Calculating the correlation between two variables
The performance of personal and business investments is measured as a percentage, return on investment. What type of variable is return on investment?
To calculate the correlation between two variables, you can follow these steps:
1. Gather the data: Collect a set of paired observations for the two variables of interest. For example, if you are interested in the relationship between the height and weight of individuals, you would need measurements of both height and weight for each person.
2. Calculate the mean: Find the mean (average) value for each variable. This involves summing all the values of a variable and dividing by the total number of observations. Let's call these means X̄ (pronounced "X bar") and Ȳ (pronounced "Y bar") for the two variables, respectively.
3. Calculate the difference from the mean: For each observation, subtract the mean of its respective variable. Let's call these differences dX and dY.
4. Square the differences: Square each difference calculated in step 3. These squared differences are denoted as dX^2 and dY^2.
5. Calculate the product of the differences: Multiply the difference for each observation from step 3 for both variables. Let's call the result dXdY.
6. Sum the values: Sum all the values of dXdY calculated in step 5. This sum is represented as ∑dXdY.
7. Calculate the correlation coefficient: Divide ∑dXdY by the square root of the product of the sums of squared differences for each variable (i.e., √(∑dX^2 * ∑dY^2)). This gives you the correlation coefficient, denoted by r.
8. Interpret the correlation coefficient: The resulting correlation coefficient, r, ranges from -1 to +1. A positive value indicates a positive relationship between the two variables, meaning that as the values of one variable increase, the values of the other variable also tend to increase. A negative value indicates a negative relationship, where as one variable increases, the other tends to decrease. A value of 0 implies no relationship or correlation between the variables.
Note: There are various methods for calculating the correlation coefficient, such as Pearson's correlation coefficient, Spearman's rank correlation coefficient, and Kendall's tau coefficient. The steps outlined above correspond to the calculation of Pearson's correlation coefficient, which is commonly used for continuous variables.