2024-08-03 16:48:57

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Concept Of Covariance

From an intuitive point of view, the covariance between 2 variables indicates the degree of linear inertia that exists in the relationship between the values of both, if such a relationship exists. It is essential to emphasize that such inertia is exclusively and utterly linear.

From a formal point of view, the covariance of variables X and Y is defined as the central moment of orders (1; 1) of the two-dimensional variable (X; Y), m 11, and is denoted by S xy:

In the previous expression, it can be seen that the covariance between X and Y is nothing but the weighted average of the products
. Note that this product will be positive when large values of X (greater than its mean) are accompanied by high values of Y (greater than its mean), or when small values of X (lower than its mean) are given together with small values of Y (less than its mean). The said product will be negative when values of X higher than its mean are accompanied by values of Y lower than its mean, or when values of X less than its mean appear together with values of Y greater than its own. Hence, the covariance shows the linear inertia existing in the relationship between the values of two variables.

What Does The Covariance Indicate?

1. The positive covariance- When one variable grows the other variable as well. They have a direct relationship.

2. The negative covariance- When one variable increases the other variable decreases. They have an inverse relationship.

Properties Of Covariance

1. The covariance is the central moment of order 1.1 of the two-dimensional distribution.

2. It is invariant to changes of origin in any of the two variables.

3. However, it depends on the unit changes. If the unit of measurement is changed in both variables, the covariance is modified proportionally to both changes:

u = a + bx v = c + dy S uv = bdS xy

4. The expression for calculating the covariance is

Where an 11 is the so-called mixed (ordinary) moment and its expression is:

If the observations are aggregated by frequencies, or:

If the observations are not aggregated by frequencies

5. If two variables are independent, their covariance is zero (the reciprocal result is not necessarily true).

6. The covariance measures the joint co-variation of two variables: If it is positive, it will give us the information that at high values of one of the variables there is a greater tendency to find high values of the other variable and low values of one of the variables, correspondingly low values. On the other hand, if the covariance is negative, the covariance of both variables will be in the opposite direction: high values will correspond to low values and high values. If the covariance is zero, there is no clear covariance in any of the However, the fact that the covariance depends on the measures of the variables does not allow comparisons between some cases and others.

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