# How to Create a Covariance Matrix in R

Covariance is a measure of how changes in one variable are associated with changes in a second variable. Specifically, it’s a measure of the degree to which two variables are linearly associated.

A covariance matrix is a square matrix that shows the covariance between many different variables. This can be a useful way to understand how different variables are related in a dataset.

The following example shows how to create a covariance matrix in R.

### How to Create a Covariance Matrix in R

Use the following steps to create a covariance matrix in R.

Step 1: Create the data frame.

First, we’ll create a data frame that contains the test scores of 10 different students for three subjects: math, science, and history.

```#create data frame
data <- data.frame(math = c(84, 82, 81, 89, 73, 94, 92, 70, 88, 95),
science = c(85, 82, 72, 77, 75, 89, 95, 84, 77, 94),
history = c(97, 94, 93, 95, 88, 82, 78, 84, 69, 78))

#view data frame
data

math science history
1    84      85      97
2    82      82      94
3    81      72      93
4    89      77      95
5    73      75      88
6    94      89      82
7    92      95      78
8    70      84      84
9    88      77      69
10   95      94      78
```

Step 2: Create the covariance matrix.

Next, we’ll create the covariance matrix for this dataset using the cov() function:

```#create covariance matrix
cov(data)

math   science   history
math     72.17778  36.88889 -27.15556
science  36.88889  62.66667 -26.77778
history -27.15556 -26.77778  83.95556
```

Step 3: Interpret the covariance matrix.

The values along the diagonals of the matrix are simply the variances of each subject. For example:

• The variance of the math scores is 72.18
• The variance of the science scores is 62.67
• The variance of the history scores is 83.96

The other values in the matrix represent the covariances between the various subjects. For example:

• The covariance between the math and science scores is 36.89
• The covariance between the math and history scores is -27.16
• The covariance between the science and history scores is -26.78

positive number for covariance indicates that two variables tend to increase or decrease in tandem. For example, math and science have a positive covariance (36.89), which indicates that students who score high on math also tend to score high on science. Conversely, students who score low on math also tend to score low on science.

negative number for covariance indicates that as one variable increases, a second variable tends to decrease. For example, math and history have a negative covariance (-27.16), which indicates that students who score high on math tend to score low on history. Conversely, students who score low on math tend to score high on history.

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