Example 3.3 Variability of Elevator Rail Diameters at Otis Elevator Measures of Variability: Variance and Standard Deviation

| 3.2 | 3.4 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Objective To calculate the variability for two suppliers and choose the one with the least variability.

| 3.2 | 3.4 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | n Suppose that Otis Elevator is going to stop manufacturing elevator rails. Instead, it is going to buy them from an outside supplier. n Otis would like each rail to have a diameter of 1 inch. n The company has obtained samples of ten elevator rails from each supplier. They are listed in columns A and B of this Excel file. OTIS4.XLS

| 3.2 | 3.4 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Which should Otis prefer? n Observe that the mean, median, and mode are all exactly 1 inch for each of the two suppliers. n Based on these measures, the two suppliers are equally good and right on the mark. However, we when we consider measures of variability, supplier 1 is somewhat better than supplier 2. Why?

| 3.2 | 3.4 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Explanation n The reason is that supplier 2’s rails exhibit more variability about the mean than do supplier 1’s rails. n If we want rails to have a diameter of 1 inch, then more variability around the mean is very undesirable!

| 3.2 | 3.4 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Variance n The most commonly used measures of variability are the variance and standard deviation. n The variance is essentially the average of the squared deviations from the mean. n We say “essentially” because there are two versions of the variance: the population variance and the sample variance.

| 3.2 | 3.4 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | More on the Variance n The variance tends to increase when there is more variability around the mean. n Indeed, large deviations from the mean contribute heavily to the variance because they are squared. n One consequence of this is that the variance is expressed in squared units (squared dollars, for example) rather than original units.

| 3.2 | 3.4 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Standard Deviation n A more intuitive measure of variability is the standard deviation. n The standard deviation is defined to be the square root of the variance. n Thus, the standard deviation is measured in original units, such as dollars, and it is much easier to interpret.

| 3.2 | 3.4 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Computing Variance and Standard Deviation in Excel n Excel has built-in functions for computing these measures of variability. n The sample variances and standard deviations of the rail diameters from the suppliers in the present example can be found by entering the following formulas: “=VAR(A5:A14)” in cell E8 and “=STDEV(A5:A14)” in cell E9.

| 3.2 | 3.4 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Computing Variances & Standard Deviations -- continued n Of course, enter similar formulas for supplier 2 in cells F8 and F9. n As we mentioned earlier, it is difficult to interpret the variances numerically because they are expressed in squared inches, not inches. n All we can say is that the variance from supplier 2 is considerably larger than the variance from supplier 1.

| 3.2 | 3.4 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 3.10 | Interpretation of the Standard Deviation n The standard deviations, on the other hand, are expressed in inches. The standard deviation for supplier 1 is approximately inch, and supplier 2’s standard deviation is approximately three times this large. n This is a considerable disparity. Hence, Otis will prefer to buy rails from supplier 1.