The coefficient of correlation between two variables X and Y is 0.48. The covariance is 36. The variance of X is 16. The standard deviation of Y is:
18.75
Calculate Standard Deviation Y from Correlation and Covariance
This problem asks us to find the standard deviation of a variable Y, given the coefficient of correlation between variables X and Y, their covariance, and the variance of variable X. We will use the formula for the coefficient of correlation to solve this.
Understanding the Given Information
We are provided with the following statistical measures:
- Coefficient of correlation between X and Y (\(r\)): 0.48
- Covariance between X and Y (\(\text{Cov}(X, Y)\)): 36
- Variance of X (\(\text{Var}(X)\)): 16
Our goal is to determine the standard deviation of Y (\(\sigma_Y\)).
Relating Correlation, Covariance, and Standard Deviations
The coefficient of correlation (\(r\)) is a measure that quantifies the linear relationship between two variables. It is defined by the formula:
\(r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}\)
Where:
- \(\text{Cov}(X, Y)\) is the covariance between X and Y.
- \(\sigma_X\) is the standard deviation of X.
- \(\sigma_Y\) is the standard deviation of Y.
We have \(r\) and \(\text{Cov}(X, Y)\). We are given the variance of X, from which we can find the standard deviation of X. Once we have \(\sigma_X\), we can rearrange the formula to solve for \(\sigma_Y\).
Step-by-Step Calculation of Standard Deviation of Y
Step 1: Find the Standard Deviation of X
The standard deviation of a variable is the square root of its variance. We are given that the variance of X (\(\text{Var}(X)\)) is 16.
\(\sigma_X = \sqrt{\text{Var}(X)}\)
Substituting the given value:
\(\sigma_X = \sqrt{16}\)
\(\sigma_X = 4\)
So, the standard deviation of X is 4.
Step 2: Rearrange the Correlation Formula to Solve for \(\sigma_Y\)
We use the formula for the coefficient of correlation:
\(r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}\)
We want to find \(\sigma_Y\). We can rearrange the formula by multiplying both sides by \(\sigma_X \sigma_Y\) and then dividing by \(r\):
\(r \times \sigma_X \times \sigma_Y = \text{Cov}(X, Y)\)
\(\sigma_Y = \frac{\text{Cov}(X, Y)}{r \times \sigma_X}\)
This rearranged formula allows us to calculate \(\sigma_Y\) using the known values.
Step 3: Substitute Values and Calculate \(\sigma_Y\)
Substitute the given values into the rearranged formula:
- \(\text{Cov}(X, Y) = 36\)
- \(r = 0.48\)
- \(\sigma_X = 4\) (calculated in Step 1)
\(\sigma_Y = \frac{36}{0.48 \times 4}\)
First, calculate the denominator:
\(0.48 \times 4 = 1.92\)
Now, perform the division:
\(\sigma_Y = \frac{36}{1.92}\)
\(\sigma_Y = 18.75\)
Thus, the standard deviation of Y is 18.75.
Summary of Results
Given the coefficient of correlation (0.48), covariance (36), and variance of X (16), we found the standard deviation of X to be 4 and subsequently calculated the standard deviation of Y to be 18.75.
| Statistical Measure | Value | Derived or Given |
|---|---|---|
| Coefficient of Correlation (\(r\)) | 0.48 | Given |
| Covariance (\(\text{Cov}(X, Y)\)) | 36 | Given |
| Variance of X (\(\text{Var}(X)\)) | 16 | Given |
| Standard Deviation of X (\(\sigma_X\)) | 4 | Derived (\(\sqrt{16}\)) |
| Standard Deviation of Y (\(\sigma_Y\)) | 18.75 | Derived (\(\frac{36}{0.48 \times 4}\)) |
The standard deviation of Y is 18.75.
Revision Table: Key Statistical Relationships
Understanding the relationships between different statistical measures is crucial for solving problems like this. Here's a quick review:
| Measure | Relationship to Variance/Standard Deviation | Relationship to Covariance/Correlation |
|---|---|---|
| Variance (\(\text{Var}\)) | Standard deviation squared (\(\sigma^2\)) | Part of correlation denominator indirectly |
| Standard Deviation (\(\sigma\)) | Square root of variance (\(\sqrt{\text{Var}}\)) | Denominator in correlation formula |
| Covariance (\(\text{Cov}(X, Y)\)) | Numerator in correlation formula | |
| Coefficient of Correlation (\(r\)) | Relates covariance to product of standard deviations (\(\frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}\)) |
Additional Information: Interpreting Correlation and Standard Deviation
- Standard Deviation: The standard deviation (\(\sigma\)) measures the dispersion or spread of data points around the mean. A higher standard deviation indicates greater variability in the data.
- Variance: Variance (\(\sigma^2\)) is the average of the squared differences from the mean. It is the standard deviation squared.
- Covariance: Covariance (\(\text{Cov}(X, Y)\)) measures the extent to which two variables change together. A positive covariance means they tend to increase or decrease together, while a negative covariance means one tends to increase as the other decreases.
- Coefficient of Correlation: The coefficient of correlation (\(r\)) standardizes covariance by dividing by the product of the standard deviations. This results in a value between -1 and +1, making it easier to interpret the strength and direction of the linear relationship regardless of the variables' scales. \(r=1\) indicates perfect positive linear correlation, \(r=-1\) indicates perfect negative linear correlation, and \(r=0\) indicates no linear correlation. In this problem, \(r = 0.48\) suggests a moderate positive linear relationship.
By using the relationships between these measures, as shown in the formula \(r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}\), we can find missing values if enough information is provided.
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