What is covariance? - Minitab
Covariance and Correlation are two mathematical concepts which are quite commonly used in business statistics. Both of these two determine. Covariance measures the linear relationship between two variables. The covariance is similar to the correlation between two variables, however, they differ in. The difference in terms of covariance and correlation, when looking at the of analogies and heuristics I suspect would obsfucate two relatively.
So, at first instead of completely ignoring the formula and speaking of it in some magical and heuristic types of analogies, lets just look at the formula and try to explain the individual components in small steps. The difference in terms of covariance and correlation, when looking at the formulas, should become clear.
Whereas speaking in terms of analogies and heuristics I suspect would obsfucate two relatively simple concepts and their differences in many situations.
What is covariance?
At this point, I might introduce a simple example, to put a face on the elements and operations so to speak. One would likely make these examples more specific e. One can then just take this process one operation at a time. Hence when an observation is further from the mean, this operation will be given a higher value. As gung points out in the comments, this is frequently called the cross product perhaps a useful example to bring back up if one were introducing basic matrix algebra for statistics.
Take note of what happens when multiplying, if two observations are both a large distance above the mean, the resulting observation will have an even larger positive value the same is true if both observations are a large distance below the mean, as multiplying two negatives equals a positive. Also note that if one observation is high above the mean and the other is well below the mean, the resulting value will be large in absolute terms and negative as a positive times a negative equals a negative number.
Finally note that when a value is very near the mean for either observation, multiplying the two values will result in a small number. Again we can just present this operation in a table. We can see all the seperate elements of what a covariance is, and how it is calculated come into play.
Now, the covariance in and of itself does not tell us much it can, but it is needless at this point to go into any interesting examples without resorting to magically, undefined references to the audience.
Let us understand the correlation matrix and covariance matrix
In a good case scenario, you won't really need to sell why we should care what the covariance is, in other circumstances, you may just have to hope your audience is captive and will take your word for it. But, continuing on to develop the difference between what the covariance is and what the correlation is, we can just refer back to the formula for correlation. Statistical Sampling and Regression: Convariance and Correlation Covariance and correlation describe how two variables are related.
Variables are positively related if they move in the same direction. Variables are inversely related if they move in opposite directions. Both covariance and correlation indicate whether variables are positively or inversely related.
Correlation also tells you the degree to which the variables tend to move together. You are probably already familiar with statements about covariance and correlation that appear in the news almost daily.
For example, you might hear that as economic growth increases, stock market returns tend to increase as well. These variables are said to be positively related because they move in the same direction. You may also hear that as world oil production increases, gasoline prices fall. These variables are said to be negatively, or inversely, related because they move in opposite directions.
The relationship between two variables can be illustrated in a graph.
Difference Between Covariance and Correlation
In the examples below, the graph on the left illustrates how the positive relationship between economic growth and market returns might appear.
The graph indicates that as economic growth increases, stock market returns also increase.
The graph on the right is an example of how the inverse relationship between oil production and gasoline prices might appear. It illustrates that as oil production increases, gas prices fall. To determine the actual relationships of these variables, you would use the formulas for covariance and correlation.
Covariance Covariance indicates how two variables are related. A positive covariance means the variables are positively related, while a negative covariance means the variables are inversely related. The formula for calculating covariance of sample data is shown below.
Before you compute the covariance, calculate the mean of x and y. The Summary Measures topic of the Discrete Probability Distributions section explains the mean formula in detail.
Now you can identify the variables for the covariance formula as follows. Since the covariance is positive, the variables are positively related—they move together in the same direction. Correlation Correlation is another way to determine how two variables are related. In addition to telling you whether variables are positively or inversely related, correlation also tells you the degree to which the variables tend to move together.
As stated above, covariance measures variables that have different units of measurement.
Using covariance, you could determine whether units were increasing or decreasing, but it was impossible to measure the degree to which the variables moved together because covariance does not use one standard unit of measurement.