Example 3.3 Variability of Elevator Rail Diameters at Otis Elevator

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

2. Objective To calculate the variability for two suppliers and choose the one with the least variability.

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

4. Which should Otis prefer? • Observe that the mean, median, and mode are all exactly 1 inch for each of the two suppliers. • 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?

5. Explanation • The reason is that supplier 2’s rails exhibit more variability about the mean than do supplier 1’s rails. • If we want rails to have a diameter of 1 inch, then more variability around the mean is very undesirable!

6. Variance • The most commonly used measures of variability are the variance and standard deviation. • The variance is essentially the average of the squared deviations from the mean. • We say “essentially” because there are two versions of the variance: the population variance and the sample variance.

7. More on the Variance • The variance tends to increase when there is more variability around the mean. • Indeed, large deviations from the mean contribute heavily to the variance because they are squared. • One consequence of this is that the variance is expressed in squared units (squared dollars, for example) rather than original units.

8. Standard Deviation • A more intuitive measure of variability is the standard deviation. • The standard deviation is defined to be the square root of the variance. • Thus, the standard deviation is measured in original units, such as dollars, and it is much easier to interpret.

9. Computing Variance and Standard Deviation in Excel • Excel has built-in functions for computing these measures of variability. • 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.

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

11. Interpretation of the Standard Deviation • The standard deviations, on the other hand, are expressed in inches. The standard deviation for supplier 1 is approximately 0.012 inch, and supplier 2’s standard deviation is approximately three times this large. • This is a considerable disparity. Hence, Otis will prefer to buy rails from supplier 1.