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Consider two random variables \(X\) and \(Y\) Here and we want to measure of the connection between two random variables and to what amount, they change together.
Covariance
We define the covariance between \(X\) and \(Y\) , written \(\text{Cov}(X,Y)\)
\[\text{Cov}(X, Y) = \frac{\sum_{i=1}^{n}(X_i - \bar{X})(Y_i - \bar{Y})}{N}\]| \(X\) | \(Y\) | \(X - \bar{X}\) | \(Y - \bar{Y}\) | \((X - \bar{X})(Y - \bar{Y})\) |
|---|---|---|---|---|
| 2 | 3 | -4 | -7 | 28 |
| 4 | 7 | -2 | -3 | 6 |
| 6 | 10 | 0 | 0 | 0 |
| 8 | 13 | 2 | 3 | 6 |
| 10 | 17 | 4 | 7 | 28 |
| \(\bar{X} = 6\) | \(\bar{Y} = 10\) | \(\sum_{i=1}^{n}(X_i - \bar{X})(Y_i - \bar{Y})= 68\) |
Here we can see covariance is 13.6. By which indicates a positive relationship between \(X\) and \(Y\). However,covariance is that it doesn’t provide a standardized measure of the relationship it doesn’t give information about the strength of the relationship or how dependent one variable is on the other.
For a standardized measure of the strength and direction of the linear relationship, we have to use correlation.
Correlation
The correlation coefficient (r) scale between -1 and 1.
\[\text{Cor}(X, Y) = \frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{\sqrt{\sum{(X_i - \bar{X})^2} \sum{(Y_i - \bar{Y})^2}}}\] \[\implies \text{Cor}(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y}\]where standard deviation \((\sigma)\)
\[\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N}(X_i - \bar{X})^2}\]Thats why correlation is often preferred over covariance due to its standardized nature, unit independence, and ease of interpretation.
Comparison:
| Covariance | Correlation |
|---|---|
| Measures the degree of joint variability between two variables. | Standardized measure of the linear relationship between two variables. |
| \(\text{Cov}(X, Y) = \frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{n-1}\) | \(\text{Corr}(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y}\) |
| Positive: Variables move in the same direction. Negative: Variables move in opposite directions. Zero: No linear relationship. | \(\rho = 1\): Perfect positive correlation. \(\rho = -1\): Perfect negative correlation. \(\rho = 0\): No linear correlation. |
| Not normalized. In the units of the product of the original variables. | Normalized to a range of -1 to 1, making it unitless and easier to interpret. |
| Depends on the units of the variables. | Scale-independent, suitable for comparing relationships between variables measured in different units. |
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