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Statistical Process Control

Variation

Causes of variation

• No process can be variation free. There will be some sort of variation in a process. Major causes of variation in a process can be classified into two categories.

Common (chance) causes of variation

• Inherent or natural in a process • Small in magnitude • Difficult to identify or remove from the process

Special causes of variation

• Variation due to some special causes • Large in magnitude • Easy to identify and remove from the process



Stable process

• A process is said to be a stable process, if there exists no special causes of variation.

• A stable process runs under common causes only. • For a stable process having normal distribution, the variation will be

Mean ± 3 X Standard Deviation.

Components of a control chart

• Lower Control Limit (LCL) • Center Line (CL) • Upper Control Limit (UCL)

• A process is said to be stable or in control, if the values lie between UCL and LCL.

• For a normally distributed data : • UCL = Mean + 3 X Standard Deviation • CL = Mean • LCL = Mean – 3 X Standard Deviation

• For non-normal continuous data, use Central Limit Theorem (CLT).



Rational sub grouping

• It is the process of creating subgroups. Rational subgroups are subsets of data that are categorized under some particular factors such as timeframe, stratifying factors, etc. Rational subgroups comprise units that are basically produced under the same conditions or produced consecutively.

• In rational sub grouping, the average and range for each subgroup is calculated separately for the purpose of charting them on the control charts. Rational sub grouping is basically used to identify and distinguish special causes of variation from common causes of variation.

• In rational sub grouping, items within a subgroup should be taken rationally and not randomly.



Types of control charts (Variables control chart)

unpaired chart paired chart

Data

Continuous or variables data

Sample size of 10 or

above

Sample size of 2 to 9 Sample size of 1

Non-normal

data

Normal data

Run chart I-MR chart

Averages and

ranges (X-bar and R)

control charts

Averages and sigma

(X-bar and S) control

charts



Types of control charts (Attributes control chart)

Data

Attributes data

Count of defectives Count of defects

Fixed

subgroup

size

Variable

subgroup

size

U-chart C-chart

Variable

subgroup

size

Fixed

subgroup

size

P-chart Np-chart



X-bar and R chart

• Averages and ranges control charts are used to analyze the central tendency and the dispersion of a process.

• Sampling should be rational not random. • The average and range of each subgroup is calculated separately and

plotted on the control chart. To analyze the variation patterns between the subgroups, the statistics of each subgroup is compared to the control limits.

Methodology

• Collect data and create rational sub grouping • Calculate the subgroup mean and ranges • Calculate control limits for R chart

• CL= R-bar = sum of all range (R) values / total number of subgroups

• UCL = D4R-bar • LCL = D3R-bar



• Calculate control limits for X-bar chart • CL = Mean = X-double bar = sum of all means / total number of

values • UCL = X-double bar + A2R-bar • LCL = X-double bar – A2R-bar

• Example: Construct a control chart for mean and range for the following data. Comment on whether the production seems to be under control or not.

42 42 19 36 42 51 60 18 15 69 64 61

65 45 24 54 51 74 60 20 30 10 9

90 78

75 68 80 69 57 75 72 27 39 11 3

93 94

78 72 81 77 59 78 95 42 62 11 8

10 9

10 9

87 90 81 84 78 13 2

13 8

60 84 15 3

11 2

13 6



Solution:

Subgroup No

Sample Observation (Subgroup size: 5)

Sample Mean X-bar)

Sample Range (R)

1 42 65 75 78 87 69.4 45

2 42 45 68 72 90 63.4 48

3 19 24 80 81 81 57 62

4 36 54 69 77 84 64 48

5 42 51 57 59 78 57.4 36

6 51 74 75 78 132 82 81

7 60 60 72 95 138 85 78

8 18 20 27 42 60 33.4 42

9 15 30 39 62 84 46 69

10 69 109 113 118 153 112.4 84

11 64 90 93 109 112 93.6 48

12 61 78 94 109 136 95.6 75

Total 859.2 716

No of Sub groups

• From the above table, we get X-double bar = 859.2 / 12 = 71.6, R-bar = 59.67

For, n = 5 we have A2 = 0.577, D3 = 0, D4 = 2.114

X chart: LCL averages = 71.6 – 0.577*59.67 = 37.17

UCL averages = 71.6 + 0.577*59.67 = 106.03

R chart: LCL ranges = 0

UCL ranges = 2.114*59.67 = 126.14



• X-bar and R chart in Minitab: Enter the data in the Minitab worksheet in a single column and the data entry should be in order (i.e., according to the sample numbers and observations given).



• Select Stat > Control Charts > Variables Charts for Subgroups > X-bar-R.

• Enter the subgroup size as 5. • From the X-bar chart, the process is out of control as the points

corresponding to the 8th and 10th samples lie outside the control limit. • Since all the sample points fall within the control limits, R chart shows

that the process is in control. • Although R-chart depicts control, the process can’t be regarded as

being in statistical control as X-bar chart shows lack of control.



X bar & S Chart

Averages and standard deviation (Sigma) control charts

• The averages and standard deviation control charts and the averages and ranges control charts are conceptually very similar, except for the fact that the subgroup standard deviation measures dispersion and not range.

• Generally, the averages and ranges charts are recommended for the subgroup sizes below ten, as the calculation and understanding of this tool is much easier for the common people. But people who are acquainted with statistics prefer the averages and standard deviation control charts for all subgroups, as these charts are much more efficient than the averages and ranges control charts with respect to the subgroup sizes larger than 10.

• The averages and sigma method is more accurate than averages and range method (as standard deviation is a better measure of dispersion than range).

• Calculate control limits for S chart • CL= Mean = S-bar = sum of subgroup sigmas / number of

subgroups • UCL = B4S-bar • LCL = B3S-bar

• Calculate control limits for X-bar chart • CL = Mean = X-double bar = sum of all means / total number of

values • UCL = X-double bar + A3S-bar • LCL = X-double bar – A3S-bar



• Remark: In small samples, the standard deviation (s) and range (R) are likely to fluctuate together, i.e., if s is small (large) R is likely to be small (large). However for large samples, a single extreme observation will have a significantly large effect on range while its effect on standard deviation will be comparatively much less. Hence, for analyzing or controlling variability, if we use small samples, the range may be used as a substitute for standard deviation with little loss in efficiency. Since range is almost as efficient as standard deviation in small samples, it is usually preferred to standard deviation in quality control analysis because of its ease of calculation.

• Note that the difference between the control limits for X-bar chart in averages and sigma method is less than the difference between the control limits for X-bar chart in averages and range method.



Individual measurements control chart (Individual X and Moving range chart)

• Individual X and Moving range (I-MR) charts are used by analysts to analyze the central tendency of a process over a prescribed period of time.

• Individual control charts are mostly used to gauge those batch processes which cannot be analyzed using standard averages charts, as the control limits on them would be too close to one another while analyzing the batch processes, where in-batch variations are too minute compared to between-batch variations. So, individual control charts are resorted to when it is deemed that the use of averages for process control is not viable.

• Example: I-MR chart. • The weekly debit outstanding (in US $ 100) is given in the table below:

Day Data Day Data

1 8.1 11 8.1

2 9.5 12 8.2

3 7.2 13 8.3

4 6 14 9.1

5 9.1 15 8

6 5.8 16 7.3

7 8.1 17 8.4

8 7.2 18 4.5

9 5.1 19 7.1

10 8 20 5.1


Statistical Process Control 9.5-8.1 = 1.4

• Step 1: Calculate MR

• Step 2: Calculate control limits for MR chart • Center Line (CL) = MR-bar • = Sum of all MR values / total no. of values • = 32/19 = 1.684211 • UCL = D4 *MR-bar = 3.276 * 1.684211 = 5.517474 • LCL = D3 *MR-bar = 0 • [for n=2 – as MR is calc based on 2 values,

D4 = 3.276 & D3 = 0] • Step 3: Calculate control limits for individual X chart • CL = X-bar = 148.2/20 = 7.41 • UCL = X-bar + E2 *MR-bar = 7.41 + 2.66 * 1.684211 • = 11.89 • LCL = X-bar – E2 *MR-bar = 7.41 – 2.66 * 1.644211 • = 2.93 • Step 4: Plot the control chart by using Minitab • Select Stat > Control Charts > Variables Charts for

Individuals > I-MR

Day Data MR

1 8.1

2 9.5 1.4

3 7.2 2.3

4 6 1.2

5 9.1 3.1

6 5.8 3.3

7 8.1 2.3

8 7.2 0.9

9 5.1 2.1

10 8 2.9

11 8.1 0.1

12 8.2 0.1

13 8.3 0.1

14 9.1 0.8

15 8 1.1

16 7.3 0.7

17 8.4 1.1

18 4.5 3.9

19 7.1 2.6

20 5.1 2

Total 148.2 32



• Since all the points are within the control limits for both the control charts, we conclude that the process is in control.



Attribute chart (p chart)

• Control chart for proportions defectives (p-chart) – when sub group size is not fixed.

Calculations

• CL = Mean = p-bar • UCL = Mean + 3 X SD = p-bar + 3 X √ p-bar(1 – p-bar) / ni • LCL = Mean - 3 X SD = p-bar - 3 X √ p-bar(1 – p-bar) / ni • p-bar = sum of defectives / total number inspected



Example of a proportions defectives chart:

Daily inspection results for the purchase order (PO) processing are given below in the table. Construct a control chart to ensure that the defectives are kept under control day by day.

Day No. Inspected No. of Defectives Day No. Inspected No. of Defectives 1 171 31 11 181 38 2 167 6 12 115 33 3 170 8 13 165 26 4 135 13 14 189 15 5 137 26 15 165 16 6 170 30 16 170 35 7 45 3 17 175 12

8 155 11 18 167 6 9 195 30 19 141 50

10 180 36 20 159 26



• Calculation of control limits • P-bar = 451 / 3152 = 0.143083756

Day No. Inspected No. of Defectives p LCL UCL 1 171 31 0.18128655 0.062751946 0.223415567 2 167 6 0.035928144 0.061795581 0.224371931 3 170 8 0.047058824 0.062516023 0.22365149 4 135 13 0.096296296 0.052673287 0.233494226 5 137 26 0.189781022 0.053335643 0.232831869 6 170 30 0.176470588 0.062516023 0.22365149 7 45 3 0.066666667 -0.01351177 0.299679283 8 155 11 0.070967742 0.058707596 0.227459917 9 195 30 0.153846154 0.067857699 0.218309814

10 180 36 0.2 0.064785993 0.22138152 11 181 38 0.209944751 0.065002585 0.221164928 12 115 33 0.286956522 0.045126483 0.24104103 13 165 26 0.157575758 0.06130441 0.224863103 14 189 15 0.079365079 0.066672963 0.219494549 15 165 16 0.096969697 0.06130441 0.224863103 16 170 35 0.205882353 0.062516023 0.22365149 17 175 12 0.068571429 0.063675331 0.222492182 18 167 6 0.035928144 0.061795581 0.224371931 19 141 50 0.354609929 0.054617825 0.231549687 20 159 26 0.163522013 0.059775691 0.226391821

Total 3152 451



• Plot the control chart by using Minitab • Select Stat > Control Charts > Attributes Charts > P. • Select No. of Defectives as variables and No. Inspected as subgroup

sizes. • From the control chart, we can see that 5 points are out of control

limits, which indicates the presence of assignable causes. • Hence, the process is not in control.



Pointers for using proportions defectives charts

• If the variation in sample size is not too much, we can use average sample size (total number inspected divided by the number of subgroups). But in the case that we have discussed before, we can’t use average sample size as there is quite a large variation in sample sizes.

Control charts for count of defectives (np charts) - when sub group size is fixed

Calculations

• CL = Mean = n(p-bar) • UCL = Mean + 3 X SD = n(p-bar) + 3 X √ n(p-bar)(1 – p-bar) • LCL = Mean - 3 X SD = n(p-bar) - 3 X √ n(p-bar)(1 – p-bar)



Example of a np chart

• The following are the data on defectives in payment of life insurance claims. Devise a mechanism to ensure that the defectives are under control. Sample size is 300.

Sample Number No. of Defectives Sample Number No. of Defectives 1 3 11 6 2 6 12 9 3 4 13 5 4 6 14 6 5 20 15 7 6 2 16 4 7 6 17 5 8 7 18 7 9 3 19 5

10 0 20 0

• Calculation of control limits • Given n = 300 • np = 111 / 20 = 5.55 • p = 5.55 / 300 = 0.0185 • LCL = 5.55 – 3 * √(5.55*(1-0.0185)) = -1.451851541 • UCL = 5.55 + 3 * √(5.55*(1-0.0185)) = 12.55185154



• Plot the control chart by using Minitab. • Select Stat > Control Charts > Attributes Charts > np. • Select No. of Defectives as variables and enter 300 as subgroup

sizes. • From the control chart we can see that only one point is out of control

limits, which indicates the presence of assignable cause.



U Chart

Control charts for defects - when sample size is not constant (u charts)

Calculations

• CL = Mean = u-bar = sum of defects / total number inspected • UCL = Mean + 3 X SD = u-bar + 3 X √(u-bar / ni) • LCL = Mean - 3 X SD = u-bar - 3 X √(u-bar / ni)



• In terms of selecting the appropriate control chart among P and u charts for a particular given data, the center line equation for a control chart has to be analyzed. If the numerator and the denominator have similar units of measure, then a P chart has to be selected. But if the numerator and the denominator have dissimilar units of measure, the u chart is best suited.

Analysis of u charts

• Analysis of the u charts can get very complex if there are variations in sample size. But still, u charts do shed some light on the presence or absence of special causes of variation.

Example of a u chart

• Construct a suitable control mechanism to ensure that the errors in payoff process are under control.

Sample No.

No. of Payoffs inspected

No. of defects

Sample No.


No. of defects

1 40 45 11 52 55 2 40 40 12 52 74 3 40 33 13 52 43 4 40 43 14 52 61 5 40 62 15 40 43 6 52 79 16 40 32 7 52 60 17 40 45 8 52 50 18 40 33 9 52 73 19 40 50

10 52 54 20 52 28



• Calculation of control limits • U-bar = sum of defects / total number inspected = 1003 / 920 =

1.090217391

Sample No.


No. of defects

u LCL UCL

1 40 45 1.125 0.594940789 1.585493994 2 40 40 1 0.594940789 1.585493994 3 40 33 0.825 0.594940789 1.585493994 4 40 43 1.075 0.594940789 1.585493994 5 40 62 1.55 0.594940789 1.585493994 6 52 79 1.519230769 0.655831075 1.524603707 7 52 60 1.153846154 0.655831075 1.524603707 8 52 50 0.961538462 0.655831075 1.524603707 9 52 73 1.403846154 0.655831075 1.524603707

10 52 54 1.038461538 0.655831075 1.524603707 11 52 55 1.057692308 0.655831075 1.524603707 12 52 74 1.423076923 0.655831075 1.524603707 13 52 43 0.826923077 0.655831075 1.524603707 14 52 61 1.173076923 0.655831075 1.524603707 15 40 43 1.075 0.594940789 1.585493994 16 40 32 0.8 0.594940789 1.585493994 17 40 45 1.125 0.594940789 1.585493994 18 40 33 0.825 0.594940789 1.585493994 19 40 50 1.25 0.594940789 1.585493994 20 52 28 0.538461538 0.655831075 1.524603707

Total 920 1003



• Plot the u chart using Minitab. • Select Stat > Control Charts > Attributes Charts > u. • Select No. of defects as variables and No. of Payoffs inspected as

subgroup sizes in the Minitab dialog box. • A point is going below the LCL, which indicates the presence of the

assignable cause.



Control charts for defects - when sample size is constant (c charts)

• The c charts can be used to analyze any variable, where the sample size remains fixed and the count of how many times a particular event occurs is the appropriate performance measure. Through the analysis of the c charts, one can know whether the central tendency of a process has been influenced by a special cause of variation to produce unusually many or few occurrences over the prescribed period of time.

• Calculations • CL = Mean = c-bar = average number of defects • UCL = Mean + 3 X SD = c-bar + 3 X √c-bar • LCL = Mean - 3 X SD = c-bar - 3 X √c-bar



• Example of a c chart

• 100 machines are inspected every day for fatal errors. The data for the past 20 days is given below. Construct a suitable control chart to ensure that the fatal errors are under control.

Day No. of Fatal Errors Day No. of Fatal Errors 1 22 11 15 2 29 12 10 3 25 13 33 4 17 14 23 5 20 15 27 6 16 16 15 7 34 17 17 8 11 18 17 9 31 19 19

10 29 20 22

• Since the number of sample sizes remains fixed and the data given involves counting the number of defects (errors), we choose c chart for analysis.

• Calculation of control limits • Mean = c = 432 / 20 = 21.6, SD = √ c = 4.647580015 • UCL = Mean + 3*SD = 35. 54274005 • LCL = Mean - 3*SD = 7.657259954



• Plot the c chart by using Minitab. • Select Stat > Control Charts > Attributes Charts > c. • Select the No. of Fatal Errors as variable in Minitab’s dialog box. • Since all the points are within the control limits, we conclude that the

process is in statistical control.



Points to remember in the selection of a control chart:

To select an appropriate control chart for a given data, one has to decide whether the given data is continuous or discrete (attribute). If it is found that the given data is attribute, then one can decide which control chart to choose among n, np, c, and u charts depending on whether the given attributes data is of counting defects or defectives and whether the sample size of the given data is fixed or varies.

On the contrary, if it is found that the given data is continuous, then one can choose any of the following control charts: averages-ranges chart, individual measurements chart, run chart or average-sigma chart. In selecting any of the above charts, due consideration has to be given to sample size and distribution of the given data, i.e., whether the data are normally distributed or not.



Conclusion

• Special and common cause of variation • Rational sub grouping • Variable control charts (X-bar and R, X-bar and s, I-MR) • Attribute control charts (p, np, u, c)

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Six Sigma Black Belt Study Guides - PMTUTOR1)_Statistical_Process_Control.pdf · Six Sigma Black Belt – Study Guides . ... items within a subgroup should be taken ... • Since