Example 3.3 Variability of Elevator Rail Diameters at Otis Elevator

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Example 3.3Variability of Elevator Rail
Diameters at Otis Elevator

• Measures of Variability Variance and Standard
  Deviation

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

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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?

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Explanation

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

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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.

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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.

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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.

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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(A5A14) in cell E8 and
  STDEV(A5A14) in cell E9.

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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.

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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 2s standard deviation is approximately
  three times this large.
• This is a considerable disparity. Hence, Otis
  will prefer to buy rails from supplier 1.