PERATIONS ANAGEMENT – Arkansas State University – Jonesboro | Home

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Sixth Edition

Jay Heizer
Barry Render
T1   Statistical Tools for Managers
T2   Acceptance Sampling
T3   The Simplex Method of Linear Programming
T4   The MODI and VAM Methods of Solving Transportation Problems
          C       D               T    U   T       O    R     I   A   L   1


DISTRIBUTIONS                          KEY TERMS
  Expected Value of a Discrete         DISCUSSION QUESTIONS
    Probability Distribution
  Variance of a Discrete Probability
  The Normal Distribution

T1-2 CD T U T O R I A L 1   S TAT I S T I C A L T O O L S   FOR   M A NAG E R S

                            Statistical applications permeate the subject of operations management because so much
                            of decision making depends on probabilities that are based on limited or uncertain infor-
                            mation. This tutorial provides a review of several important statistical tools that are useful
                            in many chapters of the text. An understanding of the concepts of probability distribu-
                            tions, expected values, and variances is needed in the study of decision trees, quality con-
                            trol, forecasting, queuing models, work measurement, learning curves, inventory, simula-
                            tion, project management, and maintenance.

                            In this section, we explore the properties of discrete probability distributions, that is,
                            distributions in which outcomes are not continuous. When we deal with discrete variables,
                            there is a probability value assigned to each event. These values must be between 0 and 1,
                            and they must sum to 1. Example T1 relates to a sampling of student grades.

    EXAMPLE T1               The dean at East Florida University, Nancy Beals, is concerned about the undergraduate statistics
                             training of new MBA students. In a sampling of 100 applicants for next year's MBA class, she
                             asked each student to supply his or her final grade in the course in statistics taken as a sophomore
                             or junior. To translate from letter grades to a numeric score, the dean used the following system:

                                                             5. A                    4. B     3. C       2. D    1. F

                             The responses to this query of the 100 potential students are summarized in the table below. Also
                             shown is the probability for each possible grade outcome. This discrete probability distribution is
                             computed using the relative frequency approach. Probability values are also often shown in
                             graph form as in Figure T1.1.

                                    Probability Distribution for Grades
                                                                                                       Number of
                             Grade Letter                       Score                                   Students        Probability,
                              Outcome                         Variable (x)                             Responding          P(x)
                                     A                                           5                        10            0.1  10/100
                                     B                                           4                        20            0.2  20/100
                                     C                                           3                        30            0.3  30/100
                                     D                                           2                        30            0.3  30/100
                                     F                                           1                        10            0.1  10/100
                                                                                                  Total  100            1.0  100/100





                                                                                        1    2    3   4 5
                                                                                       Possible values for the
                                                                                          grade variable x

                             FIGURE T1.1 s Probability Function for Grades
            D I S C R E T E P RO BA B I L I T Y D I S T R I B U T I O N S                                                     T1-3

      This distribution follows the three rules required of all probability distributions:
      1. the events are mutually exclusive and collectively exhaustive
      2. the individual probability values are between 0 and 1 inclusive
      3. the total of the probability values sum to 1
    The graph of the probability distribution in Example T1 gives us a picture of its
shape. It helps us identify the central tendency of the distribution (called the expected
value) and the amount of variability or spread of the distribution (called the variance).
Expected value and variance are discussed next.

Expected Value of a Discrete Probability Distribution
Once we have established distribution, the first characteristic we are usually interested
in is the "central tendency" or average of the distribution.1 We computed the expected
value, a measure of central tendency, as a weighted average of the values of the

                    E(x)          x P(x )  x P(x )  x P(x )  ...  x P(x )
                                         i   i          1       1       2       2               n   n       (T1.1)

            where        xi  variable's possible values
                      P(xi)  probability of each of the variable's possible values

     The expected value of any discrete probability distribution can be computed by:
(1) multiplying each possible value of the variable xi by the probability P(xi) that outcome
will occur, and (2) summing the results, indicated by the summation sign, . Example T2
shows such a calculation.

 Here is how the expected grade value can be computed for the question in Example T1.                                EXAMPLE T2
              E(x)       x P(x )  x P(x )  x P(x )  x P(x )  x P(x )  x P(x )
                             i       i       1      1       2       2       3       3   4   4       5   5

                     (5)(.1)  (4)(.2)  (3)(.3)  (2)(.3)  (1)(.1)
 The expected grade of 2.9 implies that the mean statistics grade is between D (2) and C (3), and
 that the average response is closer to a C, which is 3. Looking at Figure T1.1, we see that this is
 consistent with the shape of the probability function.

Variance of a Discrete Probability Distribution
In addition to the central tendency of a probability distribution, most decision makers
are interested in the variability or the spread of the distribution. The variance of a prob-
ability distribution is a number that reveals the overall spread or dispersion of the distri-

1   If the data we are dealing with have not been grouped into a probability distribution, the measure of central
    tendency is called the arithmetic mean, or simply, the average. Here is the mean of the following seven num-
    bers: 10, 12, 18, 6, 4, 5, 15.
                   Arithmetic mean, X  n         
                                          10  12  18  6  4  5  15
T1-4 CD T U T O R I A L 1   S TAT I S T I C A L T O O L S      FOR   M A NAG E R S

                            bution.2 For a discrete probability distribution, it can be computed using the following

                                                                   Variance       (x
                                                                                        i    E(x))2P(x i)                               (T1.2)

                                        where          xi  variable's possible values
                                                    E(x)  expected value of the variable
                                                    P(xi)  probability of each possible value of the variable

                                 To compute the preceding variance, the expected value is subtracted from each value
                            of the variable squared, and multiplied by the probability of occurrence of that value. The
                            results are then summed to obtain the variance.
                                 A related measure of dispersion or spread is the standard deviation. This quantity is
                            also used in many computations involved with probability distributions. The standard de-
                            viation, , is just the square root of the variance:

                                                                                variance                                                (T1.3)

                            Example T3 shows a variance and standard deviation calculation.

    EXAMPLE T3               Here is how this procedure is done for the statistics grade survey question:
                                     Variance         (x
                                                           i    E(x))2P(xi)

                                                   (5  2.9)2(.1)  (4  2.9)2(.2)  (3  2.9)2(.3)  (2  2.9)2(.3) 
                                                     (1  2.9)2(.1)
                                                   (2.1)2(.1)  (1.1)2(.2)  (.1)2(.3)  (  .9)2(.3)  (  1.9)2(.1)
                                                   .441  .242  .003  .243  .361
                                The standard deviation for the grade question is
                                                                           s  variance
                                                                               1.29  1.14

                            2   Just as the variance of a probability distribution shows the dispersion of the data, so does the variance of un-
                                grouped data, that is, data not formed into a probability distribution. The formula is: Variance  (X  X )2/n.
                                Using the numbers 10, 12, 18, 6, 4, 5, and 15, we find that X  10. Here are the variance computations:
                                                (1  10)2 + (12  10)2 + (18  10)2 + (6  10)2 + (4  10)2 + (5  10)2 + (15  10)2
                                   Variance =
                                                0 + 4 + 64 + 16 + 36 + 25 + 25
                                            =       = 24.28
                             We should also note that when the data we are looking at represent a sample of a whole set of data, we use the
                             term n  1 in the denominator, instead of n, in the variance formula.
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