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
The two regression lines y = c + mx and x = a + dy always pass through:
If the correlation coefficient is the geometric mean of regression coefficients, then:
The limits of a multiple correlation coefficient R1.23 are:
For three variables X1, X2 and X3, the following information is available:
s1 = 10, r12 = 0.90
s2 = 5, r13 = 0.75
s3 = 3, r23 = 0.70
Then partial regression coefficient (b13,2) of X2 on X1 and X3 is:If the correlation coefficient between two variables is zero, then the two lines of regression will be:
Using usual notations, the correlation coefficient between two random variables, X and Y, is given by:
In Spearman's rank correlation coefficient, squared differences of ranks are used to:
In Spearman's rank correlation coefficient:
\[ r_s = 1 - \frac{6\sum_{i=1}^n d_i^2}{n(n^2-1)} \]The maximum value of \(\sum_{i=1}^n d_i^2\) in case of untied ranks 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
If the correlation between X and Y is 0.3, then correlation coefficient between 2X and 3Y is:
Important Questions from Correlation and Regression
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?
Given below are two statements:
Statement l: One of the assumptions under OLS method states that the regression model is linear in the parameters, though it may or may not be linear in the variables.
Statement ll: The variance of the error, or disturbance, term is not the same regardless of the value of the explanatory variable under OLS.
In light of the above statements, choose the most appropriate answer from the options given below
If two regression coefficients are -0.8 and -0.2, then the value of coefficient of correlation is
Which of the following statements relating to Correlation and Regression are true?
(a) The Coefficient of Correlation is independent of change of origin and scale.
(b) The Coefficient of Correlation between the two variables is the arithmetic average of the two Regression Coefficients.
(c) The probable error of the Coefficient Correlation is 0.6745 times its standard error.
(d) Coefficient of Correlation multiplied by the ratio between the standard deviations of the two variables denotes the slope of the regression line.
Code:
If two regression coefficients are 0.8 and 1.2, which one of the following is the value of coefficient of correlation?
