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.

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.