MGT 6421
Quality Management II
Statistical Thinking

Many of the tenets of quality management focus on understanding processes
better. Process understanding usually comes from measuring and analyzing
data from the process. The term “statistical thinking” has emerged * to describe
a mind-set that is helpful in better understanding processes.

So what is statistical thinking? What does your training to this point suggest?
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Quality Management II
From Hare, Hoerl, Hromi and Snee (1995). “The Role of Statistical Thinking
in Management,” Quality Progress, Feb. 1995, pp. 53-60


Process Orientation
–
All work is a series of interconnected processes
–
The majority of problems are in the process
–
Managers must focus on fixing processes, not blaming people
–
Managers must ensure that the needed processes are in place
Understanding Variation
–
Variation is present in everything
–
The existence of variation represents an opportunity for improvement
–
Improvement comes from reducing variation
–
Managerial action takes variation into account in the decision-making
process
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Quality Management II
Using Data to Guide Actions
–
Facts, based on the right amount of the right kind of data, are friendly
–
Data, interpreted in light of subject matter knowledge, should drive behavior
–
Trends in key measures should be displayed graphically
–
Using data is key to improving processes
H.G. Wells:
“Statistical Thinking will one day be as necessary for efficient citizenship as the
ability to read and write.”
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Quality Management II
 Many of these principles were illustrated with W. Edwards Deming’s Red
Bead Experiment.
 Deming referred to these principles as a system of profound knowledge.
1. System Thinking
2. Variation
3. Theory of knowledge
4. Psychology
 In this class we will use statistical process control (SPC) charts and some
concepts of design to enhance our statistical thinking abilities. In terms of
Deming’s system, we will emphasize understanding variation (using SPC
charts mainly) and the theory of knowledge (using experimental design).
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Quality Management II
Probability Review

Before we begin our discussion of SPC, I would like to review a few
concepts from probability.

All processes exhibit variation. We are interested in quantifying the
variation to be able to understand it, and hopefully reduce it. Probability
gives us the necessary tools.

The variables we will use are called random variables. Random variables
are variables which describe all possible outcomes of a probabilistic
experiment.

There are two main types of random variables:
Discrete random variables take on a countable number of values. For
example, the number of defective items in a production batch is a discrete
random variable.
A discrete random variable, X, has an associated probability mass function
(pmf), f(x), where f(x) = P(X=x). The pmf must meet the following:
a. 0  f(x)  1,
b. f(x) = 1.
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Quality Management II
Continuous random variables take on continuous or interval values (there
are an infinite number of possibilities). For example, the width of an
extruded bar is a continuous random variable.
A continuous random variable, Y, has an associated probability density
function (pdf), f(y). The pdf must meet the following:
a. f(y)  0,
b.
 f ( y)dy  1 .
all y
The probability that a continuous random variable is exactly equal to a
single value is 0. Instead, we quantify the probability that the random
variable falls within a certain interval:
b
P(a  Y  b)   f ( y )dy .
a
This is equivalent to saying that the probability is equal to the area under
the pdf curve between a and b.

Examples:
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Quality Management II
Expected Value and Variance

Probability distributions are often summarized using 2 measures referred
to as the expected value and the variance. The expected value gives us
information about the center of the distribution (it is often referred to as the
average or mean), and the variance tells us about the spread.

Mathematical representation:

The expected value is denoted E(X) or more commonly as . The
variance is denoted Var(X), or more commonly as 2.
Expected Value and Variance for a discrete random variable:
 = E( X) 
 xf ( x) ,
all x
2


2
= Var ( X )   ( x   ) f ( x)    x f ( x)    2 .
all x

all x


2
Expected Value and Variance for a continuous random variable:
 = E (Y )   yf ( y)dy ,
all y
2



2

= Var (Y )   ( y   ) f ( y )dy   y f ( y )dy    2 .


all y
all y

2
Another measure often used instead of the variance is the standard
deviation, which is the square root of the variance, and is denoted .
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Quality Management II
Examples:
Common Probability Distributions

There are a number of probability distributions which are often
encountered in practical applications. We will discuss 3 of these, namely
the binomial and poisson (discrete) distributions, and the normal
(continuous) distribution.

The Binomial Distribution
The binomial distribution describes probabilistic experiments with 2
possible outcomes. For example, if we are inspecting finished goods, they
can be classified as either good or bad. Here are the assumptions of the
binomial distribution:
1. We have n independent trials;
2. There are only 2 possible outcomes for each trial;
3. The probability of a "success," p, is constant from trial to trial.
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Quality Management II
If we will let X = the number of successes in n independent trials, then
 n
P( X  x )    p x (1  p) n  x , for x=0,1,...,n,
 x
 n
n!
where   
.
 x x!(n  x )!

For the binomial distribution, = np, and 2 = np(1-p).

Example: In a population of sales invoices, 5 percent have no shipping
document attached. If an auditor takes a random sample of 50 invoices,
what is the likelihood that 3 will have missing shipping documents? What
is the probability that there will be more than 3 with missing documents?
What are the expected number and standard deviation of invoices with
missing documents?

Excel will find binomial probabilities. The formula is
=BINOMDIST(x,n,p,I) where I is 0 if we want P(X=x) and I=1 if we want
P(Xx).
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Quality Management II
The Poisson Distribution
This poisson distribution is used when we interested in the number of
occurrences per space or time (for example, the number of flaws per square
foot of material, or the number of arrivals of customers to a bank per hour).
If we let X = the number of occurrences per space or time, then
P ( X  x) 
xe  
, x=0,1,....
x!
where  is the average number of occurrences per space or time.
NOTE: When solving problems with the poisson distribution, always be
sure that  has the same units as X.

For the Poisson distribution, = , and 2 = .

For the Poisson Distribution the Excel formula is =POISSON(x, , I) where
I is 0 if we want P(X=x) and I=1 if you want P(Xx).
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Quality Management II

Example: An article in the Wall Street Journal (May 10, 1991) reported on
Nucor Steel’s efficiency in increasing output. The article reported,
however, “Its worker death rate since 1980 is the highest in the industry--in
fact, more than double the industry average. Since 1980 11 Nucor workers
have died....” If the industry average death rate is 0.4 deaths per year, what
is the likelihood of observing 11 or more deaths in 11 years? What does
your answer say about safety at Nucor Steel from 1980 to 1991?

The Poisson Approximation to the Binomial

If in a binomial distribution n is large and p is small, we can approximate
the probability with the poisson distribution. We will assume that if p  .05
and n  20 the approximation will be good enough.

All that is required to use the approximation is to let  = np, and then use
the poisson formulas as we did above.
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Quality Management II
The Normal Distribution

The normal (or gaussian) distribution is the most commonly used
continuous distribution, and is assumed for many problems in statistical
process control.
The normal probability density function is as follows:

f ( y) 
1
e
2
1 y
 

2  
2
,   y   .
where  is the mean and  is the standard deviation of the distribution.
The normal pdf is a symmetric bell shaped curve as follows:

0.4
0.35
0.3
0.25
f(y)
0.2
0.15
0.1
0.00135
0.00135
0.3413
0.3413
0.05
0.1359
0.1359
0.0214
0.0214
0
-4
-3
-2
-1
0
1
2
Number of standard deviations from the mean of y


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Quality Management II

Using the Normal Distribution:

Example:
Suppose we have a process which produces rods with a mean diameter of
.625 in. and a standard deviation of .01 in. The customer requires that no
diameter exceed .65 in., or be smaller than .6 in. What proportion of parts
will not meet these specifications?

In the spreadsheet normal probabilities of the form P(X<x) can be found
using =NORMDIST(x,,,1).
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Quality Management II
Sample Statistics and Sampling Distributions

In statistical process control, we usually take samples and make
conclusions about the process based on the samples. We are therefore
interested in the distributions of the summary statistics calculated from
these samples.

Samples from the binomial distribution:

If we are sampling from a binomial distribution, we can estimate the
probability of a successes from the sample. If X is the number of successes
we observe in a sample of size n, then the estimated probability of success,
p , is
X
p .
n
p(1 p)
if we are
n
sampling from an infinite population. The standard deviation of the sample
p(1  p)
is referred to as the standard error, and in this case is
.
n
The expected value of p is p and the variance is

Example:
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Quality Management II
Samples from the poisson distribution:
For this discussion, we will assume that we are only interested in the
number of occurrences per space. The space we will refer to as an area of
opportunity. Further, suppose we are sampling from a poisson distribution
where the area of opportunity can change with each item sampled. If we let
c be the number of occurrences in an area of size ai, then we can estimate
the number of occurrences per unit area, , as
u
c
.
ai
The expected value of u is u and its standard error is
u
, where u
ai
is the true average number of occurrences in a unit area.

Example:
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
Quality Management II
Samples from the normal distribution:
When sampling from a normal distribution for the purposes of statistical
process control, there are 3 sample statistics that are of interest. These are
the sample average, the sample range, and the sample standard deviation.

The Sample Average:
If we take a sample of size n, the mean can be estimated as
n
 Xi
X
i 1
n
.
The expected value of X is , and the standard error is


.
n
The Sample Range:
The sample range is a measure of the variability of the process. It is very
simple to calculate. If we take a sample of size n, the sample range is the
difference between the largest and smallest values in the sample:
R = Xmax - Xmin.
If the distribution being sampled from is normal, the expected value of R
is d2, and the standard error is d3, where d2 and d3 are constants which
will be discussed later.
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Quality Management II
The Sample Standard Deviation:
The sample standard deviation is also a measure of the variability of the
process. It is more difficult to calculate, but for large sample sizes, it is a
better estimator of the process standard deviation than R. If we take a
sample of size n, the sample standard deviation is
n
 ( Xi  X ) 2
s
i 1
n 1
.
If the distribution being sampled from is normal, the expected value of s
is c4, and the standard error is  1  c24 , where c4 is a constant which will
be discussed later.

Examples:
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Quality Management II
The Central Limit Theorem:
Suppose we have n independent variables, X1, X2,..., Xn, each with the
n
 Xi
same distribution. Then the sample mean, X 
i 1
, standardized by
n
subtracting its mean and dividing by its standard deviation (i.e., by
x x
), has a distribution that approaches a standard normal
n
distribution as n gets large.
(NOTE that the original distribution of the X values can be unknown.)

Examples:
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