model is more effective at predicting outcomes. For example, an R² of 0.85 suggests the model explains 85% of the variation in the dependent variable, leaving only 15% unexplained. However, R² should be interpreted alongside other metrics, as a high R² does not guarantee causation or account for overfitting in complex models.