Covariance Formula
Covariance is one of the statistical measurement to know the relationship of the variance between the two variables. The Covariance indicates how two variables are related and also helps to know whether the two variables vary together or change together.
The Covariance is denoted as Cov(X,Y) and here is the formula.
\[\LARGE Cov(X,Y)= \frac{\sum (x_{i}-\overline{x})(y_{i}-\overline{y})}{N}\]
xi = data value of x
yi = data value of y
$\overline{x}$ = mean of x
$\overline{y}$ = mean of y
N = number of data values.
Solved Examples
Economic Growth % ($x_{i}$) | S&P 500 Returns % ($y_{i}$) |
2.1 | 8 |
2.5 | 12 |
4.0 | 14 |
3.6 | 10 |
x = 2.1, 2.5, 4.0, and 3.6 (economic growth)
y = 8, 12, 14, and 10 (S&P 500 returns)
Find $\overline{X}$ and $\overline{Y}$
$\overline{X}=\frac{2.1+2.5+4+3.6}{4}$
$\overline{X}=\frac{12.2}{4}$
$\overline{X}=3.1$
$\overline{Y}=\frac{8+12+14+10}{4}$
$\overline{Y}=\frac{44}{4}$
$\overline{Y}=11$
Now, $\overline{X}=3.1$ and $\overline{Y}=11$
Now, Substitute these values into the covariance formula to determine the relationship between economic growth and S&P 500 returns.
$Cov(X,Y)= \frac{\sum (x_{i}-\overline{x})(y_{i}-\overline{y})}{N}$
$Cov(X,Y)= \frac{(2.1-3.1)(8-11)}{4-1}$
$Cov(X,Y)= \frac{(-1)(-3)+(-0.6)(1)+(0.9)(3)}{3}$
$Cov(X,Y)= \frac{3+(-0.6)+2.7+(0.5)}{3}$
$Cov(X,Y)= \frac{4.6}{3}$
$Cov(X,Y)= 1.53$