Question

Correlation coefficient between two random variables x and y denoted by ${{\mathbf{r}}_{{\mathbf{xy}}}}{\text{ or }}{\mathbf{r}}({\mathbf{x}},{\mathbf{y}})$
Option
A. ${{\mathbf{r}}_{{\text{xy}}}} = {{\mathbf{r}}_{{\text{yx}}}}$
B. Correlation coefficient(r) does not change its magnitude under the change of origin and scale.
C. r(x, y) =1
D. r lies between -1 and 1, that is $- 1 \leqslant r \leqslant 1$

Answer 1

The correlation coefficient is a statistical term that aids in the establishment of a connection between expected and actual values in a statistical experiment. The correlation coefficient's estimated value explains the exactness of the expected and real values.

Complete step-by-step answer:
The linear correlation coefficient, abbreviated as "r," expresses the degree of relationship between two variables. Since it predicts the relationship between two variables, it is also known as the Cross correlation coefficient. Let's move on to estimating the correlation coefficient in a mathematical manner.
If the two variables under consideration are x and y, the correlation coefficient can be determined using the formula.
$r = \dfrac{{n\left( {\sum x y} \right) - \left( {\sum x } \right)\left( {\sum y } \right)}}{{\left[ {n\sum {{x^2}} - {{\left( {\sum x } \right)}^2}} \right]\left[ {n\sum {{y^2}} - {{\left( {\sum y } \right)}^2}} \right]}}$
${\text{n}} =$Number of values or elements
$\sum x =$Sum of the 1 st values list
$\sum y =$Sum of the 2nd values list
$\sum {{\text{xy}}} =$Sum of the product of 1 st and 2 nd values
$\sum {{{\text{x}}^2}} =$Sum of the squares of 1st values
$\sum {{{\text{y}}^2}} =$Sum of the squares of 2nd values
The aim of the correlation coefficient is to define correlations between two variables. The below are some of the properties of the correlation coefficient:

1. The correlation coefficient is calculated in the same unit as the two variables.
2. The symbol of coefficient correlations will still be the same as the variance.
3. The correlation coefficient's numerical value will range from -1 to 1. It's referred to as a real number worth.
4. If the coefficient of correlation approaches zero, it indicates a weak correlation. We can deduce that the relationship is weak when ‘r' is close to 0.
5. A negative value for the coefficient indicates a good and negative correlation. And if ‘r' continues to hit -1, it indicates that the relationship is heading in the wrong direction. As the value of ‘r' reaches the value of + 1, the relationship is good and constructive. We should conclude that if the correlation outcome is +1, the relationship is in a positive state.
6. The correlation coefficient is risky and we don't know if the subjects are telling the truth or not. When the two factors are swapped, the coefficient of correlation remains unchanged.
7. The coefficient of correlation is a pure number that is unaffected by units. When we apply the same number to all the values of one element, it has no effect. All of the factors can be multiplied by the same positive integer. The correlation coefficient is unaffected. As previously said, since ‘r' is a scale invariant, it is unaffected by any unit.
8. We use inference to measure the relationship, although this does not imply that we are about causation. This basically means that if two variables are correlated, there's a chance the third variable is affecting them.

So, the correct answer is “Option A”.

Note: The term homoscedastic comes from the Greek and means "ability to scatter." Homoscedasticity is a statistical term that means "equal variances." The error term is the same for all values of the independent variable. If the error term is smaller for one set of independent variable values and greater for another set of values, homoscedasticity is broken. A scatter plot can be used to visually verify it. If the points lie similarly on both sides of the line of best fit, the data is said to be homoscedastic.