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:
Karl Pearson's correlation in terms of original values is
\[r=\dfrac{\sum (X-\bar{X})(Y-\bar{Y})}{\sqrt{\sum (X-\bar{X})^2\,\sum (Y-\bar{Y})^2}}\]
Substituting \(x=X-\bar{X}\) and \(y=Y-\bar{Y}\) gives
\[r=\dfrac{\sum xy}{\sqrt{\sum x^2\,\sum y^2}}\]
Note that \(n\) cancels because it appears in both numerator and denominator when expressed via covariances and standard deviations. Hence the correct expression is \(\dfrac{\sum xy}{\sqrt{\sum x^2\sum y^2}}\).
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:
The given table shows the ranking of ten students in two subjects mathematics and statistics.
Mathematics
Statistics
3
6
5
4
8
9
4
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1
10
2
2
3
1
10
6
5
9
7
The coefficient of rank correlation is:
The Karl Pearson’s correlation coefficient (r) between two random variables, X and Y, is computed by:
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
If the correlation coefficient between X and Y is 0.7, then the correlation coefficient between U and V, where U = X - 5 and V = Y + 2, is:
For Spearman's rank correlation, if the correlation coefficient is 0.7 and \(\rm \Sigma_{i=1}^n d_i^2=49.5\) then the value of sample size 'n' is:
If the regression coefficient bxy > 1 then byx is :
The limits of a multiple correlation coefficient R1.23 are:
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?
