STATISTICAL PROCESS CONTROL_ CONTROL CHARTS by hcj

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									                                STATISTICAL PROCESS CONTROL: CONTROL CHARTS
                                                Richard Mojena


INTRODUCTION

SPC is the application of statistical methodology to the control of processes.
Also called Statistical Quality Control (SQC)

    Process is in statistical control… Variations due to natural causes (sampling error, “white” noise).
    Process is not in statistical control…. Variations due to both natural and assignable causes.

Control charts are the most popular means of implementing SPC.



TYPES OF CONTROL CHARTS

Control charts for variables (continuous measure such as weight, volume, length, strength, …) track central tendency
(mean) and dispersion or variation (range) to determine whether a process is in or out of statistical control.
-
x -charts… track mean.
R-charts… track Range.




                                                                             (a) Mean and range in
                                                                             statistical control with natural
                                                                             causes
                                        Frequency

              Lower control limit                               Upper control limit

                                                                 (a) Mean but not range in control
                                                                 with natural causes




                                                               (c) Out of control with assignable
                                                               causes
                     Size
          Weight, length, speed, etc.



Control charts for attributes (binary category such as defective or not defective) track percent defectives or a count of
defectives to determine whether a process is in or out of statistical control.

p-charts… track percent defectives.
c-charts… track count for number of defects.
REVIEW OF DESCRIPTIVE STATISTICS IN SAMPLING

Mean
                n                                                                             Sample mean x is an estimate of
                xi                                                                           population mean  .
        x     i 1
                             where xi is data item i and n is sample size.
                    n
                m
               x
        x     i1
                             where m is number of samples each with sample size n. Note that x is a mean of means or
                m
                             grand mean.

Range

        R = Max(xi) – Min(xi)


Standard Deviation
                                                   Sample standard deviation s is an estimate of
                 n                                 population standard deviation σ .
                ( xi  x ) 2
        s      i 1
                         n-1
                                                                                                                    Six-sigma quality
                                                                                                                    Normal curve fits 12
Standard Error of the Mean
                                                         Sample standard error   sx   is an estimate of             standard deviations or
                                                         population standard error    x.                           ±6σ within defined
                        s
          sx                                                                                                       upper and lower
                         n                                                                                          tolerance limits (UTL,
                                                                                                                    LTL).
Central Limit Theorem (CLT)
                                                                                                                    Amounts to about 2
                                                                                                                    defects per million.
                                                Confidence Interval (Limits)                                        6σ process capability
                                                                                            99.7% of all x fall     ratio or Cp :
           95.5% of all x fall
                                                                                            within  3
              within  2                                                                                 x            (UTL - LTL)/12σ
                               x

                                                                                                                    Six-sigma quality if
                                                                                                                              Cp ≥ 1
                                                                                                                    where σ refers to
                                                                                                                    measurement of
                                                                                                                    interest.


                                                                                                                  x
                                                          x  
                        LTL                                                                                   UTL

                                                                                                      SPC     Richard Mojena   9/27/10   Page 2 of 5
  Example 1
  Calculate mean, range, standard deviation, standard error of the mean, and 2sigma confidence limits for the
  following bar thicknesses in mm. Interpret the meaning of this confidence interval.

  xi
  9
  7
  8




X-BAR-CHARTS USING STANDARD DEVIATION
                                                                           Uppper control limit
       UCL        x  z
             x              x
                                where z is the number of standard deviations (2 for 95.5% and 3 for 99.7%).
       LCL        x  z
             x              x                                                Lower control limit

  Example 2
  Suppose we have 6 samples of size 3 for the application in the previous example, with sample means: 8.0, 8.5,
  7.2, 8.8, 9.0, 10.0. Assume a robotic process set to a standard deviation of 1.0, but with unknown mean.
  Calculate the lower and upper 2-sigma control limits and plot the control chart. Conclusions? What if we were
  to use 3-sigma limits? What would change if the process is known to have a mean of 8.50?



        UCL

        mean

        LCL
        ==

                                1      2       3       4       5       6




  What’s the difference between confidence, control, and tolerance limits?


                                                                                    SPC   Richard Mojena   9/27/10   Page 3 of 5
X-BAR-CHARTS USING RANGE

     UCL        xA R
            x       2
                             where A2 is a 3-sigma mean factor from TABLE 4.5, p.94.
     LCL        xA R
            x       2
            m
             Ri
      R    i1
                             Note that R-bar is a mean of ranges for the m samples.
                m

  Example 3
  Suppose in the previous example that the population standard deviation is not known. Ranges for the six
  samples are 2, 3, 2, 3, 4, 5. Recalculate 3-sigma control limits and draw conclusions.




R-CHARTS

     UCLR  D 4 R


     LCL R  D 3 R           where 3-sigma factors D3 and D4 are taken from TABLE 4.5, p.94.
            m
             Ri
      R    i1
                m

   Example 4
   Find the R-chart control limits for the previous example. Conclusion?




                                                                             SPC      Richard Mojena   9/27/10   Page 4 of 5
P-CHARTS

If we define p as proportion defectives in a random sample of large (>30) size n, then the CLT says that the sampling
distribution of p is normal with mean p (calculated on m samples) and standard error sp.
              p (1  p )
        sp                                   m
                  n                           pi
        UCL p  p  zsp         where p      i1
                                                       or p = Average defects / n
                                                  m
        LCL p  p  zsp

     Example 5
     Suppose that 5 random samples of 1000 insurance claims are taken to audit for errors. The number of errors
     in each sample are 40, 30, 50, 35, and 45. Determine 3-sigma control limits. Conclusion?




C-CHARTS

Here the number of defects (count c) per unit (usually time, but could be other continuoums such as volume, distance,
…) is determined across a sample of k units. Assuming a Poisson distribution, the mean is c and the standard
deviation isc . Thus,
        UCL c  c  zsc                       k
                                              ci
                                where c     i 1
                                                      , sc =   c.
                                                  k
        LCL c  c  zsc

   Example 6
   Consider a large corporation with many PCs. The event “PC requires repair” is assessed on a daily basis.
   Suppose that over a 5-day period the number of PC repairs is 20, 30, 10, 65, and 15. Calculate 3-sigma
   control limits and draw a conclusion.




EXCEL IMPLEMENTATIONS

Class software demo: SPC.xls.
Sample commercial software as Excel add-ins (Googling Excel SPC yielded over 600k results):
 KnowWare International
 spcforexcel.com

                                                                                    SPC   Richard Mojena   9/27/10   Page 5 of 5

								
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