What covariance isCovariance is a measure of LINEAR association (relationship) between two variables. That is why it is also called a measure of linear dependence: it tells us whether the two variables tend to rise together, move in opposite directions, or have no linear relationship.Independent variables have zero covariance. The converse, however, is not true: zero covariance only means the absence of a linear relationship. Two variables can have zero covariance and a strong curved relationship (for example, y = x² with x symmetric around zero).Calculating covariance for ungrouped dataImagine two data sets:X: 2, 4, 6, 8, 10Y: 10, 8, 6, 4, 2We can see right away that X and Y have a negative relationship: the larger X, the smaller Y.To calculate the covariance, each value of X must be linked to a value of Y — both were observed on the same individual, event or moment (here, the values in the same position of the lists form a pair). It is this link that lets us answer the question we care about: when X increases, does Y decrease, stay the same or increase? That is why we will work with the original pairs of data, and not with separate frequency distributions of each variable, which lose this link.The simplest way to understand covariance is to look at the deviations from the mean (both means are 6):Deviations of X: −4, −2, 0, 2, 4Deviations of Y: 4, 2, 0, −2, −4When one set has a negative deviation, the other has a positive one, and vice versa: this is what reveals the negative relationship. Covariance turns this observation into a number.Covariance is the mean of the products of the deviations: the deviation of each value of X multiplied by the deviation of the corresponding value of Y.As a mathematical formula:\[ cov(X, Y) = \frac{\sum{(x - \bar{x})(y - \bar{y})}}{n} \]This is the population covariance. When the data are a sample and we want to estimate the population covariance, we divide by n − 1 instead of n — exactly as with the variance. In this article we will divide by n.For the previous example:XYDeviations of XDeviations of YProduct of deviations210−44−1648−22−466000842−2−41024−4−16\[ cov(X, Y) = \frac{-40}{5} = -8 \]The covariance is negative and equals −8. The number 8 on its own does not say much: it depends on the units of the variables (if X were measured in centimetres instead of metres, the covariance would be 100 times larger). What matters here is the sign: when one variable increases, the other decreases. To measure the strength of the relationship, we will use correlation, in the next article.The logic of the sign becomes clear when we look at the possible combinations:When both deviations are negative, the product is positive: if the two variables tend to be below their means together, the covariance tends to be positive.When both deviations are positive, the product is also positive: if the two variables tend to be above their means together, the covariance tends to be positive.When the deviations have opposite signs, the product is negative: if one variable tends to be above its mean when the other is below, the covariance tends to be negative.For five pairs of data, the products of the deviations were 6, −2, 0, 3 and 8. What is the covariance (dividing by n) and what does it indicate?Answer: 3, a positive relationship — the sum of the products is 6 − 2 + 0 + 3 + 8 = 15, and 15 / 5 = 3. The positive sign indicates that the variables tend to be above (or below) their means together. The option 15 forgets to divide by n.Another formula for calculating covarianceCovariance can also be calculated another way. Starting from its definition as the expected value (mean) of the product of the deviations:\[ cov(X, Y) = \frac{\sum{(x - \bar{x})(y - \bar{y})}}{n} \]\[ cov(X, Y) = E[(x - \bar{x})(y - \bar{y})] \]\[ cov(X, Y) = E[x · y - \bar{x} · y - \bar{y} · x + \bar{x} · \bar{y}] \]\[ cov(X, Y) = E[x · y] - E[\bar{x} · y] - E[\bar{y} · x] + E[\bar{x} · \bar{y}] \]\[ cov(X, Y) = E[x · y] - \bar{x} · E[y] - \bar{y} · E[x] + \bar{x} · \bar{y} \]\[ cov(X, Y) = E[x · y] - \bar{x} · \bar{y} - \bar{y} · \bar{x} + \bar{x} · \bar{y} \]\[ cov(X, Y) = E[x · y] - \bar{x} · \bar{y} \]\[ cov(X, Y) = \frac{\sum{x · y}}{n}- \bar{x} · \bar{y} \]In words: covariance is also the mean of the products x · y minus the product of the means.Examples1st exampleXYX · Y2102048326636843210220\[ cov(X, Y) = \frac{\sum{x · y}}{n}- \bar{x} · \bar{y} = \frac{140}{5} - 6 · 6 = 28 - 36 = -8 \]The same result we got with the deviations.2nd exampleHere \( \bar{x} = 4 \) and \( \bar{y} = 5 \):XYDeviations of XDeviations of YProduct of 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:\[ cov(X, Y) = \frac{\sum{x · y}}{n}- \bar{x} · \bar{y} = \frac{95}{5} - 4 · 5 = 19 - 20 = -1 \]For 4 pairs of data, the sum of the products x · y is 120, the mean of X is 3 and the mean of Y is 8. What is the covariance?Answer: 6 — cov = 120 / 4 − 3 · 8 = 30 − 24 = 6. The option 30 stops at the first term and forgets to subtract the product of the means.Properties of covariance1) The covariance of a variable with itself is its variance\[ cov(X, X) = var(X) \]2) The order of the variables does not matter\[ cov(X, Y) = cov(Y, X) \]3) Adding constants to the variables does not change the covariance\[ cov(X + a, Y + b) = cov(X, Y) \]Adding a constant shifts the mean by the same amount, so the deviations stay the same.4) Multiplying the variables by constants multiplies the covariance\[ cov(a · X, b · Y) = a · b · cov(X, Y) \]The covariance between the price of a product in dollars (X) and the units sold (Y) is −40. If the price is converted to cents (X multiplied by 100) and every day's sales increase by 5 units, what is the new covariance?Answer: −4000 — multiplying X by 100 multiplies the covariance by 100, and adding 5 to Y changes nothing: −40 · 100 = −4000. The option −4005 treats adding a constant as if it affected the covariance.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 has two data sets, X and Y, whose covariance is 10, and he decided to transform X only. First he subtracted the mean from every value of X, so that the mean became zero. Right after that he added 50 to every value, so that the mean became 50. Finally, he multiplied every value of X by 100. What is the covariance between X and Y now?5) Calculate the covariance of the following data sets (prices of two stocks over seven days):BradescoCanon9.7735.049.6234.939.4634.799.4135.079.4334.349.4534.259.3634.35Answer keyAnswers dividing by n (population covariance):1) 232 (with n − 1: 290);2) −60 (with n − 1: −75);3) −0.04 (with n − 1: −0.05);4) 1000 — subtracting the mean and adding 50 do not change the covariance; multiplying X by 100 multiplies it by 100: 10 · 100 = 1000;5) 0.0256 (with n − 1: 0.0299).
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