Statistical Process Control

Statistical Process Control jl.sixsigma@gmail.com 716-830-4620
John Lupienski SSAT John Lupienski

Control Chart Basics - Objectives
At the end of this section, you will be able to: Input data into all types of SPC Charts Recognize the various types of statistical process control (SPC) Select the appropriate MT- 17 control chart for the given data Interpret control charts and recommend the appropriate action to be taken on a process Understand the difference between “Common cause” vs “Special cause” variation Notes to the Instructor: This module is a follow-up to the monitoring plan element of control plans. Some of this will be a review from the Measure phase, but many of the charts will be new.

What is Common Cause Variation?
Variation which is natural in the process as it is designed to operate Seen as the normal, expected, usually random, variation that is inherent in the process factors: Materials, methods, people, machines, environment, and measurement If only common cause variation is present The process is stable and therefore, predictable and controllable Notes to the Instructor: No specific response is necessary when all the observed variation is common cause variation, unless the process capability is undesirable.

What is Special Cause Variation?
Variation which is not designed to be part of the process The result of undue or extraordinary influence of factors Seen as abnormal, unexpected, unusual indications – may appear as non-random (patterned) variation If special cause variation is present, The process is unstable or out of statistical control and the causes must be investigated Notes to the Instructor: Special cause variation deserves a response of some sort because it is an unusual event. It may be a good event to be repeated or a bad event to be avoided. In either case the unusual event should be recognized and the cost-benefit of a response should be weighed. Recall the acronym; AIR it out: Acknowledge special causes Document the occurrence Investigate the cause Document the findings Incorporate beneficial causes Take action to replicate causes of desired performance Remove detrimental causes Take action to eliminate causes of undesired performance

Key Elements of a Control Chart
Courier Delivery Time 3  Upper Control Limit Data Plotted over Time Notes to the Instructor: Data: Initially simply plotted over time on a run chart and connected by lines Sufficient quantity (often approximated as n=30) required Regular periodicity (data collected at logical process intervals) Sound rationale used (process knowledge as to representative samples) Centerline: Calculated from the representative data Usually the average of the data Control Limits: Statistically derived from the representative data Equivalent of 3 standard deviations set above / below the average 3  Lower Control Limit Centerline

Identify Signals of Special Cause Variation: 3 Primary Rules
One point outside the 3-sigma limit 7 points in a row on one side of centerline 6 points in a row consistently increasing or decreasing Point outside the limit 10 20 30 40 50 60 70 80 90 Run Trend Notes to the Instructor: The validity of rule 2 requires a normal distribution. Rules 1 and 3 are valid regardless of the shape of the distribution.

Identify Signals of Special Cause Variation: 3 Secondary Rules
Hugging Approaching Control Limits Cycle 15 points in a row hugging centerline 2 out of 3 points in a row beyond 2 std. deviations from centerline Cycles (seasonal, time zone close-out) Notes to the Instructor: Rule 4 is almost impossible to achieve in a non-normal distribution; it is not valid for leptokurtic distributions. Rule 5 is not valid if the distribution is skewed. Rule 6 is valid regardless of the shape of the distribution.

Developing a Control Chart
Select the appropriate critical variable to chart Determine data type, frequency, quantity (sub-group size), and rationale for sampling plan Select the type of control chart to use – See Roadmap Gather and record sufficient data – in the absence of anything unusual Calculate the centerline and control limits using Minitab Interpret the chart – take appropriate action Notes to the Instructor: Based on your monitoring plan, determine what chart will be necessary. Use Minitab to create the charts.

Recalculating Control Limits
Recalculated control limits when: Trial limits (calculated with less than sufficient data) are replaced Signals of special cause variation (in original data only) are explained and removed The process has changed, and the old limits no longer characterize the new process Notes to the Instructor: Be aware of the occasional need to recalculate control limits. It is necessary and recommended only under the prescribed conditions above.

Discrete Data (Attribute) Continuous Data (Variables)
Control Chart Roadmap I, mR Chart np Chart Constant sample size Variable sample size Defectives Defects Discrete Data (Attribute) Continuous Data (Variables) Control Charts Subgroup size = 2 to 10 p Chart c Chart u Chart Subgroup Size > 10 Empirical Rule Subgroup size = 1 Poisson Distribution Binomial Distribution x, R Chart x, s Chart Notes to the Instructor: The x-bar,s chart is preferable if sample size is not an issue. Standard deviation is a better estimate of variability. Also, the larger the sample size, the more robust the chart is to deviations from normality (central limit theorem – as sample size increases, the distribution of the averages will be more normal, regardless of the shape of the parent distribution).

Charts for Continuous Data
X Bar & R Chart: Average and Range Chart Multiple observations taken with regular periodicity Each period results in a sub-group of observations The Average chart tracks the sub-group average from the process and the range chart tracks the range within each sub-group I-MR Chart: Individual Values and Moving Ranges Chart Used for single observations of continuous data Either infrequent, or insufficient for sub-grouping The Individuals chart tracks individual measurements from the process and the range chart tracks the moving range between individual measurements _ Notes to the Instructor: The x-bar, R chart plots averages of subgroups. It is applicable where logical subgroups exist in chronological order. It eliminates the need to collect every data point but often requires a sampling plan to execute. Used effectively, it can effectively demonstrate the difference between short term (within group) and long term (between group) variation. This provides an estimate of the entitlement (within group) level of variation – if the variation between the groups could be controlled.

Average and Range Chart Calculations
Calculate the statistics ( , R Chart) Centerline Control Limits Where: x: Average of the subgroup averages, it becomes the centerline of the control chart Xi: Average of each subgroup k: Number of subgroups Ri : Range of each subgroup (Maximum observation – Minimum observation) R: The average range of the subgroups, the centerline on the range chart UCLx: Upper control limit on average chart LCLx: Lower control limit on average chart UCLR: Upper control limit on range chart LCLR : Lower control limit range chart A2, D3, D4: Constants that vary according to the subgroup sample size Notes to the Instructor: The control limit calculations are based on Shewhart constants to approximate 3 standard deviations from the mean.

Average & Standard Deviation Chart Calculations
Calculate the statistics ( , s Chart) Centerline Control Limits Where: x : Average of the subgroup averages, it becomes the centerline of the average chart xi : Average of each subgroup k : Number of subgroups si : Standard deviation of each subgroup s : Average of the subgroup standard deviations, the centerline on the s chart UCLx: Upper control limit on average chart LCLx: Lower control limit on average chart UCLs: Upper control limit on s chart LCLs : Lower control limit s chart A3, B3, B4: Constants that vary according to the subgroup sample size Notes to the Instructor: Rigorous understanding of the formulas or derivation of the constants is not important as theses calculations are all performed by Minitab.

Constants Table X-bar Chart for sigma R Chart Constants S Chart Constants Constants estimate Sample Size = n A2 A3 d2 D3 D4 B3 B4 2 1.880 2.659 1.128 -- 3.267 3 1.023 1.954 1.693 2.574 2.568 4 0.729 1.628 2.059 2.282 2.266 5 0.577 1.427 2.326 2.114 2.089 6 0.483 1.287 2.534 2.004 0.030 1.970 7 0.419 1.182 2.704 0.076 1.924 0.118 1.882 8 0.373 1.099 2.847 0.136 1.864 0.185 1.815 9 0.337 1.032 2.970 0.184 1.816 0.239 1.761 10 0.308 0.975 3.078 0.223 1.777 0.284 1.716 John Lupienski

Basic Control charts - dialogs & menus
Menus are better organized by category… Let go into the Control Chart Menus and look at the various types of information available John Lupienski

Basic Control chart - X Bar & R Chart
Open file: Camshaft.mtw Stats> Control Charts>Var. charts for SubGrp> XBar R chart John Lupienski

Basic Control chart - dialogs & menus Lets now review Scale & Option Do you have C5 Time Stamp?
John Lupienski

X Bar & R Control Chart file Camshaft
X Bar & R Control Chart file Camshaft.mtw subgroup size = 5 & time dated John Lupienski

Control Chart Tests stats>control charts>var
Control Chart Tests stats>control charts>var. charts for sub grp>X Bar R> Options>Tests John Lupienski

Control Charts: Update Define Tests
Select Tools>Options>Control Charts and Quality Tools>Define Tests Change Test 2 from 9 to 7 John Lupienski

Control Chart Tests Select Tools>Options>Control Charts and Quality Tools> Tests to Perform Select Perform all eight tests John Lupienski

X Bar & R Control Chart for Camshaft
X Bar & R Control Chart for Camshaft.mtw data subgroup size = 5, with time dated and all tests clicked John Lupienski

Example- I-MR Chart for the Camshaft
Example- I-MR Chart for the Camshaft.mtw same data Try to make this chart check options and tests John Lupienski

What Else? X Bar, R Chart The chart is best used when
Data are available in sufficient quantity and frequency for small samples to be pulled with regularity Process knowledge is the basis of rationale for meaningful interpretation Ideally, 30 subgroups (k = 30) have been collected Trial limits can be calculated with less than k = 30 Examples of uses (sampled hourly, by shift, etc.) Time to process transactions, response times Inquiry handling times, time from request to receipt Dollar metrics – value, sales, costs, usages, variances Variable dimensions or environmental readings or conditions What Else? Notes to the Instructor: The x-bar,R chart is useful when the users (operators) are unfamiliar with the concept of standard deviation or when the data points must be calculated by hand.

Summary of Variable Charts
Chart Type Purpose Subgroup Notes X & R Average X Range (R) Monitor the average of a characteristic over time Monitor the variability of a characteristic / time n > 1 2 n 10 Use for sets of measurements Standard Deviation (S) M onitor the variability Used with the X charts when the sample size is > 10 Increased sensitivity due to increased sample size. Use when tight control is necessary or sampling size cost is a factor Individual & Moving Range (I - MR) of an individual None For example use for sales, costs, variances, customer satisfaction score, total Exponentially Weighted Moving Average (EWMA) Monitor small shifts in the process 1 Smoothes data to emphasize trends. Uses weights to emphasize importance of recent data Cumulative Sum (CUSUM) Used for specialized applications (similar to EWMA) Same sensitivity as EWMA. John Lupienski

Questions? John Lupienski

Discrete Control Chart Roadmap
I, mR Chart np Chart Constant sample size Variable sample size Defectives Defects Discrete Data (Attribute) Continuous Data (Variables) Control Charts Subgroup size = 2 to 10 p Chart c Chart u Chart Subgroup Size > 10 Empirical Rule Subgroup size = 1 Poisson Distribution Binomial Distribution x, R Chart x, s Chart Notes to the Instructor: The x-bar,s chart is preferable if sample size is not an issue. Standard deviation is a better estimate of variability. Also, the larger the sample size, the more robust the chart is to deviations from normality (central limit theorem – as sample size increases, the distribution of the averages will be more normal, regardless of the shape of the parent distribution).

Charts for Discrete Data
np Chart: number of defective units Must have equal subgroup size p Chart: proportion of defective units Can have unequal subgroup size c Chart: number of defects u Chart: number of defects per unit Notes to the Instructor: We will not cover the p-chart as it was addressed in the Measure phase. It is included in the material as a reference.

np & p Control Chart Roadmap

np Chart The np chart will record, and plot, the count of the number of defective (non-conforming) units As with all discrete charts, larger subgroup size (n = 50 or greater) is desirable – though not always possible As with all discrete charts, the possibility exists that the computed lower control limit would be negative In that case the chart will simply bottom out at zero Only an upper control limit will be set and interpreted Notes to the Instructor: Recall that the p-chart plotted ratios. The np chart plots counts of defective items. Because it is a count, not a ratio, it is logical that the sample size must be constant.

np Chart Examples What Else? Examples of applications of an np chart:
The number of unresolved trouble calls out of 50 sampled each day at the help desk The number of trades with incorrect broker codes, out of 100 sampled each day The number of incomplete credit applications, out of 60 sampled each week The number of accounts not balanced out of 200 that are tracked each billing period What Else? Notes to the Instructor: In each case, a constant sample size is chosen. This requires a sampling plan to be administered.

Example np Chart You work in a toy manufacturing company and your job is to inspect the number of defective bicycle tires. You inspect 200 samples in each lot and then decide to create an NP chart to monitor the number of defectives. To make the np chart easier to present at the next staff meeting, you decide to split the chart by every 10 inspection lots. 1 Open the worksheet TOYS.mtw. 2 Choose Stat > Control Charts > Attributes Charts > NP. 3 In Variables, enter Rejects. 4 In Subgroup sizes, enter Inspected. 5 Click NP Chart Options, then click the Display tab. 6 Under Split chart into a series of segments for display purposes, choose Each segment contains __ subgroups and enter10. 7 Click OK in each dialog box. Review Session window output, What is your interpretations and conclusion? Notes to the Instructor: Run through this example in Minitab for the class.

Example np Chart for Rejects for Toys.mtw
Interpreting the results Inspection lots 9 and 20 fall above the upper control limit, indicating that special causes may have affected the number of defectives for these lots. You should investigate what special causes may have influenced the out-of-control number of bicycle tire defectives for inspection lots 9 and 20. Notes to the Instructor: Column interpretation:

p Chart The p chart will record the number of defectives and the sample size - but calculate and plot the proportion of defectives As with all discrete charts, larger subgroup size (n=>50) is desirable – though not always possible As with all discrete charts, the possibility exists that the computed lower control limit would be negative In that case the chart will simply bottom out at zero Only an upper control limit will be set and interpreted Notes to the Instructor: Skip – reference only

p Chart – Variable Subgroup Size
The control limits vary depending on the subgroup size UCL LCL Notes to the Instructor: Skip – reference only

If Then use Where p Chart – 25% Rule
In other words, if the sample size of any given sample (ni) is within 25% of the mean of all your sample sizes (n), then just use the sample size mean (n) to calculate your control limits. Notes to the Instructor: Skip – reference only

p Chart – 25% Rule The control limits can be constant when the 25% rule is used Notes to the Instructor: Skip – reference only

P Chart Example - with unequal subgroup sizes
Suppose you work in a plant that manufactures picture tubes for televisions. For each lot, you pull some of the tubes and do a visual inspection. If a tube has scratches on the inside, you reject it. If a lot has too many rejects, you do a 100% inspection on that lot. A P chart can define when you need to inspect the whole lot. 1 Open the worksheet EXH_QC.mtw. 2 Choose Stat > Control Charts >Attributes Charts > P. 3 In Variables, enter Rejects. 4 In Subgroup sizes, enter Sampled. Click OK. Review Session window output and P Chart Graph. What is your interpretation and conclusion? Notes to the Instructor:

Example : P Chart for Rejects from Exh_qc.mtw Data
Notes to the Instructor: Interpreting the results Sample 6 is outside the upper control limit. Consider inspecting the lot.

C & U Control Chart Roadmap

c Chart Used to track number of defects given a constant area of opportunity (sample size) The c chart is based on the Poisson distribution which stipulates that the potential for a defect is essentially infinite, while the relative occurrence is rare given the opportunity The number of defects will be plotted and recorded on the c chart Notes to the Instructor: The c chart is like the np chart for defectives. Use when a constant sample size is indicated.

c Chart Calculations Calculate the C Chart statistics Control Limits
Where: ci: Number of nonconformities for any given sample of units c: Average number of nonconformities per unit k: Number of subgroups LCLc: Lower control limit on u chart. UCLc: Upper control limit on u chart. Centerline Control Limits Notes to the Instructor: Minitab makes these calculations for us. The control limit calculations are fairly straight-forward.

Example c Chart Data Suppose you work for a linen manufacturer. Each 100 square yards of fabric can contain a certain number of blemishes before it is rejected. You want to track the number of blemishes per 100 square yards over a period of several days, to see if your process is stable & normal. You also want the control chart to show control limits at 2, and 3 standard deviations above and below the center line. 1 Open the worksheet EXH_QC.MTW. 2 Choose Stat > Control Charts > Attributes Charts > C. 3 In Variables, enter Blemish. 4 Click C Chart Options, then click the S Limits tab. 5 Under Display control limits at, enter in These multiples of the standard deviation. 6 Under Place bounds on control limits, check Lower standard deviation limit bound and enter 0. 7 Click OK in each dialog box. Graph window output

Example c Chart for Blemishes/100 sq. ft.
Are the Blemishes stable and in control? Notes to the Instructor: Process is stable even within 2 sigma limits. Interpreting the results Because the points fall in a random pattern, within the bounds of the 3s and 2s control limits, you conclude the process is behaving predictably and is in control.

u Chart The u chart is an attribute control chart that is used with non-conformities or defects per unit The u chart is based on the Poisson distribution Like the p chart, the u chart is a proportion and can handle a varying subgroup size; the u chart is to the c chart as the p chart is to the np chart The u chart is typically applied to more complex processes such as continuous processes Notes to the Instructor: Of course in a continuous process the definition of a “unit” is arbitrary. The assumption is that any single unit of output has nearly infinite opportunities for failure. In the transactional world, while it is not continuous, the process of reviewing contracts would be a good example of a u chart opportunity.

u Chart The u chart is an attribute control chart that is used with nonconformities or defects per unit Based on the Poisson distribution Used for unequal sample sizes Like the p chart, the u chart is a proportion and can handle a varying subgroup size; the u chart is to the c chart as the p chart is to the np chart Typically applied to more complex processes such as continuous processes Notes to the Instructor: Of course in a continuous process the definition of a “unit” is arbitrary. The assumption is that any single unit of output has nearly infinite opportunities for failure. In the transactional world, while it is not continuous, the process of reviewing contracts would be a good example of a u chart opportunity.

u Chart Calculations Calculate the u Chart statistics Control Limits
Where: u: Average number of nonconformities per unit. ni: Number inspected in each subgroup LCLu: Lower control limit on u chart. UCLu: Upper control limit on u chart. Centerline Control Limits Notes to the Instructor: These calculations are performed by Minitab. Since the control limits are a function of sample size, they will vary with sample size like the p chart.

Example u Chart with variable Subgroup size
The number of defects will be recorded, but due to the varying sample size, a ratio of defects per unit must be calculated and plotted on the u chart As production manager of a toy manufacturing company, you want to monitor the number of defects per unit of motorized toy cars. You inspect 20 units of toys and create a U chart to examine the number of defects in each unit of toys. You want the U chart to feature straight control limits, so you fix a subgroup size of 100 (the average number of toy cars sampled). 1 Open the worksheet TOYS.MTW. 2 Choose Stat > Control Charts > Attributes Charts > U. 3 In Variables, enter Defects. In Subgroup sizes, enter Sample. 5 Look at the chart. Click U Chart Options, then click the S Limits tab. 6 Under When subgroup sizes are unequal, calculate control limits, Recalculate Assuming all subgroups have size then enter 100. 7 Click OK in each dialog box. See session window for OOC pts. Notes to the Instructor: The u chart plots dpu.

u Chart – Toy defects with Variable Subgroup Size
Notes to the Instructor: Note that the control limits vary based on size of the subgroup. Ask the class, is this process in control? The answer is no. Now under u chart option click Tests tab and click all tests. What do you find now?

u Chart –Toy defects with the 25% Rule of Subgroup size Average n=100
Interpreting the results Units 5 and 6 are still above the upper control limit line, indicating that special causes may have affected the number of defects in these units. You should investigate what special causes may have influenced the out-of-control toy car defects for these units. After clicking all tests you find a good trend that should also be investigated-WHY Notes to the Instructor: Notice that Minitab flags the rule violation when the control limits are flat.

u Chart –Toy defects with Modified Stage limits
Now let’s assume after point # 6 we found the root cause and fixed the problem and you want to compute new Limits starting with point # 7. We must first go to worksheet: Toys and create a new column “C8 Variable” and set the first 6 rows equal to “1” and the remaining “14” rows down to row 20 equal to “2”. Under “U” Chart Options click on “Stages” then enter “ C8 Variable”. Now look at the chart and what conclusions can you draw? John Lupienski

Summary of Attribute Charts
John Lupienski

Control Charts Summary
At this point, you should be able to: Recognize the elements of statistical process control (SPC) Understand the basis of process stability studies Select the appropriate control chart for a given process metric Interpret control charts and recommend the appropriate action to be taken on a process Notes to the Instructor: The intent of this module was to give an overview of the charts not covered in the Measure phase. It should equip the Black Belts with a complete toolbox as they develop their monitoring plan.

Questions? John Lupienski

Statistical Process Control

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