Covariance of two vectors $X$ and $y$ gives the relation between them, that is, how much the vectors point in the same direction.
$cov(X,Y)$ = $\frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{N-1}$
When a distribution table is given with the frequency of each data, the covariance is given by the given formula,
$cov(X,Y)$ = $\frac{\sum f_ix_iy_i}{N}$ $-\bar{x}\bar{y}$
Word Problems
Problem 1:
Find the covariance between the given two sets of data.
$X:\ 2,\ 5,\ 8,\ 11$
$Y:\ 5,\ 9,\ 1,\ 4$
Solution:
The mean of both the vectors can be given as,
$\bar{x}$ = $6.5$
$\bar{y}$ = $4.75$
The covariance can be calculated by using the formula,
$cov(X,Y)$ = $\frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{N-1}$
Putting the values and calculating we get,
$cov(X,Y)$ = $\frac{(2-6.5)+(5-6.5)+(8-6.5)+(11-6.5))((5-4.75)+(9-4.75)+(1-4.75)+(4-4.75))}{4-1}$ = $-5.5$
Problem 2:
The given table shows the profit of two companies (in million dollars) per annum. Find the covariance between them.
Year
|
Ageis Technologies
|
Ajanata Technologies
|
| 2005 |
11 |
9 |
| 2006 |
8 |
4 |
| 2007 |
7 |
12 |
| 2008 |
9 |
11 |
Solution:
The mean of both the data set is given as,
Mean for Ageis technologies, $\bar{x}$ = $8.75$
Mean for Ajanata technologies, $\bar{y}$ = $9$
Total number of observations, $N$ = $4$
The covariance can be calculated by using the formula,
$cov(X,Y)$ = $\frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{N-1}$
Putting the values and calculating we get,
$cov(X,Y)$ = $\frac{(11-8.75)+(8-8.75)+(8-8.75)+(7-8.75))((9-9)+(4-9)+(12-9)+(11-9))}{4-1}$ = -$0.333$
Problem 3:
The marks of $6$ students of a class for Mathematics and English is given here. Find the covariance between them.
| English |
65 |
76 |
61 |
86 |
88 |
70 |
| Mathematics |
72 |
67 |
56 |
94 |
81 |
98 |
Solution:
The mean of both the data set is given as,
English, $\bar{x}$ = $74.33$
Mathematics, $\bar{y}$ = $78$
Total number of observations, $N$ = $6$
The covariance can be calculated by using the formula,
$cov(X,Y)$ = $\frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{N-1}$
Putting the values and calculating we get,
$cov(X,Y)$ = $94.4$