The correlation coefficient computed form a set of 30 observations is 0.8 then the percentage of variation not explained by linear regression is
36%
Understanding Variation in Linear Regression
In linear regression, we often want to understand how much of the variation in the dependent variable can be attributed to the linear relationship with the independent variable. The correlation coefficient (denoted by \(r\)) is a measure of the linear association between two variables. While the correlation coefficient itself tells us the strength and direction of the linear relationship, its square, known as the coefficient of determination (\(r^2\)), tells us the proportion of the variance in the dependent variable that is predictable from the independent variable.
Correlation Coefficient and Explained Variation
The value of the correlation coefficient \(r\) is given as 0.8. The coefficient of determination, \(r^2\), represents the proportion of the total variation in the dependent variable that is explained by the linear relationship with the independent variable.
The formula for the coefficient of determination is:
\(r^2 = (\text{correlation coefficient})^2\)
Given \(r = 0.8\), we calculate \(r^2\) as:
\(r^2 = (0.8)^2 = 0.64\)
This means that 0.64, or 64%, of the variation in the dependent variable is explained by the linear regression model.
Calculating Unexplained Variation
The total variation in the dependent variable can be split into two parts:
- Variation explained by the regression model.
- Variation not explained by the regression model (also called residual variation).
The proportion of variation not explained by the linear regression model is \(1 - r^2\). This represents the part of the variation that is due to other factors or random chance, not accounted for by the linear relationship with the independent variable.
Using the calculated \(r^2\) value of 0.64:
Proportion of unexplained variation = \(1 - r^2 = 1 - 0.64 = 0.36\)
Percentage of Unexplained Variation
To express this proportion as a percentage, we multiply by 100%:
Percentage of unexplained variation = \(0.36 \times 100\% = 36\%\)
Therefore, 36% of the variation is not explained by the linear regression using the given data with a correlation coefficient of 0.8.
Let's summarise the key terms:
| Term | Symbol | What it Represents |
|---|---|---|
| Correlation Coefficient | \(r\) | Strength and direction of linear relationship (-1 to +1) |
| Coefficient of Determination | \(r^2\) | Proportion of variance explained by the model (0 to 1) |
| Proportion of Unexplained Variation | \(1 - r^2\) | Proportion of variance not explained by the model (0 to 1) |
Conclusion
Given a correlation coefficient of 0.8, the coefficient of determination is 0.64. This means 64% of the variation is explained by the linear regression. Consequently, the percentage of variation not explained by linear regression is \(100\% - 64\% = 36\%\).
Revision Table: Correlation and Regression Concepts
| Concept | Definition | Relation to \(r\) |
|---|---|---|
| Correlation Coefficient (\(r\)) | Measures the linear relationship's strength and direction. | Given directly or calculated from data. |
| Coefficient of Determination (\(r^2\)) | Proportion of dependent variable variance explained by the model. | \(r^2 = (\text{Correlation Coefficient})^2\) |
| Unexplained Variation | Proportion of dependent variable variance not explained by the model. | \(1 - r^2\) |
Additional Information: Importance of Explained Variation
Understanding the percentage of explained variation (\(r^2\)) and unexplained variation (\(1 - r^2\)) is crucial in evaluating the effectiveness of a linear regression model. A high \(r^2\) indicates that the model does a good job of predicting the dependent variable based on the independent variable. A low \(r^2\) suggests that either the linear relationship is weak, or other variables not included in the model significantly influence the dependent variable. The unexplained variation highlights the portion of the variability in the data that remains unaccounted for by the current linear model, prompting further investigation into other potential predictors or non-linear relationships.
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Code:
If two regression coefficients are 0.8 and 1.2, which one of the following is the value of coefficient of correlation?
