#covariance

Analysis of Covariance PROC MIXED, PROC GLM, Chapter 15, SAS Essentials, Alan Elliott & Wayne Woodward LEARNING … source

Difference between Variance and Covariance

Variance is the measure of, the spread between numbers, in a given data set. In other words, it means, how far each number in the data set is, from the mean of this data set.

2. Covariance is the measure of, the directional relationship between, two random variables. In other words, covariance measures, how much, two random variables vary together. The word, Directional, means positive or…

Both Correlation and Covariance are very closely related to each other and yet they differ a lot.

When it comes to choosing between Covariance vs Correlation, the latter stands to be the first choice as it remains unaffected by the change in dimensions, location, and scale, and can also be used to make a comparison between two pairs of variables. Since it is limited to a range of -1 to +1, it is useful to draw comparisons between variables across domains. However, an important limitation is that both these concepts measure the only linear relationship.

original source : https://m.blog.naver.com/PostView.nhn?blogId=sw4r&logNo=221025662499&proxyReferer=https%3A%2F%2Fwww.google.com%2F

Suppose you have taken the data for heights and weights of students in class and you want to figure out the correlation between heights and weights of students. The relation between these two parameters is defined mathematically by one of the 3 ways 1) Covariance 2) Pearson Correlation Coefficient 3) Spearman's rank correlation coefficient Each of these metrics has its own pros and cons so let's dive deeper into them.   Covariance Covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values, (i.e., the variables tend to show similar behavior), the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other, (i.e., the variables tend to show the opposite behavior), the covariance is negative. The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables  

In case we want to the covariance of a variable with respect to itself, it is always zero.   A simple way to understand covariance is by using this graph as an example

In this we can see stock market returns increase as economic growth increases and vice versa, hence we can say these two are positively correlated. Further gasoline prices and world oil production decrease as the other increase and we can say they are negatively correlated. The reason why monotonically increasing seems to have positive covariance is because for any point they will be either above mean or below mean and hence make overall covariance +tive. Note 1) The magnitude of covariance has nothing to do with the amount of overlap. Let's say something has a covariance of 5 doesn't mean anything. In fact, even if we change the units of heights and weights from cms to feet, lbs to kgs the covariance for the same dataset will change. What if we standardize the datasets before applying covariance, that becomes correlation and that can tell how much the data is correlated. 2) However, if there are outliers in the dataset, we may have a situation where covariance is -time for monotonically increasing relation. Pearson correlation coefficient The Pearson correlation coefficient (PCC), also referred to as Pearson's r, the Pearson product-moment correlation coefficient (PPMCC) or the bivariate correlation, is a measure of the linear correlation between two variables X and Y. Owing to the Cauchy–Schwarz inequality it has a value between +1 and −1, where 1 is the total positive linear correlation, 0 is no linear correlation, and −1 is the total negative linear correlation

Suppose you have taken the data for heights and weights of students in class and you want to figure out the correlation between heights and weights of students. The relation between these two parameters is defined mathematically by one of the 3 ways 1) Covariance 2) Pearson Correlation Coefficient 3) Spearman's rank correlation coefficient Each of these metrics has its own pros and cons so let's dive deeper into them.   Covariance Covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values, (i.e., the variables tend to show similar behavior), the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other, (i.e., the variables tend to show the opposite behavior), the covariance is negative. The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables  

In case we want to the covariance of a variable with respect to itself, it is always zero.   A simple way to understand covariance is by using this graph as an example

In this we can see stock market returns increase as economic growth increases and vice versa, hence we can say these two are positively correlated. Further gasoline prices and world oil production decrease as the other increase and we can say they are negatively correlated. The reason why monotonically increasing seems to have positive covariance is because for any point they will be either above mean or below mean and hence make overall covariance +tive. Note 1) The magnitude of covariance has nothing to do with the amount of overlap. Let's say something has a covariance of 5 doesn't mean anything. In fact, even if we change the units of heights and weights from cms to feet, lbs to kgs the covariance for the same dataset will change. What if we standardize the datasets before applying covariance, that becomes correlation and that can tell how much the data is correlated. 2) However, if there are outliers in the dataset, we may have a situation where covariance is -time for monotonically increasing relation. Pearson correlation coefficient The Pearson correlation coefficient (PCC), also referred to as Pearson's r, the Pearson product-moment correlation coefficient (PPMCC) or the bivariate correlation, is a measure of the linear correlation between two variables X and Y. Owing to the Cauchy–Schwarz inequality it has a value between +1 and −1, where 1 is the total positive linear correlation, 0 is no linear correlation, and −1 is the total negative linear correlation

I couldn’t find a better explanation of this. So I had to blog it for posterity. The original article may be found here. 

For a background on the topic. See related article. (Different source, but simple explanation)

Covariance and Correlation are two mathematical concepts which are commonly used in the field of probability and statistics. Both concepts describe the relationship between…