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Statistical Quality Control
                                                                                            • Statistical Quality Control can broadly be divided into
                                                                                              the following
                                                                                               – Acceptance Sampling
                 Statistical Quality Control                                                      • Acceptance sampling is the middle of the road approach between
                                                                                                    no inspection and 100% inspection.
                                                                                                  • The main attributes of acceptance sampling are the go and no-go
                                       Presented by:                                                variable.
                                                               Anupam Kumar                       • This method is employed when testing is destructive and 100%
                                                                      Reader                        inspection is not possible.
                                                                 SMS Varanasi                     • The major disadvantage of acceptance sampling was that it builds
                                                  Email: anupamkr@gmail.com                         in incompetency in the production process.
                                                                                               – Statistical Process Control
                                   © Copyright 2013 Anupam Kumar                        7                            © Copyright 2013 Anupam Kumar                     8




                Statistical Process Control                                                             Quality Control Chart
       • Statistical Process Control is a method used to                                    • A quality control chart consists of the following:
         determine whether the process of                                                      – Points representing measurements of a quality
                                                                                                 characteristic in samples taken from the process at
         manufacturing is within control or not.                                                 different times.
       • It can broadly be divided into the followings                                         – A center line, drawn at the process characteristic mean
         aspects.                                                                                which is calculated from data
                                                                                               – Upper and lower control limits that indicate the threshold
          – Control Chart for Variables                                                          at which the process output is considered statistically
          – Control Chart for Attributes                                                         ‘unlikely’.
          – Control Chart for Defectives or Fraction Defects.                                  – Upper and lower warning limits may also be drawn as
                                                                                                 separate line, usually 2 standard deviations above and
                                                                                                 below the centre line.
                                   © Copyright 2013 Anupam Kumar                        9                            © Copyright 2013 Anupam Kumar                  10




                 Standard Error of Means                                                               Quality Control Charts
       • The sample mean is the usual estimator of                                          • Quality Control Charts for Variables
         a population mean.                                                                    – Standard Deviation Chart or σ Chart.
       • However, different samples drawn from that same                                          • When standards of manufacture and allowable or tolerance limits
                                                                                                    are not specified a standard deviation chart is prepared.
         population would in general have different values of                                     • Standard Deviation Chart takes into account the standard errors of
         the sample mean.                                                                           mean as the determining limits.
       • The standard error of the mean is the standard                                           • Upper Control Limit: M + 3(σ/√n)
         deviation of those sample means over all possible                                        • Mean Line: M
         samples drawn from the population.                                                       • Lower Control Limit: M - 3(σ/√n)
                                                                                               – Where, M is the population mean, σ is the Standard
          Standard Error of mean = (Population Standard Deviation)/(Sample Size)^0.5             Deviation of population and n is the sample size.
          Standard Error of mean = σ /√n
                                   © Copyright 2013 Anupam Kumar                       11                            © Copyright 2013 Anupam Kumar                  12




© Copyright 2013 Anupam Kumar                                                                                                                                              1
Quality Control Charts                                                                                                  Illustrations
       • Quality Control Charts for Variables                                                                        • A quality control sample of 100 items was taken and the
          – Standard Deviation Chart or σ Chart.                                                                       standard deviation of the same was estimated as 0.25 cm.
                                                                                                                         – Estimate the standard error of the mean.
                  • When standards of manufacture and allowable or tolerance limits
                    are not specified a standard deviation chart is prepared.                                            – Draw the control chart for the sample if the mean is 10 cm.
          – Mean or X Chart                                                                                          • Draw X chart and R chart for the given data when the value
                  • This is used to measure the central tendency and shows erratic or                                  for A2 = 0.58, D3 = 0 and D4 = 2.11 for value of 5.
                    cyclic shifts in the process. It detects steady progress changes like                                    Sample       X          R         Sample             X     R
                    wear.                                                                                                      No.                               No.
                  UCL / LCL = X +/- A2 R                                                                                        1        7.0         2             6             11.0   4
          – Range or R Chart                                                                                                    2        7.5         3             7             11.5   3
                  • It controls the general variability of the process and is affected by                                       3        8.0         2             8             4.0    2
                    changes in process variability. It measures the spread.                                                     4        10.0        2             9             3.5    3
                  LCL = D3R                                                                                                     5        9.5         3            10             4.0    2
                                       © Copyright 2013 Anupam Kumar                      13                                                                                                14
                  UCL = D4R                                                                                                                      © Copyright 2013 Anupam Kumar




        Quality Control Chart for Attribute                                                                              Fraction Defect Chart or P Chart
       • These are used when the quality is defined in                                                               • The object of this chart is to control the proportion of
                                                                                                                        defective according to specified attribute.
         terms of product attributes which may not be
                                                                                                                     • Even when the attribute standard is not specified, it is inferred
         measured.                                                                                                      by finding the average fraction defectives.
       • Types of Quality Control Chart for Attributes                                                               • This chart is used to control the general quality of the
                                                                                                                        component parts.
         are:
                                                                                                                     • It checks whether the fluctuations in the product quality are
          – Fraction Defect Chart or p – defective chart                                                                due to chance causes or some assignable reasons.
          – Number Defective or np- chart                                                                                          UCL / LCL = p +/- 3 ((p*q) /n )1/2; where
          – Number Defective per unit chart or c chart.                                                              p is average fraction defectives,
                                                                                                                     q is average fraction non – defectives (q = 1-p), and
                                           © Copyright 2013 Anupam Kumar                                        15
                                                                                                                     n is number of item in each sample.                               16
                                                                                                                                                 © Copyright 2013 Anupam Kumar




                                     Illustration 3                                                                                   Illustration 3 (Soln.)
       • A quality controller examines a sample of 1000 items                                                        •   Average probability = p = 236.7/1000 = 0.2367
         each and finds the number of defectives as follows.                                                         •   q = 1-p = 1-0.2367 = 0.7633
         Draw the control chart for the sample.                                                                      •   n is the number of items in each sample = 1000
         Sample      Number       Sample      Number       Sample          Number        Sample    Number
         No.         of           No.         of           No.             of            No.       of                •   Control limits = p +/- 3 ((p*q) /n )1/2
                     defectives               defectives                   defectives              defectives
            1           215          6           240           11            285            16        190            •   UCL = 0.2367 + 3*((0.2367*0.7633)/1000)^0.5
            2           227          7           218           12            287            17        215                ⇒UCL = 0.2367+3*0.013441 = 0.2367+0.040324
            3           220          8           195           13            312            18        210                ⇒UCL = 0.277024
            4           180          9           280           14            195            19        215
            5           320          10          310           15            180            20        240
                                                                                                                     • LCL = 0.2367 - 3*((0.2367*0.7633)/1000)^0.5
                                                Total                                                4734                ⇒LCL = 0.2367+3*0.013441 = 0.2367+0.040324
                                              Average                                   Solution    236.7                ⇒LCL = 0.196376
                                           © Copyright 2013 Anupam Kumar                                        17                               © Copyright 2013 Anupam Kumar              18




© Copyright 2013 Anupam Kumar                                                                                                                                                                    2
Number Defectives or np Chart                                                                              Illustration 4
                                                                                              •   In a manufacturing process, the number of defectives found in the
       • This chart is similar to the p chart                                                     inspection of 15 lots of 400 items each is given below. Determine the trial
       • It is used when the calculation for proportion                                           control limits for np chart and state whether the process is in control or
                                                                                                  not.
         of fraction of defectives is complex.                                                     – What will be new value of mean fraction defective if some obvious points
                                                                                                     outside the control limits are eliminated?
       • The limits for the np chart is as follows.                                                – What will be the corresponding upper and lower control limits and examine
                                                                                                     whether the process is still in control or not?
                UCL / LCL = np +/- 3*(np*(1-p))1/2 ;                                                                                                          Solution
                                                                                                    Lot No.     No. of       Lot No.         No. of           Lot No.       No. of
       Where p = Σnp / Σn                                                                                       defectives                   defectives                     defectives
                                                                                                        1            2             6               0              11              6
                                                                                                        2            5             7               1              12              0
                                                                                                        3            0             8               0              13              3
                                                                                                        4           14             9              18              14              0
                                    © Copyright 2013 Anupam Kumar                        19             5            3            10                8
                                                                                                                              © Copyright 2013 Anupam Kumar       15              6              20




                                                                                                  Number of Defects Per Unit Chart
                            Illustration 4 (Soln.)
                                                                                                            or c – chart
       •    Average probability = p = 66/(15*400) = 0.011                                     • In case the sample is such that there are no uniform
       •    q = 1-p = 1-0.011 = 0.989                                                           number of items:
       •    n = number of item in each sample = 400                                                – the limits on the control chart cannot be laid out as n
                                                                                                     would vary and thus
       •    Control limits = np +/- 3*(np*(1-p))1/2                                                    • Fraction defective chart cannot be used
       •    UCL = (400*0.011) + 3*((400*0.011*0.989)^0.5)                                              • Number defective chart also cannot be used.

             ⇒UCL = 4.4 + 3*2.086049 = 4.4 + 6.258147 = 10.658147                             • In such a case the number of defectives per unit is
       • LCL = (400*0.011) - 3*((400*0.011*0.989)^0.5)                                          plotted.
             ⇒LCL = 4.4 - 3*2.086049 = 4.4 - 6.258147 = -1.858147                             • It controls the number of defects observed per unit
             ⇒LCL = 0                                                                           or per sample.
                                                                                                               UCL / LCL = C +/- 3 C1/2 ,
                                    © Copyright 2013 Anupam Kumar                        21                                                                      © Copyright 2013 Anupam Kumar   22
                                                                                              Where C is average no. of defect per sample




                                   Illustrations
       • The following figures show the number of defective
         items discovered in 10 samples taken on 10
         consecutive days in a month.
           Date                1      2      3       4       5      6   7   8   9   10
                                                                                                                             Six Sigma …
           Nos. Defective      1      2      4       16      4      3   5   20 16   4

       • 10 castings were inspected in order to locate defects
         in them. Every casting was found to contain certain
         number of defects as given below. Plot the relevant
         chart and draw your conclusions.
           Date                1      2      3       4       5      6   7   8   9   10
           Nos. Defective      2      4      1       5       5      6   3   4   0   7
                                                                                         23                                   © Copyright 2013 Anupam Kumar                                      24
                                    © Copyright 2013 Anupam Kumar




© Copyright 2013 Anupam Kumar                                                                                                                                                                         3

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Statistical quality control

  • 1. Statistical Quality Control • Statistical Quality Control can broadly be divided into the following – Acceptance Sampling Statistical Quality Control • Acceptance sampling is the middle of the road approach between no inspection and 100% inspection. • The main attributes of acceptance sampling are the go and no-go Presented by: variable. Anupam Kumar • This method is employed when testing is destructive and 100% Reader inspection is not possible. SMS Varanasi • The major disadvantage of acceptance sampling was that it builds Email: anupamkr@gmail.com in incompetency in the production process. – Statistical Process Control © Copyright 2013 Anupam Kumar 7 © Copyright 2013 Anupam Kumar 8 Statistical Process Control Quality Control Chart • Statistical Process Control is a method used to • A quality control chart consists of the following: determine whether the process of – Points representing measurements of a quality characteristic in samples taken from the process at manufacturing is within control or not. different times. • It can broadly be divided into the followings – A center line, drawn at the process characteristic mean aspects. which is calculated from data – Upper and lower control limits that indicate the threshold – Control Chart for Variables at which the process output is considered statistically – Control Chart for Attributes ‘unlikely’. – Control Chart for Defectives or Fraction Defects. – Upper and lower warning limits may also be drawn as separate line, usually 2 standard deviations above and below the centre line. © Copyright 2013 Anupam Kumar 9 © Copyright 2013 Anupam Kumar 10 Standard Error of Means Quality Control Charts • The sample mean is the usual estimator of • Quality Control Charts for Variables a population mean. – Standard Deviation Chart or σ Chart. • However, different samples drawn from that same • When standards of manufacture and allowable or tolerance limits are not specified a standard deviation chart is prepared. population would in general have different values of • Standard Deviation Chart takes into account the standard errors of the sample mean. mean as the determining limits. • The standard error of the mean is the standard • Upper Control Limit: M + 3(σ/√n) deviation of those sample means over all possible • Mean Line: M samples drawn from the population. • Lower Control Limit: M - 3(σ/√n) – Where, M is the population mean, σ is the Standard Standard Error of mean = (Population Standard Deviation)/(Sample Size)^0.5 Deviation of population and n is the sample size. Standard Error of mean = σ /√n © Copyright 2013 Anupam Kumar 11 © Copyright 2013 Anupam Kumar 12 © Copyright 2013 Anupam Kumar 1
  • 2. Quality Control Charts Illustrations • Quality Control Charts for Variables • A quality control sample of 100 items was taken and the – Standard Deviation Chart or σ Chart. standard deviation of the same was estimated as 0.25 cm. – Estimate the standard error of the mean. • When standards of manufacture and allowable or tolerance limits are not specified a standard deviation chart is prepared. – Draw the control chart for the sample if the mean is 10 cm. – Mean or X Chart • Draw X chart and R chart for the given data when the value • This is used to measure the central tendency and shows erratic or for A2 = 0.58, D3 = 0 and D4 = 2.11 for value of 5. cyclic shifts in the process. It detects steady progress changes like Sample X R Sample X R wear. No. No. UCL / LCL = X +/- A2 R 1 7.0 2 6 11.0 4 – Range or R Chart 2 7.5 3 7 11.5 3 • It controls the general variability of the process and is affected by 3 8.0 2 8 4.0 2 changes in process variability. It measures the spread. 4 10.0 2 9 3.5 3 LCL = D3R 5 9.5 3 10 4.0 2 © Copyright 2013 Anupam Kumar 13 14 UCL = D4R © Copyright 2013 Anupam Kumar Quality Control Chart for Attribute Fraction Defect Chart or P Chart • These are used when the quality is defined in • The object of this chart is to control the proportion of defective according to specified attribute. terms of product attributes which may not be • Even when the attribute standard is not specified, it is inferred measured. by finding the average fraction defectives. • Types of Quality Control Chart for Attributes • This chart is used to control the general quality of the component parts. are: • It checks whether the fluctuations in the product quality are – Fraction Defect Chart or p – defective chart due to chance causes or some assignable reasons. – Number Defective or np- chart UCL / LCL = p +/- 3 ((p*q) /n )1/2; where – Number Defective per unit chart or c chart. p is average fraction defectives, q is average fraction non – defectives (q = 1-p), and © Copyright 2013 Anupam Kumar 15 n is number of item in each sample. 16 © Copyright 2013 Anupam Kumar Illustration 3 Illustration 3 (Soln.) • A quality controller examines a sample of 1000 items • Average probability = p = 236.7/1000 = 0.2367 each and finds the number of defectives as follows. • q = 1-p = 1-0.2367 = 0.7633 Draw the control chart for the sample. • n is the number of items in each sample = 1000 Sample Number Sample Number Sample Number Sample Number No. of No. of No. of No. of • Control limits = p +/- 3 ((p*q) /n )1/2 defectives defectives defectives defectives 1 215 6 240 11 285 16 190 • UCL = 0.2367 + 3*((0.2367*0.7633)/1000)^0.5 2 227 7 218 12 287 17 215 ⇒UCL = 0.2367+3*0.013441 = 0.2367+0.040324 3 220 8 195 13 312 18 210 ⇒UCL = 0.277024 4 180 9 280 14 195 19 215 5 320 10 310 15 180 20 240 • LCL = 0.2367 - 3*((0.2367*0.7633)/1000)^0.5 Total 4734 ⇒LCL = 0.2367+3*0.013441 = 0.2367+0.040324 Average Solution 236.7 ⇒LCL = 0.196376 © Copyright 2013 Anupam Kumar 17 © Copyright 2013 Anupam Kumar 18 © Copyright 2013 Anupam Kumar 2
  • 3. Number Defectives or np Chart Illustration 4 • In a manufacturing process, the number of defectives found in the • This chart is similar to the p chart inspection of 15 lots of 400 items each is given below. Determine the trial • It is used when the calculation for proportion control limits for np chart and state whether the process is in control or not. of fraction of defectives is complex. – What will be new value of mean fraction defective if some obvious points outside the control limits are eliminated? • The limits for the np chart is as follows. – What will be the corresponding upper and lower control limits and examine whether the process is still in control or not? UCL / LCL = np +/- 3*(np*(1-p))1/2 ; Solution Lot No. No. of Lot No. No. of Lot No. No. of Where p = Σnp / Σn defectives defectives defectives 1 2 6 0 11 6 2 5 7 1 12 0 3 0 8 0 13 3 4 14 9 18 14 0 © Copyright 2013 Anupam Kumar 19 5 3 10 8 © Copyright 2013 Anupam Kumar 15 6 20 Number of Defects Per Unit Chart Illustration 4 (Soln.) or c – chart • Average probability = p = 66/(15*400) = 0.011 • In case the sample is such that there are no uniform • q = 1-p = 1-0.011 = 0.989 number of items: • n = number of item in each sample = 400 – the limits on the control chart cannot be laid out as n would vary and thus • Control limits = np +/- 3*(np*(1-p))1/2 • Fraction defective chart cannot be used • UCL = (400*0.011) + 3*((400*0.011*0.989)^0.5) • Number defective chart also cannot be used. ⇒UCL = 4.4 + 3*2.086049 = 4.4 + 6.258147 = 10.658147 • In such a case the number of defectives per unit is • LCL = (400*0.011) - 3*((400*0.011*0.989)^0.5) plotted. ⇒LCL = 4.4 - 3*2.086049 = 4.4 - 6.258147 = -1.858147 • It controls the number of defects observed per unit ⇒LCL = 0 or per sample. UCL / LCL = C +/- 3 C1/2 , © Copyright 2013 Anupam Kumar 21 © Copyright 2013 Anupam Kumar 22 Where C is average no. of defect per sample Illustrations • The following figures show the number of defective items discovered in 10 samples taken on 10 consecutive days in a month. Date 1 2 3 4 5 6 7 8 9 10 Six Sigma … Nos. Defective 1 2 4 16 4 3 5 20 16 4 • 10 castings were inspected in order to locate defects in them. Every casting was found to contain certain number of defects as given below. Plot the relevant chart and draw your conclusions. Date 1 2 3 4 5 6 7 8 9 10 Nos. Defective 2 4 1 5 5 6 3 4 0 7 23 © Copyright 2013 Anupam Kumar 24 © Copyright 2013 Anupam Kumar © Copyright 2013 Anupam Kumar 3