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Example 3.3

Example 3.3. Measures of Variability: Variance and Standard Deviation. 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.

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Example 3.3

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  1. Example 3.3 Measures of Variability: Variance and Standard Deviation

  2. 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.

  3. 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?

  4. 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!

  5. 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.

  6. 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.

  7. 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.

  8. 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(Supplier1)” in cell E8 and “=STDEV(Supplier1)” in cell E9.

  9. 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.

  10. 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 quite a disparity. Hence, Otis will prefer to buy rails from supplier 1.

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