# PROBABILITY AND STATISTICS REVIEW

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PROBABILITY AND STATISTICS
REVIEW
Describing & Summarizing Data

The Binomial Distribution

The Poisson Distribution

The Normal Distribution

Sampling Distributions

OPS 465 - Qual Mgmt
RANDOM VARIABLES AND
REALITY
• All processes are subject to variability

• Unassignable variation
– Variation inherent in the process which cannot be reduced
without process redesign
– “Randomness”

• Assignable variation
– Process variation which represents deviation from expectations
and may be traced (“assigned”) to an external factor

OPS 465 - Qual Mgmt
RANDOM VARIABLES AND
REALITY
• A key objective in QM is separating assignable from
unassignable variation
– Assignable variation can be quickly eliminated

• First need to model the unassignable variation
– A reference probability distribution is selected whose behavior
matches that of the quality characteristic generated by the
process (coin flipping)
– If the process produces a different distribution, we know that
assignable variation is present
– Selecting the correct distribution for a quality characteristic is
a critical step

OPS 465 - Qual Mgmt
SPOTTING PROBLEMS USING A
REFERENCE DISTRIBUTION
(HYPOTHESIS TESTING)
• After a reference distribution describing the population
is selected, samples of the quality characteristic are
checked periodically for assignable variation
– Quality data is collected
– Sample statistics are calculated
– The sample statistics are compared with those from the
reference distribution

•   Does this data come from this distribution?
– If yes, conclude there is no assignable variation (in control)
– If no, conclude there is assignable variation (out of control)
OPS 465 - Qual Mgmt
DESCRIBING & SUMMARIZING
DATA
• How can data (Xi) from a population of size N or from a
sample of size n be summarily described?

• Graphically
– Histogram

• Numerically
– Mean
– Variance = (Standard Deviation)2

OPS 465 - Qual Mgmt
DESCRIBING & SUMMARIZING
DATA
• Mean                N                                    n
 Xi                                 Xi
i 1                                i 1
                                x
N                                     n

• Variance = (Standard Deviation)2
n
 Xi  x 
N
 X i    2                                                  2

i 1                                            i 1
2                                             s2 
N                                                 n 1

OPS 465 - Qual Mgmt
Excel
THE BINOMIAL DISTRIBUTION
• Models "go/no-go" attribute quality characteristics
– n -- Sample size
– p -- Probability of a nonconformity
– X -- Number of nonconformities in sample {X = 0, 1, . , n}

• X is a binomial random variable with following
distribution:
n!
f ( X)              p X (1  p )n  X
X! (n  X)!
• The mean and variance are:
np  np                      np
2
 np(1  p)

OPS 465 - Qual Mgmt
THE BINOMIAL DISTRIBUTION
• Example: improperly sealed orange juice cans
– n = 100
– p = 0.02

n!
f ( X)              p X (1  p )n  X
X! (n  X)!

100!
f ( 0)                0.020 (1  0.02)1000  0.98100  0.1326
0! (100  0)!
100!
f (1)                  0.021 (1  0.02)1001  (100)(0.02)(0.9899 )  0.2707
1! (100  1)!

OPS 465 - Qual Mgmt
Excel
THE BINOMIAL DISTRIBUTION
• Example: improperly sealed orange juice cans
– n = 100
– p = 0.02

• The mean and variance are:

np  np                   np  np(1  p)
2

np  (100)(0.02)  2       np  (100)(0.02)(1  0.02)  1.96
2

OPS 465 - Qual Mgmt
Excel
THE POISSON DISTRIBUTION
• Models integer-valued quality characteristics that range
from 0 to infinity
– c -- Constant rate of nonconformities per item
– X -- number of nonconformities per item{x = 0, 1, 2, . . . }

• X is a Poisson random variable with following
distribution:
c Xe c
f ( X) 
X!
• The mean and variance are:
c  c                               c  c
2

OPS 465 - Qual Mgmt
THE POISSON DISTRIBUTION
• Example: scratches per table top
– c=2

c Xe c
f ( X) 
X!

20 e  2
f ( 0)            e  2  0.1353
0!

21 e  2
f (1)            e  2  2(0.1353)  0.2707
1!

OPS 465 - Qual Mgmt
Excel
THE POISSON DISTRIBUTION
• Example: scratches per table top
– c=2

• The mean and variance are:

c  c                          c  c
2

c  2                          c  2
2

OPS 465 - Qual Mgmt
Excel
THE POISSON DISTRIBUTION

• For future reference:

• When n is large and p is small and np < 5,

• The Poisson can be used to approximate the binomial
distribution if we set:
c = np
2 = (100)(0.02)

• Why do this? Binomial is unwieldy for large n

OPS 465 - Qual Mgmt
Excel
THE NORMAL DISTRIBUTION
• Models continuous quality characteristics that range
from negative to positive infinity
–   µ -- Mean
–   2 -- Variance
–   X -- Value of quality characteristic { -  < X < }

• X is a normal random variable with the following
distribution:
1  X   2
1                
2  
f ( X)       e
 2

OPS 465 - Qual Mgmt
THE NORMAL DISTRIBUTION
• All normal distributions have same shape

Same Probability

µ                                                     0            z
µ +z 
Quality Characteristic Y                                   Standard Normal X

• If X is a standard normal random variable
• And Y is any normal random variable with mean µ and
standard deviation :

PROB(X  z) = PROB(Y  µ + z)
OPS 465 - Qual Mgmt
THE NORMAL DISTRIBUTION

OPS 465 - Qual Mgmt
THE NORMAL DISTRIBUTION
• Since: PROB(X  z) = PROB(Y  µ + z)

• If Y ~ N[, ] AND X ~ N[0, 1], then:
– Let A be any constant
– Let z be chosen so that A =  + z (I.E. SET z = (A - )/)

• Then PROB(Y  A) = PROB(Y   + z)
= PROB(X  z)

OPS 465 - Qual Mgmt
THE NORMAL DISTRIBUTION
• Example: weighing filled bags of sugar
  = 10
  = 0.1

• What is the likelihood of a bag weighing more than 10.2
pounds?

• PROB(Y  A) = PROB(Y   + z) = PROB(X  z)
• PROB(Y  10.2) = PROB(Y  10+ 2(0.1)) = PROB(X  2)

A   10.2  10
• Or we could calculate:     z                2
       0.1
OPS 465 - Qual Mgmt
SAMPLING DISTRIBUTIONS
• If we know the distribution of a random variable --

• We can construct a distribution of any function of the
random variables

– Averages

– Ranges

– Proportions

– Standard deviations

OPS 465 - Qual Mgmt
SAMPLING DISTRIBUTIONS: THE
BINOMIAL DISTRIBUTION
• If X is binomially distributed with
– n -- Sample size
– p -- Probability of a nonconformity

• We know the mean and variance are then:
np  np                    np  np(1  p)
2

• But the sample proportion pi = Xi/n has the following
mean and variance:
p(1  p)
p  p                p 
2
n
OPS 465 - Qual Mgmt
SAMPLING DISTRIBUTIONS: THE
BINOMIAL DISTRIBUTION
• Example: improperly sealed orange juice cans
– n = 100
– p = 0.02

• The mean and variance of the number nonconforming
are:
np  np  2          np  np(1  p)  1.96
2

• The mean and variance of the fraction nonconforming
are:
p(1  p)
p  p  0.02        p             0.000196
2
n

OPS 465 - Qual Mgmt
Excel
SAMPLING DISTRIBUTIONS: THE
POISSON DISTRIBUTION
• If X is Poisson distributed with
– c -- Constant rate of nonconformities per item
– n – Size of item
– u – Constant rate of nonconformities per unit of size

• We know the mean and variance are then:
c  c                            c  c
2

• But the number of nonconformities per unit of size ui =
Xi/n has the following mean and variance:
c                      u
u  u                    u 
2
n                      n
OPS 465 - Qual Mgmt
SAMPLING DISTRIBUTIONS: THE
POISSON DISTRIBUTION
• Example: scratches per square foot of table top
– c=2
– n = 10

• The mean and variance of the number nonconforming
per item are:
c  c  2               c  c  2
2

• The mean and variance of the number nonconforming
per unit of size are:
c                             u
u  u   0.2                u
2
  0.02
n                             n
OPS 465 - Qual Mgmt
Excel
SAMPLING DISTRIBUTIONS: THE
NORMAL DISTRIBUTION
• If X is normally distributed with
–   µ -- Mean
–   2 -- Variance

• Let samples of size n be collected
X   Sample mean
R   Sample range
   Sample s tan dard deviation

• Each of these statistics has a distribution with a mean
and standard deviation

OPS 465 - Qual Mgmt
SAMPLING DISTRIBUTIONS: THE
NORMAL DISTRIBUTION
• For the sample means:

X                   X 
n
• For the sample ranges:
 R  d 2              R  d 3

• For the sample standard deviations:

S  c4             S   1  c4
2

OPS 465 - Qual Mgmt
SAMPLING DISTRIBUTIONS: THE
NORMAL DISTRIBUTION

OPS 465 - Qual Mgmt
SAMPLING DISTRIBUTIONS: THE
NORMAL DISTRIBUTION
• Example: weighing filled bags of sugar
  = 10
  = 0.1
– n=5

• The distribution of sample means is as follows:

 0 .1
 X    10            X              .045
n   5

OPS 465 - Qual Mgmt
Excel
SAMPLING DISTRIBUTIONS: THE
NORMAL DISTRIBUTION
• Example: weighing filled bags of sugar
  = 10
  = 0.1
– n=5

• The distribution of sample ranges is as follows:

R  d 2  0.1( 2.326)  0.233

R  d 3  0.1(0.864)  0.086

OPS 465 - Qual Mgmt
Excel
SAMPLING DISTRIBUTIONS: THE
NORMAL DISTRIBUTION
• Example: weighing filled bags of sugar
  = 10
  = 0.1
– n=5

• The distribution of sample standard deviations is as
follows:

S  c4  0.1(0.94)  0.094

S   1  c4  0.1 1  0.942  0.034
2

OPS 465 - Qual Mgmt
Excel

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