Question

Calculate the coefficient of correlation between X and Y series from the following data:
\[n = 15,\bar x = 25,\bar y = 18,{\sigma _x} = 3.01,{\sigma _y} = 3.03,\sum {\left( {{X_i} - \bar X} \right)\left( {{Y_i} - \bar Y} \right)} = 122\]

Answer 1

Hint:
Here we will use the concept of the coefficient of correlation. Firstly we will find the value of the covariance of the \[X\] and \[Y\] series using the formula. Then we will put the value of covariance in the formula of the coefficient of correlation to get its value. Coefficient of correlation is the parameter which is used to measure the strength of the relationship between the two variables that varies relatively.

Formula used:
We will use the following formulas:
1) Coefficient of correlation \[ = \rho = \dfrac{{Cov\left( {X,Y} \right)}}{{{\sigma _x}{\sigma _y}}}\]
2) Covariance, \[Cov\left( {X,Y} \right) = \sum\limits_{i = 1}^n {\dfrac{{\left( {{X_i} - \bar X} \right)\left( {{Y_i} - \bar Y} \right)}}{{n - 1}}} \]

Complete Step by Step Solution:
Given values are \[n = 15,\bar x = 25,\bar y = 18,{\sigma _x} = 3.01,{\sigma _y} = 3.03,\sum {\left( {{X_i} - \bar X} \right)\left( {{Y_i} - \bar Y} \right)} = 122\]
Firstly we will find the value of the correlation of the X and Y series of the data. Therefore, by using the formula of the covariance, we get
Covariance, \[Cov\left( {X,Y} \right) = \sum\limits_{i = 1}^n {\dfrac{{\left( {{X_i} - \bar X} \right)\left( {{Y_i} - \bar Y} \right)}}{{n - 1}}} \]
Now we will put the values given in the formula of the covariance. Therefore, we get
\[ \Rightarrow \] Covariance, \[Cov\left( {X,Y} \right) = \dfrac{{122}}{{15 - 1}}\]
Subtracting the terms in the denominator, we get
\[ \Rightarrow \] Covariance, \[Cov\left( {X,Y} \right) = \dfrac{{122}}{{14}} = \dfrac{{61}}{7}\]
Now we will use the formula of the coefficient of the correlation to get its value. Therefore, we get
Coefficient of correlation \[ = \rho = \dfrac{{Cov\left( {X,Y} \right)}}{{{\sigma _x}{\sigma _y}}}\]
Now by putting all the values in the formula of the coefficient of correlation we will get its value. Therefore, we get
\[ \Rightarrow \] Coefficient of correlation \[ = \rho = \dfrac{{\dfrac{{61}}{7}}}{{3.01 \times 3.03}}\]
\[ \Rightarrow \] Coefficient of correlation \[ = \rho = 0.955\]

Hence the coefficient of correlation between \[X\] and \[Y\] series is \[0.955\].

Note:
The value of the coefficient of the correlation general varies from the negative one to the positive one i.e. \[ - 1\] to 1. To calculate the value of the coefficient of correlation we have to calculate the value of the covariance between those two variables as it measures the variation of the two variables relatively. Positive value of the covariance means that the change in the relative value moves together but the negative value of the covariance means that the change in the relative value varies inversely.