OPERATIONS
MANAGEMENT
Sixth Edition
Jay Heizer
Barry Render
CD TUTORIALS
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
STATISTICAL TOOLS
F O R MANAGERS
TUTORIAL OUTLINE
DISCRETE PROBABILITY SUMMARY
DISTRIBUTIONS KEY TERMS
Expected Value of a Discrete DISCUSSION QUESTIONS
Probability Distribution
PROBLEMS
Variance of a Discrete Probability
BIBLIOGRAPHY
Distribution
CONTINUOUS PROBABILITY
DISTRIBUTIONS
The Normal Distribution
T1-1
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.
DISCRETE PROBABILITY DISTRIBUTIONS
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
0.4
Probability
0.3
0.2
0.1
0.0
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
variable:
n
E(x) x P(x ) x P(x ) x P(x ) ... x P(x )
i1
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
5
E(x) x P(x ) x P(x ) x P(x ) x P(x ) x P(x ) x P(x )
i1
i i 1 1 2 2 3 3 4 4 5 5
(5)(.1) (4)(.2) (3)(.3) (2)(.3) (1)(.1)
2.9
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.
X
Arithmetic mean, X n
10 12 18 6 4 5 15
10
7
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
equation:
n
Variance (x
i1
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:
5
Variance (x
i1
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
1.29
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 =
7
0 + 4 + 64 + 16 + 36 + 25 + 25
=
7
170
= = 24.28
7
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.
Norfolk Public Schools Preliminary Spring 2011 SOL Results School Board Meeting July 20, 2011 Draft 7-18-11 SEAS Table of Contents 1) Executive Summary District Highlights 2) District Results 3) English by School 4) Mathematics by School 5) History by School 6) Science by School Executive …
Officer RECORD MANAGEMENT The following information is gouge on how to keep your record up-to-speed. The blue links go to the NPC website. All these items are handled by different offices, so do not wait. Start getting your ship in order NOW! BUPERS ONLINE (BOL) …
School Counseling Program Practicum and Internship Manual School Counseling Program Department of Counseling Gallaudet University 800 Florida Ave., NE Washington, DC 20002 Revised June 2007/Copyright: Department of Counseling, Gallaudet University Table of Contents Overview of the Manual 4 Introduction to the Department of Counseling 5 …
District Adopted Materials List 2008 2009 Elementary Education Language Arts Title Publisher Online Edline L.V. Grammar, Gr. 2-5 Harcourt Language, 2002 Harcourt Or (School Choice) Write Source Language Program, 2002 Great Source Reading, Grades K - 5 MacMillan McGraw-Hill, Treasures Macmillan Intervention Reading, 3-5 Break …
