Covariance Calculator between X and Y with examples

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:

Descriptive Statistics

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.

VRCBuzz co-founder and passionate about making every day the greatest day of life. Raju is nerd at heart with a background in Statistics. Raju looks after overseeing day to day operations as well as focusing on strategic planning and growth of VRCBuzz products and services. Raju has more than 25 years of experience in Teaching fields. He gain energy by helping people to reach their goal and motivate to align to their passion. Raju holds a Ph.D. degree in Statistics. Raju loves to spend his leisure time on reading and implementing AI and machine learning concepts using statistical models.

Leave a Comment