\begin{tabular}{|c|c|c|c|c|c|}\hline Hours Studying & 0 & 1 & 2 & 4 & 5 \\\hline Midterm Grades & 67 & 77 & 83 & 97 & 99 \\\hline\end{tabular}Find The Value Of The Coefficient Of Determination. Round Your Answer To Three Decimal
Introduction
As students, we often wonder how much our studying habits affect our grades. A common question is: "How much of the variation in my grades can be explained by the number of hours I study?" This is where the concept of the coefficient of determination comes in. In this article, we will explore the coefficient of determination, its significance, and how to calculate it using a given dataset.
What is the Coefficient of Determination?
The coefficient of determination, often denoted as R-squared (R²), is a statistical measure that indicates the proportion of the variance in the dependent variable (in this case, midterm grades) that is predictable from the independent variable (hours studying). In other words, it measures how well a linear regression model fits the data. A high R-squared value indicates that the model is a good fit, while a low value suggests that the model is not a good fit.
Calculating the Coefficient of Determination
To calculate the coefficient of determination, we need to follow these steps:
- Calculate the mean of the dependent variable (midterm grades): We need to find the average midterm grade for each group of hours studying.
- Calculate the deviations from the mean: We need to find the difference between each midterm grade and the mean midterm grade for each group of hours studying.
- Calculate the sum of the squared deviations: We need to square each deviation and add them up for each group of hours studying.
- Calculate the total sum of squares: We need to find the sum of the squared deviations for all groups of hours studying.
- Calculate the regression sum of squares: We need to find the sum of the squared deviations that can be explained by the linear regression model.
- Calculate the coefficient of determination: We need to divide the regression sum of squares by the total sum of squares.
Dataset
| Hours Studying | Midterm Grades |
|---|---|
| 0 | 67 |
| 1 | 77 |
| 2 | 83 |
| 4 | 97 |
| 5 | 99 |
Step 1: Calculate the Mean of the Dependent Variable
To calculate the mean midterm grade for each group of hours studying, we need to add up the midterm grades for each group and divide by the number of observations.
| Hours Studying | Midterm Grades | Mean |
|---|---|---|
| 0 | 67 | 67 |
| 1 | 77 | 77 |
| 2 | 83 | 83 |
| 4 | 97 | 97 |
| 5 | 99 | 99 |
Step 2: Calculate the Deviations from the Mean
To calculate the deviations from the mean, we need to subtract the mean midterm grade for each group from the actual midterm grade.
| Hours Studying | Midterm Grades | Mean | Deviation |
|---|---|---|---|
| 0 | 67 | 67 | 0 |
| 1 | 77 | 77 | 0 |
| 2 | 83 | 83 | 0 |
| 4 | 97 | 97 | 0 |
| 5 | 99 | 99 | 0 |
Step 3: Calculate the Sum of the Squared Deviations
To calculate the sum of the squared deviations, we need to square each deviation and add them up for each group of hours studying.
| Hours Studying | Midterm Grades | Mean | Deviation | Squared Deviation |
|---|---|---|---|---|
| 0 | 67 | 67 | 0 | 0 |
| 1 | 77 | 77 | 0 | 0 |
| 2 | 83 | 83 | 0 | 0 |
| 4 | 97 | 97 | 0 | 0 |
| 5 | 99 | 99 | 0 | 0 |
Step 4: Calculate the Total Sum of Squares
To calculate the total sum of squares, we need to find the sum of the squared deviations for all groups of hours studying.
Total Sum of Squares = 0 + 0 + 0 + 0 + 0 = 0
Step 5: Calculate the Regression Sum of Squares
To calculate the regression sum of squares, we need to find the sum of the squared deviations that can be explained by the linear regression model.
Regression Sum of Squares = 0
Step 6: Calculate the Coefficient of Determination
To calculate the coefficient of determination, we need to divide the regression sum of squares by the total sum of squares.
Coefficient of Determination = Regression Sum of Squares / Total Sum of Squares = 0 / 0
Since the total sum of squares is zero, the coefficient of determination is undefined. This is because the linear regression model does not explain any of the variation in the midterm grades.
Conclusion
Q: What is the coefficient of determination?
A: The coefficient of determination, often denoted as R-squared (R²), is a statistical measure that indicates the proportion of the variance in the dependent variable (in this case, midterm grades) that is predictable from the independent variable (hours studying).
Q: Why is the coefficient of determination important?
A: The coefficient of determination is important because it helps us understand how well a linear regression model fits the data. A high R-squared value indicates that the model is a good fit, while a low value suggests that the model is not a good fit.
Q: How do I calculate the coefficient of determination?
A: To calculate the coefficient of determination, you need to follow these steps:
- Calculate the mean of the dependent variable (midterm grades): You need to find the average midterm grade for each group of hours studying.
- Calculate the deviations from the mean: You need to find the difference between each midterm grade and the mean midterm grade for each group of hours studying.
- Calculate the sum of the squared deviations: You need to square each deviation and add them up for each group of hours studying.
- Calculate the total sum of squares: You need to find the sum of the squared deviations for all groups of hours studying.
- Calculate the regression sum of squares: You need to find the sum of the squared deviations that can be explained by the linear regression model.
- Calculate the coefficient of determination: You need to divide the regression sum of squares by the total sum of squares.
Q: What does a high R-squared value indicate?
A: A high R-squared value indicates that the linear regression model is a good fit for the data. This means that the model is able to explain a large proportion of the variation in the dependent variable (midterm grades).
Q: What does a low R-squared value indicate?
A: A low R-squared value indicates that the linear regression model is not a good fit for the data. This means that the model is not able to explain a large proportion of the variation in the dependent variable (midterm grades).
Q: Can the coefficient of determination be negative?
A: No, the coefficient of determination cannot be negative. The R-squared value is always between 0 and 1, where 0 indicates that the model is not a good fit and 1 indicates that the model is a perfect fit.
Q: Can the coefficient of determination be 1?
A: Yes, the coefficient of determination can be 1. This indicates that the linear regression model is a perfect fit for the data.
Q: What are some common mistakes to avoid when calculating the coefficient of determination?
A: Some common mistakes to avoid when calculating the coefficient of determination include:
- Not checking for multicollinearity: Multicollinearity occurs when two or more independent variables are highly correlated with each other. This can lead to inaccurate estimates of the coefficient of determination.
- Not checking for outliers: Outliers are data points that are significantly different from the rest of the data. These can lead to inaccurate estimates of the coefficient of determination.
- Not using the correct formula: The formula for calculating the coefficient of determination is R² = 1 - (SSE / SST), where SSE is the sum of the squared errors and SST is the total sum of squares.
Conclusion
In this article, we answered some frequently asked questions about the coefficient of determination. We discussed what the coefficient of determination is, why it's important, and how to calculate it. We also discussed some common mistakes to avoid when calculating the coefficient of determination. By understanding the coefficient of determination, you can better understand how well a linear regression model fits the data and make more informed decisions.