How would you explain the difference between correlation and covariance? - Cross Validated
Covariance measures the linear relationship between two variables. The covariance is similar to the correlation between two variables, however, they differ in the following ways: Thus, a perfect linear relationship results in a coefficient of 1. In probability theory and statistics, the mathematical concepts of covariance and correlation are very similar. Both describe the degree to which two random variables or sets of random variables tend to deviate from their expected values in similar ways. cross-covariance and cross-correlation are functions of the time difference. Covariance can tell how the stocks move together, but to determine the strength of the relationship, we need to look at their correlation.
Variance is the difference of squaring out Random Variable at different points when we calculate Expectation. Squaring before calculating Expectation and after calculating Expectation yield very different results! The difference between these results is the Variance. What is really interesting is the only time these answers are the same is if the Sampler only outputs the same value each time, which of course intuitively corresponds to the idea of there being no Variance.
The greater the actual variation in the values coming from the Random Variable is the greater the different between the two values used to calculate Variance will be.
At this point we have a very strong, and very general sense of how we can measure Variance that doesn't rely on any assumptions our intuition may have about the behavior of the Random Variable.
Covariance - measuring the Variance between two variables Mathematically squaring something and multiplying something by itself are the same. Because of this we can rewrite our Variance equation as: But now we can ask the question "What if one of the Xs where another Random Variable?
Covariance and correlation
If Variance is a measure of how a Random Variable varies with itself then Covariance is the measure of how one variable varies with another. Correlation - normalizing the Covariance Covariance is a great tool for describing the variance between two Random Variables. But this new measure we have come up with is only really useful when talking about these variables in isolation.
Correlation between different Random Variables produce by the same event sequence The only real difference between the 3 Random Variables is just a constant multiplied against their output, but we get very different Covariance between any pairs.
The problem is that we are no longer accounting for the Variance of each individual Random Variable. The way we can solve this is to add a normalizing term that takes this into account. Putting everything we've found together we arrive at the definition of Correlation: The short answer is The Cauchy-Schwarz inequality.
What is covariance?
Exploring the relationship between Correlation and the Cauchy-Schwarz inequality deserves its own post to really develop the intuition. 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.
Understanding Variance, Covariance, and Correlation — Count Bayesie
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.
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.
Relationship between correlation and sample variance - Cross Validated
If you need to define the variance itself, you could just say that the variance is the same thing as the covariance of a series with itself i. And all the same concepts that you introduced with the covariance apply i. Maybe note here that a series can not have a negative variance as well which should logically follow from the math previously presented. So we are dividing the covariance we just calculated by the product of the variances of each series.
So again, I'm a hypocrite and resort to some, take my word for it, but at this point we can introduce all the reasons why we use the correlation coefficient. One can then relate these math lessons back to the heuristics that have been given in the other statements, such as Peter Flom's response to one of the other questions.
While this was critisized for introducing the concept in terms of causal statements, that lesson should be on the agenda at some point as well. I understand in some circumstances this level of treatment would not be appropriate. The senate needs the executive summary.