Covariance between X and Y
Covariance measures the simultaneous variability between the two variables. It indicates how the two variables are related. A positive value of covariance indicate that the two variables moves in the same direction, whereas a negative value of covariance indicate that the two variables moves on opposite direction.
Let $(x_i, y_i), i=1,2, \cdots , n$ be $n$ pairs of observations.
Formula
The sample covariance between $x$ and $y$ is denoted by $Cov(x,y)$ or $s_{xy}$ and is defined as
$Cov(x,y) =s_{xy}=\dfrac{1}{n-1}\sum_{i=1}^{n} x_i y_i -\overline{x}\overline{y}$
OR
$s_{xy} = \frac{1}{n-1}\bigg(\sum xy - \frac{(\sum x)(\sum y)}{n}\bigg)$
where,
$\overline{x}=\dfrac{1}{n}\sum_{i=1}^{n}x_i$sample mean of $x$,$\overline{y}=\dfrac{1}{n}\sum_{i=1}^{n}y_i$sample mean of $y$
CovarianceCalculator
Use this calculator to find the covariance between $X$ and $Y$ for raw data.
| Covariance Calculator | |
|---|---|
| Enter the X Values (Separated by comma,) | |
| Enter the Y Values (Separated by comma,) | |
| Results | |
| Number of Obs. (n): | |
| Sample Mean of X : ($\overline{x}$) | |
| Sample Mean of Y : ($\overline{y}$) | |
| Sample variance of X: ($s^2_x$) | |
| Sample variance of Y : ($s^2_y$) | |
| Sample covariance between X and Y : ($s_{xy}$) | |
How to calculate Covariance?
Step 1 - Enter the $X$ values separated by commas
Step 2 - Enter the $Y$ values separated by commas
Step 3 - Click calculate button to calculate covariance
Step 4 - Gives the number of pairs of observations
Step 5 - Gives the sample mean of $X$
Step 6 - Gives the sample mean of $Y$
Step 7 - Gives the sample variance of $X$
Step 8 - Gives the sample variance of $Y$
Step 9 - Gives the sample covariance between $X$ and $Y$
Covariance Example 1
A study was conducted to analyze the relationship between advertising expenditure and sales. The following data were recorded:
| X Advertising (in \$) | 20 | 24 | 30 | 32 | 35 |
|---|---|---|---|---|---|
| Y Sales (in \$) | 310 | 340 | 400 | 420 | 490 |
Compute the covariance between advertising expenditure and sales.
Solution
Let $x$ denote the advertising expenditure and $y$ denote the sales.
| $x$ | $y$ | $x^2$ | $y^2$ | $xy$ | |
|---|---|---|---|---|---|
| 1 | 20 | 310 | 400 | 96100 | 6200 |
| 2 | 24 | 340 | 576 | 115600 | 8160 |
| 3 | 30 | 400 | 900 | 160000 | 12000 |
| 4 | 32 | 420 | 1024 | 176400 | 13440 |
| 5 | 35 | 490 | 1225 | 240100 | 17150 |
| Total | 141 | 1960 | 4125 | 788200 | 56950 |
The sample mean of $X$ is
$$ \begin{aligned} \overline{x} & = \frac{\sum x}{n}\\ &= \frac{141}{5}\\ &= 28.2 \end{aligned} $$
The sample mean of $Y$ is
$$ \begin{aligned} \overline{y} & = \frac{\sum y}{n}\\ &= \frac{1960}{5}\\ &= 392 \end{aligned} $$
The sample covariance between $x$ and $y$ is
$$ \begin{aligned} s_{xy} & = \frac{1}{n-1}\bigg(\sum xy - \frac{(\sum x)(\sum y)}{n}\bigg)\\ & = \frac{1}{5-1}\bigg(56950-\frac{(141)(1960)}{5}\bigg)\\ &= \frac{1}{4}\bigg(56950-\frac{276360}{5}\bigg)\\ &= \frac{1}{4}\bigg(56950-55272\bigg)\\ &= \frac{1678}{4}\\ &= 419.5. \end{aligned} $$
The covariance between advertising expenditure and sales is $419.5$. Since the value of covariance is positive, there is a positive relationship between advertising expenditure and sales. That is the two variables moves in the same direction.
Covariance Example 2
A study of the amount of rainfall and the quantity of air pollution removed produced the following data:
| Daily Rainfall (0.01cm) | 4.3 | 4.5 | 5.9 | 5.6 | 6.1 | 5.2 | 3.8 | 2.1 | 7.5 |
|---|---|---|---|---|---|---|---|---|---|
| Particulate Removed ($\mu g/m^3$) | 126 | 121 | 116 | 118 | 114 | 118 | 132 | 141 | 108 |
Calculate covariance between daily rainfall and particulate removed,
Solution
Let $x$ denote the daily rainfall (0.01 cm) and $y$ denote the particulate removed ($\mu g/m^3$).
| $x$ | $y$ | $x^2$ | $y^2$ | $xy$ | |
|---|---|---|---|---|---|
| 1 | 4.3 | 126 | 18.49 | 15876 | 541.8 |
| 2 | 4.5 | 121 | 20.25 | 14641 | 544.5 |
| 3 | 5.9 | 116 | 34.81 | 13456 | 684.4 |
| 4 | 5.6 | 118 | 31.36 | 13924 | 660.8 |
| 5 | 6.1 | 114 | 37.21 | 12996 | 695.4 |
| 6 | 5.2 | 118 | 27.04 | 13924 | 613.6 |
| 7 | 3.8 | 132 | 14.44 | 17424 | 501.6 |
| 8 | 2.1 | 141 | 4.41 | 19881 | 296.1 |
| 9 | 7.5 | 108 | 56.25 | 11664 | 810.0 |
| Total | 45.0 | 1094 | 244.26 | 133786 | 5348.2 |
The sample mean of $X$ is
$$ \begin{aligned} \overline{x} & = \frac{\sum x}{n}\\ &= \frac{45}{9}\\ &= 5 \end{aligned} $$
The sample mean of $Y$ is
$$ \begin{aligned} \overline{y} & = \frac{\sum y}{n}\\ &= \frac{1094}{9}\\ &= 121.5556 \end{aligned} $$
The sample covariance between $x$ and $y$ is
$$ \begin{aligned} s_{xy} & = \frac{1}{n-1}\bigg(\sum xy - \frac{(\sum x)(\sum y)}{n}\bigg)\\ & = \frac{1}{9-1}\bigg(5348.2-\frac{(45)(1094)}{9}\bigg)\\ &= \frac{1}{8}\bigg(5348.2-\frac{49230}{9}\bigg)\\ &= \frac{1}{8}\bigg(5348.2-5470\bigg)\\ &= \frac{-121.8}{8}\\ &= -15.225. \end{aligned} $$
The covariance between daily rainfall and particulate removed is $-15.225$. Since the value of covariance is negative, there is a negative relationship between daily rainfall and particulate removed. That is the two variables moves in the opposite direction.
Conclusion
In this tutorial, you learned about formula for covariance between $X$ and $Y$ for raw data and how to calculate covariance between $X$ and $Y$ for raw data. You also learned about how to solve numerical problems on covariance.
To learn more about other descriptive statistics measures, please refer to the following tutorials:
Let me know in the comments if you have any questions on Covariance calculator for raw data with examples and your thought on this article.
