What Is CovarianceCovariance is a measure of LINEAR association (relationship) between two random variables. Thus, independent variables have zero covariance.Covariance is sometimes called a measure of linear dependence between the two random variables.Calculating covariance for ungrouped dataImagine two data sets:X: 2, 4, 6, 8, 10Y: 10, 8, 6, 4, 2It is immediately clear that x and y have a negative correlation: the more we have of X, the less we have of Y.In this case, the elements need to be linked to an event or time (for this case, those in the same column are occurring simultaneously). It is necessary to know that x is linked to y, because we specifically want to know whether when x increases, y will decrease, remain the same, or increase! Due to this characteristic, the only way to calculate covariance is with ungrouped data; we will no longer use frequency distributions or classes.The simplest way to understand covariance is by looking at the deviations:deviations of x: -4, -2, 0, 2, 4deviations of y: 4, 2, 0, -2, -4It can be seen that when one set has a negative deviation, the other has a positive deviation and vice versa; this shows the negative correlation. But, for now, we want to study covariance, from which correlation is studied.> Covariance is the average of the following calculation: deviations of set A multiplied by the deviations of set BIn mathematical formula:\[ cov(X, Y) = \frac{\sum{(x - \bar{x})(y - \bar{y})}}{n} \]For the case of the previous example:XYdeviations of Xdeviations of YMultiplication of the deviations210-44-1648-22-466000842-2-41024-4-16\[ cov(X, Y) = \frac{-40}{5} = -8 \]The covariance in this case is negative and equals -8. The number 8 does not tell us much in this case, but the fact that it is negative does; this indicates that when one variable increases, the other decreases.Covariance makes a lot of sense in practice when we stop to think about the following alternatives we have:- When both deviations are negative, their multiplication will be positive, so the covariance tends to be positive. In other words, when the two variables tend to fall together, the covariance tends to be positive.- When both deviations are positive, their multiplication will be positive, so the covariance tends to be positive. In other words, when the two variables tend to rise together, the covariance tends to be positive.- When the two deviations have opposite signs, the multiplication will be negative, so the covariance tends to be negative. In other words, when the two variables do not behave the same way, the covariance tends to be negative.Other formulas for calculating covarianceIt is also possible to calculate covariance in another way:\[ cov(X, Y) = \frac{\sum{(x - \bar{x})(y - \bar{y})}}{n} \]\[ cov(X, Y) = E[(x - \bar{x})(y - \bar{y})] \]\[ cov(X, Y) = E[x · y - \bar{x} · y - \bar{y} · x + \bar{x} · \bar{y}] \]\[ cov(X, Y) = E[x · y] - E[\bar{x} · y] - E[\bar{y} · x] + E[\bar{x} · \bar{y}] \]\[ cov(X, Y) = E[x · y] - \bar{x} · E[y] - \bar{y} · E[x] + \bar{x} · \bar{y} \]\[ cov(X, Y) = E[x · y] - \bar{x} · \bar{y} - \bar{y} · \bar{x} + \bar{x} · \bar{y} \]\[ cov(X, Y) = E[x · y] - \bar{x} · \bar{y} \]\[ cov(X, Y) = \frac{\sum{x · y}}{n}- \bar{x} · \bar{y} \]In other words: covariance can also be expressed as the average of the multiplication of each element of X by Y minus their multiplied means.Examples1st exampleXYX*Y2102048326636843210220\[ cov(X, Y) = \frac{\sum{x · y}}{n}- \bar{x} · \bar{y} = \frac{140}{5} - 6 · 6 = 28 - 36 = -8 \]2nd exampleXYdeviations of Xdeviations of YMultiplication of the deviationsX*Y25-2001026-21-212632-2-4185914445521-3-310\[ cov(X, Y) = \frac{\sum{(x - \bar{x})(y - \bar{y})}}{n} = \frac{-5}{5} = -1 \]Or:\[ cov(X, Y) = \frac{\sum{x · y}}{n}- \bar{x} · \bar{y} = \frac{95}{5} - 4 · 5 = 19 - 20 = -1 \]Properties of covariance1) The covariance of two identical sets is the variance of the set:\[ cov(X, X) = var(X) \]2) The covariance of A and B is the same as the covariance of B and A:\[ cov(X, Y) = cov(Y, X) \]3) Adding any elements to variables A and B does not change the covariance:\[ cov(X + a, Y + b) = cov(Y, X) \]4) Multiplying any element by variables A and B multiplies the covariance:\[ cov(X · a, Y · b) = a · b · cov(Y, X) \]Exercises1) Calculate the covariance of the following data sets:XY51010201530255035702) Calculate the covariance of the following data sets:XY254015303520104015503) Calculate the covariance of the following data sets:XY92835363424) Pedrinho saw a data set and decided to make some transformations. First, he subtracted the mean from all the data, to have a mean of zero. Right after that, he decided to add 50 to all the data so the mean would be 50. Then he multiplied all the data by 100. At the beginning, the covariance of the set had a value of 10; how much is it worth now?5) Calculate the covariance of the following data sets:BradescoCanon9.7735.049.6234.939.4634.799.4135.079.4334.349.4534.259.3634.35Answer Key1) 232;2) -60;3) -0.04;4) 1000;5) 0.0256
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