PROBABILITY AND STATISTICS REVIEW

							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