Question

What is the range of possible values of correlation coefficient?

Answer 1

We use the condition that the correlation coefficient is used to measure how strong the relation between two variables is. Here, the correlation coefficient is used when there are two variables and in order to develop their relation.
By using this definition and the use of correlation coefficient we find the possible values of correlation coefficient.

Complete step-by-step solution:
We are asked to find the possible values of correlation coefficient.
We know that the correlation coefficient is used to measure the relation between the two variables.
Here, let us rewrite the definition using the two variables.
Suppose there are two variables $x,y$ then for small change in $x$ the change in $y$ is given as the correlation coefficient. Here, we can see that this gives the relation between the variables $x,y$
Here, we can see that the correlation coefficient is termed for small changes in $x,y$ which means that the changes are very less and also the relationship between $x,y$ is linear.
So, by using this definition and use of correlation coefficient the possible values are given as the interval that is $-1\le r\le 1$ where, $'r'$ is the correlation coefficient.
Here, we can see that the above interval can be divided into 3 parts as follows:
(1) Negative correlation coefficient $\left( -1\le r<0 \right)$
Here we can see that the slope that determines the relation between $x,y$ is negative which suggests that the relation is linearly decreasing.
(2) Zero correlation coefficient $\left( r=0 \right)$
Here, the slope that determines the relation between $x,y$ is ‘0’ that suggests that there is no relation at all. This can also be taken as that $x,y$ are independent of each other.
(3) Positive correlation coefficient \left( 0Here, we can see that the slope that indicates the relation between x,y$is positive that tells us that the graph between$x,y$is linearly increasing. Therefore, we can conclude that the possible values of correlation coefficient can obtain are in the interval$\left[ -1,1 \right]$that is,$\therefore -1\le r\le 1$

Note: We need to note that sometimes that relation between two variables can be more than ‘1’ that is the relation between $x,y$ can be taken as,
$y=2x$
Here, the relation 2 doesn’t indicate the correlation coefficient because the variables $x,y$ are taken from the statistical data that have scatter plots. We use this correlation coefficient to determine one relation between two scatter plots.
If the relation is taken as $y=2x$ then there is no guarantee that the variables $x,y$ can take values in some constant difference so that points will be scattered in the plane so that finding the correlation coefficient will be always in the interval $\left[ -1,1 \right]$.