IntroductionAs with covariance, when two variables are linked by a statistical relationship, we say there is a correlation between them. The difference is that correlation measures not only the direction of the relationship but also its strength, on a scale that does not depend on the units of the data.Functional and statistical relationshipsAs we know, the perimeter and the side of a square are related. The relationship between them is perfectly defined and can be expressed by a mathematical sentence:\[ 2p = 4l \]Where 2p is the perimeter and l is the side.Given any value of l, we can determine the value of 2p exactly.Now consider the relationship between the weight and the height of a group of people. This relationship is not of the same kind as the previous one: it is much less precise. People of different heights may have the same weight, and people of the same height may have different weights. Even so, on average, the taller the person, the heavier.Relationships like perimeter–side are called functional relationships, and those like weight–height are called statistical relationships. Correlation studies statistical relationships.Scatter diagramConsider a random sample of ten of the 98 students in a class at college A, with the grades they obtained in Mathematics and in Statistics:Mathematics gradeStatistics grade5.06.08.09.07.08.010.010.06.05.07.07.09.08.03.04.08.06.02.02.0Plotting the ordered pairs \( (x_i, y_i) \) on Cartesian axes, we get a cloud of points called a scatter diagram. This diagram gives a rough but useful idea of the correlation — which, in this case, exists and is positive.What linear correlation isSince the points in this example are spread around an upward-sloping line, the correlation is called a positive linear correlation.So a correlation is:positive linear, if the points of the diagram are spread around an upward-sloping line;negative linear, if the points are spread around a downward-sloping line;non-linear, if the points are spread around a curve.If the points are scattered without forming any definite pattern, we conclude that there is no relationship between the variables under study.We then have:In a scatter diagram, the points rise and then fall, forming an arch. What type of correlation does it show?Answer: Non-linear — the points follow a curve, not a straight line. Note: Pearson's coefficient for such a data set can be close to zero even though there is a strong relationship between the variables.How to calculate the linear correlation coefficientThe tool used to measure linear correlation is the correlation coefficient. It should indicate the strength of the correlation between two variables and also its direction (positive or negative).We will use Pearson's correlation coefficient, given by:\[ r = \dfrac{\sigma_{x,y}}{\sigma_x · \sigma_y} \]The correlation coefficient is the covariance divided by the product of the standard deviations of the two variables.An equivalent formula is:\[ r = \frac{\sum{(x - \bar{x})(y - \bar{y})}}{\sqrt{\sum{(x - \bar{x})^{2}}} · \sqrt{\sum{(y - \bar{y})^{2}}}} \]In it the n cancels out, so it makes no difference whether the covariance and standard deviations use divisor n or n − 1 — as long as the same divisor is used above and below.Unlike covariance, of which we only use the sign, the correlation coefficient gives much more information, because it always lies between −1 and +1, that is, it belongs to the interval [−1, +1].So:if the correlation between two variables is perfect and positive, r = +1;if the correlation is perfect and negative, r = −1;if there is no linear correlation between the variables, r = 0.To interpret intermediate values, a common rule of thumb (there are others, and the cut-offs vary from field to field) is the following, looking at the value of r without its sign:Value of r (+ or −)Interpretation0 to 0.19Very weak correlation0.20 to 0.39Weak correlation0.40 to 0.59Moderate correlation0.60 to 0.79Strong correlation0.80 to 1Very strong correlationIt is also common to write r as a percentage (r = 0.41 = 41%), as we will do in the answer key.Properties of correlation1. The correlation coefficient does not depend on the units of measurement of the variables: it is a dimensionless number between −1 and +1, that is, −1 ≤ r ≤ +1.2. The correlation coefficient of a variable with itself is equal to +1.3. Swapping the order of the variables does not change the correlation coefficient, that is, rXY = rYX.4. Adding or subtracting a constant to one or both variables does not change the correlation coefficient.5. Multiplying or dividing one or both variables by a positive constant does not change the correlation coefficient. If the constant is negative, the value stays the same but the sign is reversed.The correlation between temperature in degrees Celsius and ice-cream sales is 0.8. If temperature is converted to Fahrenheit (F = 1.8 · C + 32), what will the correlation be?Answer: 0.8 — multiplying by a positive constant (1.8) and adding a constant (32) do not change r, which does not depend on units. Covariance, on the other hand, would be changed by the multiplication by 1.8.ExamplesCalculating the correlation coefficientLet us calculate the correlation coefficient for the values of the variables x and y:xy1241068812101412The covariance of the data set is 2.4 (with divisor n). The standard deviation of x is 2.04 and that of y is 2.83.So:\[ r = \frac{2.4}{2.04 · 2.83} = 0.42 = 42\% = \text{moderate correlation} \]The Mathematics and Statistics gradesFor the ten students in the scatter diagram, the calculations give \( \sum{(x - \bar{x})(y - \bar{y})} = 50.5 \), \( \sum{(x - \bar{x})^{2}} = 58.5 \) and \( \sum{(y - \bar{y})^{2}} = 52.5 \). So:\[ r = \frac{50.5}{\sqrt{58.5} · \sqrt{52.5}} = \frac{50.5}{55.42} = 0.91 \]A very strong positive correlation, as the diagram already suggested.Two variables have covariance 12; the standard deviation of the first is 3 and that of the second is 5. What is the correlation coefficient and how is it classified?Answer: 0.8, very strong correlation — r = 12 / (3 · 5) = 12 / 15 = 0.8, which falls in the 0.80 to 1 band. The option 4 divides the covariance by only one standard deviation; an r greater than 1 is always a sign of a calculation error.Exercises1) Draw the scatter diagrams that may be found for the following correlations: perfect positive correlation, negative correlation and non-linear correlation.2) Draw the scatter diagram of the following variables:xy50106020801005025What type of correlation do these variables show?3) What is the correlation coefficient of the variables in the previous exercise?4) What is the correlation coefficient of the following variables:xy102030515155505) What is the correlation coefficient of the following variables:xy1013021540536) Davi found that the covariance of two data sets was 10, that the standard deviation of one set is 2 and that of the other is 5. What is the correlation coefficient of the two sets?7) Once again, Davi found that the covariance of two data sets was 200, that the standard deviation of one set is 20 and that of the other is 40. After this analysis, however, he had to modify one of the sets by dividing all of its values by 10,000. What is the correlation coefficient of the two sets now?Answer key2) Positive linear correlation3) 94%4) −84%5) 0% (r ≈ −0.008: practically no linear correlation)6) 100%7) 25% — dividing one of the sets by a positive constant does not change r = 200 / (20 · 40) = 0.25.
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