Question

Prove that the coefficient of correlation lies between 1- and 1

Answer 1

Hint: ccording to given in the question we have to prove that the coefficient of correlation lies between 1- and 1 so, first of all we have to understand coefficient of correlation as explained below:
Coefficient of correlation: The coefficient of correlation is the specific measure that quantifies the strength of the linear relationship between two variables given in a correlation analysis and the coefficient is symbolised by r in a correlation report.
Now, to prove that the coefficient of correlation lies between 1- and 1 we have to use the formula to find the correlation coefficient as given below:
Formula used:
$ \Rightarrow r = \dfrac{{\sum {({x_i} - \overline x )({y_i} - \overline y )} }}{{\sqrt {\sum {{{({x_i} - \overline x )}^2}\sum {{{({y_i} - \overline y )}^2}} } } }}................(a)$
Where, r is the correlation coefficient, ${x_i}$values of the x-variable in the sample, $\overline x $is the mean of the values of the x-variable, ${y_i}$ values of the y-variable in the sample, and $\overline y $ is the mean of the values of the y-variable.
Now, we have to use the formula (A) above to determine the correlation coefficient which is basically r and we have to consider the obtained expression as expression (1)
Now, as we know that $X = ({x_i}\overline x )$and $Y = ({y_i}\overline y )$so, we have to substitute these values in the expression (1) as obtained.
Now, we have to check for the inequality using schwarz's inequality according to which our obtained expression should be less than or equal to 1 and on solving further we will obtain the required value of r.

Complete step by step solution:
Step 1: First of all we have to use the formula (a) to find the correlation coefficient as mentioned in the solution hint.
$ \Rightarrow r = \dfrac{{\sum {({x_i}\overline x )({y_i} - \overline y )} }}{{\sqrt {\sum {{{({x_i}\overline x )}^2}\sum {{{({y_i} - \overline y )}^2}} } } }}$…………………….(1)
Step 2: Now, as we know that $X = ({x_i}\overline x )$and $Y = ({y_i}\overline y )$ hence, on substituting the values in the expression (1) as obtained in the step 1
$ \Rightarrow r = \dfrac{{\sum {XY} }}{{\sqrt {\sum {{{(X)}^2}\sum {{{(Y)}^2}} } } }}$………………………………(2)
Step 3: Now, to solve the expression (2) as obtained in the step 2 we have to use the swchwarz’s inequality. Hence,
\[ \Rightarrow (S{X^2}{Y^2}) \leqslant {\sum X ^2}\sum {{Y^2}} \]
Now, taking the terms to the left side of the expression,
\[ \Rightarrow \dfrac{{(S{X^2}{Y^2})}}{{{{\sum X }^2}\sum {{Y^2}} }} \leqslant 1\]
And as we know that,
$ \Rightarrow r = \dfrac{{\sum {XY} }}{{\sqrt {\sum {{{(X)}^2}\sum {{{(Y)}^2}} } } }}$
Hence,
$
   \Rightarrow {r^2} = 1 \\
   \Rightarrow r = \pm 1 \\
 $
Hence proved.

Note: The correlation coefficient is used in statistics to determine how strong a relation is between the given two variables so it calculates the strength of the relationship between the relative moments of two given variables.