Find The Correlation Coefficient, { R $}$, Of The Data Described Below.The Counselor At Danville High School Suspects That Students Are Signed Up For Too Many Extracurricular Activities And Thinks This Negatively Affects Their Academic
The Correlation Coefficient: A Measure of Relationship Between Extracurricular Activities and Academic Performance
As a high school counselor, it's essential to understand the relationship between students' involvement in extracurricular activities and their academic performance. The counselor at Danville High School suspects that students who are signed up for too many extracurricular activities may experience a negative impact on their academic performance. To investigate this hypothesis, we need to calculate the correlation coefficient, denoted as , which measures the strength and direction of the linear relationship between two variables. In this case, the two variables are the number of extracurricular activities and academic performance.
The correlation coefficient, , is a statistical measure that calculates the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where:
- A value of 1 indicates a perfect positive linear relationship.
- A value of -1 indicates a perfect negative linear relationship.
- A value of 0 indicates no linear relationship.
To calculate the correlation coefficient, we need to follow these steps:
- Collect data: Gather data on the number of extracurricular activities and academic performance for a sample of students.
- Calculate the mean: Calculate the mean of both variables.
- Calculate the deviations: Calculate the deviations from the mean for both variables.
- Calculate the covariance: Calculate the covariance between the two variables.
- Calculate the correlation coefficient: Use the formula to calculate the correlation coefficient.
Let's assume we have the following data on the number of extracurricular activities and academic performance for a sample of 10 students:
| Student | Extracurricular Activities | Academic Performance |
|---|---|---|
| 1 | 2 | 80 |
| 2 | 3 | 70 |
| 3 | 4 | 90 |
| 4 | 5 | 60 |
| 5 | 6 | 85 |
| 6 | 7 | 75 |
| 7 | 8 | 95 |
| 8 | 9 | 65 |
| 9 | 10 | 80 |
| 10 | 11 | 70 |
To calculate the mean, we need to add up all the values and divide by the number of observations.
| Student | Extracurricular Activities | Academic Performance |
|---|---|---|
| 1 | 2 | 80 |
| 2 | 3 | 70 |
| 3 | 4 | 90 |
| 4 | 5 | 60 |
| 5 | 6 | 85 |
| 6 | 7 | 75 |
| 7 | 8 | 95 |
| 8 | 9 | 65 |
| 9 | 10 | 80 |
| 10 | 11 | 70 |
Mean of Extracurricular Activities = (2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11) / 10 = 65 Mean of Academic Performance = (80 + 70 + 90 + 60 + 85 + 75 + 95 + 65 + 80 + 70) / 10 = 80
To calculate the deviations, we need to subtract the mean from each value.
| Student | Extracurricular Activities | Deviation | Academic Performance | Deviation |
|---|---|---|---|---|
| 1 | 2 | -63 | 80 | 0 |
| 2 | 3 | -62 | 70 | -10 |
| 3 | 4 | -61 | 90 | 10 |
| 4 | 5 | -60 | 60 | -20 |
| 5 | 6 | -59 | 85 | 5 |
| 6 | 7 | -58 | 75 | -5 |
| 7 | 8 | -57 | 95 | 15 |
| 8 | 9 | -56 | 65 | -15 |
| 9 | 10 | -55 | 80 | 0 |
| 10 | 11 | -54 | 70 | -10 |
To calculate the covariance, we need to multiply the deviations and sum them up.
Covariance = ( (-63) * 0 + (-62) * (-10) + (-61) * 10 + (-60) * (-20) + (-59) * 5 + (-58) * (-5) + (-57) * 15 + (-56) * (-15) + (-55) * 0 + (-54) * (-10) ) / 10 Covariance = ( 0 + 620 + 610 + 1200 + -295 + 290 + -855 + 840 + 0 + 540 ) / 10 Covariance = 4090 / 10 Covariance = 409
To calculate the correlation coefficient, we need to use the formula .
First, we need to calculate the standard deviations of both variables.
Standard Deviation of Extracurricular Activities = sqrt( ( (-63)^2 + (-62)^2 + (-61)^2 + (-60)^2 + (-59)^2 + (-58)^2 + (-57)^2 + (-56)^2 + (-55)^2 + (-54)^2 ) / 10 ) Standard Deviation of Extracurricular Activities = sqrt( ( 3969 + 3844 + 3721 + 3600 + 3481 + 3364 + 3249 + 3136 + 3025 + 2916 ) / 10 ) Standard Deviation of Extracurricular Activities = sqrt( 31395 / 10 ) Standard Deviation of Extracurricular Activities = sqrt( 3139.5 ) Standard Deviation of Extracurricular Activities = 55.8
Standard Deviation of Academic Performance = sqrt( ( 0^2 + (-10)^2 + 10^2 + (-20)^2 + 5^2 + (-5)^2 + 15^2 + (-15)^2 + 0^2 + (-10)^2 ) / 10 ) Standard Deviation of Academic Performance = sqrt( ( 0 + 100 + 100 + 400 + 25 + 25 + 225 + 225 + 0 + 100 ) / 10 ) Standard Deviation of Academic Performance = sqrt( 1200 / 10 ) Standard Deviation of Academic Performance = sqrt( 120 ) Standard Deviation of Academic Performance = 10.95
Now, we can calculate the correlation coefficient.
Correlation Coefficient = Covariance / ( Standard Deviation of Extracurricular Activities * Standard Deviation of Academic Performance ) Correlation Coefficient = 409 / ( 55.8 * 10.95 ) Correlation Coefficient = 409 / 609.39 Correlation Coefficient = 0.67
The correlation coefficient of 0.67 indicates a moderate positive linear relationship between the number of extracurricular activities and academic performance. This means that as the number of extracurricular activities increases, academic performance also tends to increase. However, the relationship is not perfect, and there may be other factors at play.
In conclusion, the correlation coefficient of 0.67 suggests a moderate positive linear relationship between the number of extracurricular activities and academic performance. While this relationship is not perfect, it suggests that students who are involved in more extracurricular activities tend to perform better academically. However, it's essential to note that correlation does not imply causation, and there may be other factors at play. Further research is needed to fully understand the relationship between extracurricular activities and academic performance.
Frequently Asked Questions (FAQs) About the Correlation Coefficient
A: The correlation coefficient is a statistical measure that calculates the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where:
- A value of 1 indicates a perfect positive linear relationship.
- A value of -1 indicates a perfect negative linear relationship.
- A value of 0 indicates no linear relationship.
A: The correlation coefficient is calculated using the following formula:
r = cov(X, Y) / (σ_X σ_Y)
Where:
- r is the correlation coefficient
- cov(X, Y) is the covariance between the two variables
- σ_X is the standard deviation of the first variable
- σ_Y is the standard deviation of the second variable
A: Correlation does not imply causation. This means that just because two variables are related, it doesn't mean that one variable causes the other. There may be other factors at play that are influencing the relationship.
A: A correlation coefficient of 0 indicates that there is no linear relationship between the two variables. This means that the variables are not related in a way that can be described by a straight line.
A: A correlation coefficient of 1 or -1 indicates a perfect linear relationship between the two variables. This means that the variables are related in a way that can be described by a straight line.
A: The correlation coefficient can be used to make predictions about the future, but it's essential to note that correlation does not imply causation. This means that just because two variables are related, it doesn't mean that one variable causes the other.
A: The correlation coefficient can be used to identify potential causes of a problem, but it's essential to note that correlation does not imply causation. This means that just because two variables are related, it doesn't mean that one variable causes the other.
A: The correlation coefficient has many applications in various fields, including:
- Business: to analyze the relationship between sales and marketing efforts
- Medicine: to analyze the relationship between disease and risk factors
- Social sciences: to analyze the relationship between socioeconomic factors and behavior
- Economics: to analyze the relationship between economic indicators and policy decisions
A: Some common mistakes to avoid when using the correlation coefficient include:
- Assuming that correlation implies causation
- Failing to consider other factors that may influence the relationship
- Using the correlation coefficient to make predictions about the future
- Using the correlation coefficient to identify the cause of a problem
In conclusion, the correlation coefficient is a powerful statistical tool that can be used to analyze the relationship between two variables. However, it's essential to note that correlation does not imply causation, and there may be other factors at play that are influencing the relationship. By understanding the limitations and applications of the correlation coefficient, you can use it to make informed decisions and predictions about the future.