Question

Covariance \[(x,y)\] between \[x\] and \[y\] if \[\sum x = 15\], \[\sum y = 40\], \[\sum xy = 110\], \[n = 5\] is
A.\[22\]
B.\[2\]
C.\[ - 2\]
D.None of these

Answer 1

Hint: In this question, we have to calculate the value of covariance \[(x,y)\] between \[x\] and \[y\]. We will use the formula for calculation of covariance. We simply put the required values into the formula and solve to get the desired result.
Formula used: \[\operatorname{cov} (x,y) = \dfrac{{\sum xy}}{n} - \left( {\dfrac{{\sum x}}{n}} \right)\left( {\dfrac{{\sum y}}{n}} \right)\]

Complete step-by-step answer:
This question is based on covariance. Covariance in statistics is the measurement of the relationship between two variables. For example, we know that when demand rises and as a result production of goods rises. Thus we can say demand and production have some relationship. This measurement of relationship is called covariance.
When we are given the summation of some terms, then the covariance between \[x\] and \[y\], is given by the formula \[\operatorname{cov} (x,y) = \dfrac{{\sum xy}}{n} - \left( {\dfrac{{\sum x}}{n}} \right)\left( {\dfrac{{\sum y}}{n}} \right)\] Where \[n\] is the number of terms.
Consider the given question,
We are given, \[\sum x = 15\], \[\sum y = 40\], \[\sum xy = 110\], \[n = 5\]
We know that the covariance between two variable say \[x\]and \[y\] is given by the formula
\[\operatorname{cov} (x,y) = \dfrac{{\sum xy}}{n} - \left( {\dfrac{{\sum x}}{n}} \right)\left( {\dfrac{{\sum y}}{n}} \right)\] , Where \[n\] is the number of terms.
Putting the values, we get
\[ \Rightarrow \operatorname{cov} (x,y) = \dfrac{{110}}{5} - \left( {\dfrac{{15}}{5}} \right)\left( {\dfrac{{40}}{5}} \right)\]
Using the BODMAS Rule, we will first divide.
On dividing \[110\] by \[5\] we get \[22\] , \[15\] by \[5\] we get \[3\] and \[40\] by \[5\] we get \[8\]. Hence we have
 \[ \Rightarrow \operatorname{cov} (x,y) = 22 - \left( 3 \right)\left( 8 \right)\]
On multiplying, we get
\[ \Rightarrow \operatorname{cov} (x,y) = 22 - 24\]
On solving, we get
\[ \Rightarrow \operatorname{cov} (x,y) = - 2\]
Hence Option \[(C)\] is correct.
So, the correct answer is “Option C”.

Note: The general formula of covariance between two number is given by formula \[\operatorname{cov} (x,y) = \dfrac{{\sum \left( {{x_i} - \overline x } \right)\left( {{y_i} - \overline y } \right)}}{n}\] where \[{x_i}\] is the data values of \[x\], \[{y_i}\]is the data values of \[y\], \[\overline x \]is the mean of data\[{x_i}\] and \[\overline y \]is the mean of the data \[{y_i}\].
Covariance of the variable \[x\] with \[y\] is the same as the covariance of \[y\] with \[x\]. i.e. \[\operatorname{cov} (x,y) = \operatorname{cov} (y,x)\]
Covariance of the variable with itself is zero. i.e. \[\operatorname{cov} (x,x) = 0\]
Covariance may be positive, negative and zero also.