Covariance Calculator

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Covariance(X,Y) --
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Sum(Y) --
Covariance(X,Y) SUM(xi - xmean)*(yi - ymean)/(samplesize -1)

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Covariance calculator can be used to calculate the relationship between the two commonly described sets of variables X and Y. Hence, It allows us to understand the relation between two sets of data.

Apart from calculating covariance, it also calculates the mean value for a given data set. In this post, we will discuss covariance, the formula for covariance, how to find covariance with examples, and much more.

What is Covariance?

Covariance measures how many random variables (X, Y) differ in one population. When there are higher dimensions or random variables in the population, a matrix represents the relationship among the various dimensions. By defining the relationship as the relationship between increasing two random variables in the entire dimension, the covariance matrix may be simpler to understand. The smaller X values and greater Y values give a positive covariance ranking, while the greater X values and the smaller Y values give a negative covariance. When all random variables are not statistically dependent, the covariance would be negative or non-linear. These are all covariance properties.

\(X < Y \Rightarrow + ve \ covarinace\)

\(X > Y \Rightarrow -ve \ covarinace\)

Covariance may be used to quantify variables that do not have the same units of measurement. By using covariance, we can determine whether units increase or decrease. The degree to which the variables shift together cannot be consolidated. The reason behind this is: there are several measurement units used for covariance.

Covariance Formula

There are different formulas for sample and population covariance. The x and y samples both have n random values X and Y, respectively. The elements of the first sample are represented by x1, x2,..., xn, xmean refers to the mean value of these elements of the sample. On the other hand, the elements of the second sample are denoted by are y1, y2, ..., yn, and mean of these values are represented by ymean.

If two sample sizes are available, then the following covariance equation is the sample covariance formula Cov(x,y).

\(Cov_{sam}(x, y) = \dfrac{sum (x_i - x_{mean}) (y_i - y_{mean})}{n}\)

The summation proceeds to the last value of n. In this equation:

\(n\) refers to the size of the sample for both X and Y

\(x_i - x_{mean}\) refers to the difference between sample elements for X and the mean value of the sample.

\(y_i - y_{mean}\) represents the difference between sample elements for and the mean value of the sample.

How to Calculate Covariance?

We will calculate covariance using an example so that you can understand the concept completely.

Hubert is a businessman who likes to acquire running businesses if he sees an opportunity. He had invested in an oil company Green Petro recently. For the sake of diversification, he needs to invest in a few more companies. He wants to buy shares of one more company i.e., Golden Oil and Super Oil. He doesn't know which company he should go for.

It can be decided by calculating the covariance for both companies.

For stocks of the Green Petro and Golden Oil, Hubert arbitrarily picks five closing rates. Green Petro represents xi, and Golden Oil represents yi.







diff × y diff




































Mean value





Follow the below steps to calculate covariance:

Step 1: Calculate the mean value for xi by adding all values and dividing them by sample size, which is 5 in this case. \(x_{mean}= 10.81\)

Step 2: Calculate the mean value for yi by adding all values and dividing them by sample size. \(Y_{mean}= 8.718\)

Step 3: Now, calculate the diffIt can be calculated by subtracting each element of x from the mean value of x.

\(x_{diff} = x_i - x_{mean}\)

Use the above equation to find differences for all x values and place them in a column like in the above table.

Step 4: Do the same for y, calculate ydiff by subtracting all values of y from the mean value of y.

\(y_{diff} = y_i - y_{mean}\)

Step 5: Multiply all values of xdiff and ydiff and place them in a new column.

Step 6: Add the last column values, which are the product of the two differences. Divide by the sample size, which is 5, after adding the values. The value after dividing by sample size is covariance, which is -3.90 in this case.

We can assume that both companies' closing prices vary with this measured covariance value -3.90. The covariance for Green Petro and Super Oil can also be calculated by applying the same process, and then Hubert can easily decide which company he should go for.

How to use Covariance Calculator?

Covariance can be calculated manually, and we will explain the complete process in the next sections. To be honest, manual covariance calculation is a bit trickier to carry out. That's where our sample covariance calculator comes in handy. It makes the calculation very simple by just taking the values from the user. To calculate covariance using this calculator, follow the below steps:

  • Enter the data set for the X variable by separating values with a comma in the given input box.
  • Enter the data set for the Y variable in the next input box and separate values using a comma.
  • Press the Calculate button to see the result.

It will not only give you covariance for input values but also a complete breakdown of the whole process. It will show the sum of X, the sum of Y, X mean, Y mean, covariance, and the whole calculation based on the covariance equation. You can use this calculator to solve your statistics problems and complete your assignments efficiently. Let's discuss the covariance definition.

Population Covariance

We don't normally have access to the whole population data. We have only limited access to the sample sizes. Nevertheless, these tests can provide an evaluation of population covariance for random variables X and Y. Using the below formula, population covariance can be calculated with the sample values:

\(Cov_{pop}(X, Y) = \dfrac{sum (x_i - x_{mean}) (y_i - y_{mean})}{(n-1)}\)

The value n-1 gives a marginally improved value than the sample covariance formula. It is normal for one to conclude that since small samples are not actually the full variance between whole populations, the denominator n-1 is the best correction factor.

How Sample and Population Covariance Relate?

The connection between population and sample covariance can be defined as the following equation.

\(Cov_{pop} (X, Y) = \Big(\dfrac{n}{n-1}\Big) \times Cov_{sam} (X, Y)\)

But note that as the sample size increases, the gap between n and n-1 will be less. Therefore, comparable results are provided for large samples by the population covariance and the sample covariance formula.

Covariance vs. Correlation

Covariance is a function that calculates the difference of X to Y, which are two random variables, while correlation is another way of expressing the difference between two random variables X and Y. The relation between correlation and covariance can be written as:

\(Corr (X, Y) = \dfrac{Cov (X, Y)}{\sigma_x x \sigma_y}\)

In this equation:

 \(\sigma_X\) refers to the standard deviation of X, and

 \(\sigma_Y\) refers to the standard deviation of Y.

Correlation can be treated as a stable covariance form. The correlation, according to this formula, should be between 1 and -1. That is why correlation is more commonly used than covariance, although they do the same work. Covariance is also directly related to variance. You can calculate variance using our variance calculator.


What Does Covariance Tell Us?

Covariance tells us the degree of variation between two variables. It tells us how much a variable differs from another variable. In plain language, it calculates how two variables relate to each other monotonically.

What Does a Negative Covariance Mean?

A negative covariance means that if the value of one variable rises, the other variable falls, or if one variable drops, the other increases. Variables are considered to be inversely related if the covariance is negative.

For example, if the temperature decreases, the use of heaters increases. The covariance between temperature and heater is negative.

What Does a Covariance of 0 Mean?

The covariance will be zero if the two random variables are not dependent on each other. Nevertheless, a zero covariance does not imply the independence of the variables. There can still be a non-linear relationship, resulting in a covariance value of zero.

What Does a Covariance of 1 Mean?

The covariance 1 means that the two variables under observation are directly related to each other. It means if one variable goes up, others will go up too, and if one variable decreases in value, others will too.

What is a High Covariance?

A high covariance implies that the relationship between the two variables is strong. The higher the covariance, the stronger relationship between both variables. It is the opposite in the case of low covariance. A low covariance depicts the weaker relationship between two variables.

Can Covariance Be Negative?

Yes, covariance can be negative in a case where two variables are inversely related.

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