The Karl Pearson’s correlation coefficient (r) between two random variables, X and Y, is computed by:
Understanding Karl Pearson's Correlation Coefficient
Karl Pearson's correlation coefficient, often denoted by \(r\), is a statistical measure that quantifies the strength and direction of a linear relationship between two continuous variables, let's say X and Y. The value of \(r\) ranges from -1 to +1. A value of +1 indicates a perfect positive linear correlation, -1 indicates a perfect negative linear correlation, and 0 indicates no linear correlation.
Formula for Karl Pearson's Correlation Coefficient
The formula for computing Karl Pearson's correlation coefficient \(r(X,Y)\) between two random variables X and Y involves their covariance and their respective standard deviations.
The formula is defined as the covariance between X and Y divided by the product of their standard deviations:
$$ r(X,Y) = \frac{Cov(X,Y)}{\sigma_X \sigma_Y} $$
Where:
- \(Cov(X,Y)\) is the covariance between variables X and Y. Covariance measures the extent to which two variables change together. A positive covariance indicates that X and Y tend to increase or decrease together, while a negative covariance indicates that one increases as the other decreases.
- \(\sigma_X\) is the standard deviation of variable X. Standard deviation measures the dispersion or spread of the data points in variable X around its mean.
- \(\sigma_Y\) is the standard deviation of variable Y. Standard deviation measures the dispersion or spread of the data points in variable Y around its mean.
The covariance provides information about how X and Y vary together, but its magnitude is influenced by the scales of X and Y. Dividing by the product of the standard deviations normalizes the covariance, resulting in a dimensionless measure (the correlation coefficient \(r\)) that is not affected by the units of the variables.
Analyzing the Given Options
Let's compare the standard formula for Karl Pearson's correlation coefficient with the options provided:
r(X,Y) = Cov(X,Y)*σXσY
This option suggests multiplying the covariance by the standard deviations. This is incorrect as the formula requires division by the product of standard deviations.- \(\rm r(X,Y)=\frac{\sigma_X \sigma _Y}{Cov(X,Y)}\)
This option presents the reciprocal of the correct formula, with the product of standard deviations in the numerator and covariance in the denominator. This is incorrect. - \(\rm r(X,Y)=\frac{Cov(X,Y)}{\sigma_X \sigma _Y}\)
This option correctly represents the covariance divided by the product of the standard deviation of X and the standard deviation of Y. This matches the standard definition of Karl Pearson's correlation coefficient. - \(\rm r(X,Y)=\frac{Cov(X,Y)}{(\sigma_X \sigma _Y)^2}\)
This option suggests dividing the covariance by the square of the product of the standard deviations. This is incorrect.
Based on the analysis, the formula \( \rm r(X,Y)=\frac{Cov(X,Y)}{\sigma_X \sigma _Y} \) accurately defines Karl Pearson's correlation coefficient between two random variables X and Y.
Revision Table: Karl Pearson Correlation Coefficient
| Concept | Description | Formula |
|---|---|---|
| Karl Pearson's r | Measures strength and direction of linear relationship between two variables. | \(\rm r(X,Y)=\frac{Cov(X,Y)}{\sigma_X \sigma _Y}\) |
| Covariance (Cov) | Measures how much two variables change together. | \(Cov(X,Y) = E[(X-E[X])(Y-E[Y])]\) |
| Standard Deviation (\(\sigma\)) | Measures spread of a single variable. | \(\sigma_X = \sqrt{Var(X)}\) |
Additional Information: Properties of Correlation Coefficient
- The value of Karl Pearson's correlation coefficient \(r\) always lies between -1 and +1, inclusive (\(-1 \le r \le +1\)).
- A correlation coefficient close to +1 indicates a strong positive linear relationship.
- A correlation coefficient close to -1 indicates a strong negative linear relationship.
- A correlation coefficient close to 0 indicates a weak or no linear relationship. It's important to note that a zero correlation coefficient does not necessarily mean the variables are independent; they could have a non-linear relationship.
- Correlation does not imply causation. A high correlation between X and Y does not mean that X causes Y or that Y causes X.
- The correlation coefficient is symmetric, i.e., \(r(X,Y) = r(Y,X)\).
Similar Questions
If, in a series. there are m items whose ranks are common, then which factor is added for correction for each repeating value in both the series?
If X 1, X 2, … X nis a simple random sample without replacement of size n from a finite population of N units with mean μ and variance σ 2, the covariance of (X i, Xj ) will be:
If x = X- \(\bar{X}\) and y = Y - \(\bar{Y}\) and the number of pairs (X, Y) is n, then the Karl Pearson's coefficient of correlation is:
The given table shows the ranking of ten students in two subjects mathematics and statistics.
Mathematics
Statistics
3
6
5
4
8
9
4
8
7
1
10
2
2
3
1
10
6
5
9
7
The coefficient of rank correlation is:
If the correlation between X and Y is 0.3, then correlation coefficient between 2X and 3Y is:
For three random variables X1 , X2 and X3, the pairwise correlation coefficients are r12 = r13 = r23 = r. The multiple correlation coefficient \(\rm R_{1.2.3}^2\) is
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If the regression coefficient bxy > 1 then byx is :
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Important Questions from Correlation and Regression
If the regression coefficient of x on y and y on x are \(- \frac{1}{2}\) and \(- \frac{1}{8}\) respectively, then what is the correlation coefficient between x and y?
Variable Y regresses with variable X with the conditions that \(\overline X = 5.50\) , \(\overline Y = 3.50\) and b = 1.50 in the linear regression model (Y = a + bX), where \(\overline Y\) and \(\overline X\) are means of the respective variables and b refers to gradient of line of Y w. r. t X. Which one of the following values of parameter 'a' of the model is correct?
It is given that X̅ = 10, Y̅ = 90, σ X= 3, σ Y= 12 and r XY = 0.8. The regression equation of X on Y is
For 10 observations on price (x) and supply (y), the following data was obtained:
∑x = 130, ∑y = 220, ∑ x 2= 2288, ∑y 2= 5506 and ∑ xy = 3467.
What is the line of regression of y on x?If, in a series. there are m items whose ranks are common, then which factor is added for correction for each repeating value in both the series?
