Find the coefficient of correlation if given, cov(x, y) = 420, \(\sigma_x^2 = 484\) , and \(\sigma_y^2 = 441\)
0.909
Coefficient of Correlation Overview
The coefficient of correlation, often denoted as \(r\), is a statistical measure that quantifies the strength and direction of a linear relationship between two variables, typically \(x\) and \(y\). Its value ranges from -1 to +1.
- A value close to +1 indicates a strong positive linear relationship.
- A value close to -1 indicates a strong negative linear relationship.
- A value close to 0 indicates a weak or no linear relationship.
Correlation Formula Explained
To calculate the coefficient of correlation, we use the following formula, which involves the covariance between the two variables and their respective standard deviations:
\[r = \frac{\text{cov}(x, y)}{\sigma_x \sigma_y}\]
Where:
- \(\text{cov}(x, y)\) represents the covariance between variables \(x\) and \(y\). It measures how two variables change together.
- \(\sigma_x\) represents the standard deviation of variable \(x\). It is the square root of the variance of \(x\).
- \(\sigma_y\) represents the standard deviation of variable \(y\). It is the square root of the variance of \(y\).
Given Values Analysis
From the question, we are provided with the following values:
| Measurement | Value |
|---|---|
| Covariance of \(x\) and \(y\) (\(\text{cov}(x, y)\)) | \(420\) |
| Variance of \(x\) (\(\sigma_x^2\)) | \(484\) |
| Variance of \(y\) (\(\sigma_y^2\)) | \(441\) |
Standard Deviation Calculation
Before we can calculate the coefficient of correlation, we need to find the standard deviations (\(\sigma_x\) and \(\sigma_y\)) from the given variances. The standard deviation is simply the square root of the variance.
- Standard Deviation of \(x\): \[\sigma_x = \sqrt{\sigma_x^2} = \sqrt{484}\] \[\sigma_x = 22\]
- Standard Deviation of \(y\): \[\sigma_y = \sqrt{\sigma_y^2} = \sqrt{441}\] \[\sigma_y = 21\]
Coefficient Calculation Steps
Now that we have all the necessary values—covariance, standard deviation of \(x\), and standard deviation of \(y\)—we can substitute them into the coefficient of correlation formula.
Given:
- \(\text{cov}(x, y) = 420\)
- \(\sigma_x = 22\)
- \(\sigma_y = 21\)
Substitute these values into the formula:
\[r = \frac{\text{cov}(x, y)}{\sigma_x \sigma_y}\] \[r = \frac{420}{22 \times 21}\] \[r = \frac{420}{462}\]
Performing the division:
\[r \approx 0.909090...\]
Rounding to three decimal places, the coefficient of correlation is approximately \(0.909\).
Result Interpretation
A coefficient of correlation of approximately \(0.909\) indicates a strong positive linear relationship between variables \(x\) and \(y\). This means that as \(x\) increases, \(y\) also tends to increase significantly.
Important Questions from Correlation Analysis - Teaching
What type of relationship exists between two variables, if all points on the scatter diagram appear to fall on a straight line going upward from left to right?
An experiment consists of tossing a coin 20 times. Such an experiment is performed 50 times. The number of heads and the number of tails in each experiment are noted. What is the correlation coefficient between the two?
