Empirical covariance simply explained

What you need:

• statistical variables
• arithmetic mean
• Readings
• sample

Understand the statement of covariance

The empirical covariance is a non-standardized measure that describes the linear relationship between two statistical variables. You usually have a sample (x_i, y_i) given.

• The covariance is defined relatively clearly. First you need the means of the readings x_i and determine their deviation from the arithmetic mean. Proceed in the same way with the measured values ​​y_i. Now multiply these deviations of the measured values ​​from the respective arithmetic mean and add them up over i. In the end, you divide this value by n, that is, by the sample size.
• You can now interpret the covariance as follows. If the covariance is positive, then X and Y tend to have a correlation in the same direction, i. H. hits an x
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  _i for a certain i strongly upwards, then the y beats out_i also upwards. The greater the covariance, the stronger this relationship.
• If the covariance values ​​are negative, there is a tendency in the opposite direction. At 0 there is no connection at all.

Example of empirical covariance

• Suppose you have the sample (x_i, y_i) given. In this simple case i = 3 and the values ​​x₁ = 2, x₂ = 2.2, x₃ = 6,3. Likewise, you have the values ​​of y₁ = 1.1, y₂ = 1.9 and y₃ = 4.5 given.


Calculate empirical covariance

In statistics, you need empirical covariance in some places. But what …

• You can now determine the arithmetic mean by x = (2 + 2.2 + 6.3) / 3 = 3.5 and y = (1.1 + 1.9 + 4.5) / 3 = 2.5.
• You can calculate the empirical covariance as ((2-3.5) (1.1-2.5) + (2.2-3.5) (1.9-2.5) + (6.3-3, 5) (4.5-2.5)) / 3 = (2.1 + 0.78 + 5.6) / 3 = 8.48 / 3 = 2.82 (...).
• The variance is therefore relatively strongly positive, i.e. H. the linear relationship between the measured values ​​tends to be large. You can already see from the values ​​that they move in the same direction and a deflection of x₃ upwards also a deflection of y₃ follows.

As you can see, in this simple example the empirical covariance is explained very simply. These considerations are used in the design of equity portfolios that should both offer a relatively high return and promise a relatively low risk.