# 10 CONTROL CHART CONTROL CHART

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1 Module 10 CONTOL CHT CONTOL CHT 1

2 What is a Control Chart? control chart is a statistical tool used to distinguish between variation in a process resulting from common causes and variation resulting from special causes It presents a graphic display of process stability or instability over time (Viewgraph 1) very process has variation Some variation may be the result of causes which are not normally present in the process This could be special cause variation Some variation is simply the result of numerous, ever-present differences in the process This is common cause variation Control Charts differentiate between these two types of variation One goal of using a Control Chart is to achieve and maintain process stability Process stability is defined as a state in which a process has displayed a certain degree of consistency in the past and is expected to continue to do so in the future This consistency is characterized by a stream of data falling within control limits based on plus or minus 3 standard deviations (3 sigma) of the centerline [ef 6, p 82] We will discuss methods for calculating 3 sigma limits later in this module NOT: Control limits represent the limits of variation that should be expected from a process in a state of statistical control When a process is in statistical control, any variation is the result of common causes that effect the entire production in a similar way Control limits should not be confused with specification limits, which represent the desired process performance Why should teams use Control Charts? stable process is one that is consistent over time with respect to the center and the spread of the data Control Charts help you monitor the behavior of your process to determine whether it is stable Like un Charts, they display data in the time sequence in which they occurred However, Control Charts are more efficient that un Charts in assessing and achieving process stability Your team will benefit from using a Control Chart when you want to (Viewgraph 2) Monitor process variation over time Differentiate between special cause and common cause variation ssess the effectiveness of changes to improve a process Communicate how a process performed during a specific period 2 CONTOL CHT

3 What Is a Control Chart? statistical tool used to distinguish between process variation resulting from common causes and variation resulting from special causes CONTOL CHT VIWGPH 1 Why Use Control Charts? Monitor process variation over time Differentiate between special cause and common cause variation ssess effectiveness of changes Communicate process performance CONTOL CHT VIWGPH 2 CONTOL CHT 3

4 What are the types of Control Charts? There are two main categories of Control Charts, those that display attribute data, and those that display variables data ttribute Data: This category of Control Chart displays data that result from counting the number of occurrences or items in a single category of similar items or occurrences These count data may be expressed as pass/fail, yes/no, or presence/absence of a defect Variables Data: This category of Control Chart displays values resulting from the measurement of a continuous variable xamples of variables data are elapsed time, temperature, and radiation dose While these two categories encompass a number of different types of Control Charts (Viewgraph 3), there are three types that will work for the majority of the data analysis cases you will encounter In this module, we will study the construction and application in these three types of Control Charts: X-Bar and Chart Individual X and Moving ange Chart for Variables Data Individual X and Moving ange Chart for ttribute Data Viewgraph 4 provides a decision tree to help you determine when to use these three types of Control Charts In this module, we will study only the Individual X and Moving ange Control Chart for handling attribute data, although there are several others that could be used, such as the np, p, c, and u charts These other charts require an understanding of probability distribution theory and specific control limit calculation formulas which will not be covered here To avoid the possibility of generating faulty results by improperly using these charts, we recommend that you stick with the Individual X and Moving ange chart for attribute data The following six types of charts will not be covered in this module: X-Bar and S Chart Median X and Chart c Chart u Chart p Chart np Chart 4 CONTOL CHT

5 What re the Control Chart Types? Chart types studied in this module: X-Bar and Chart Individual X and Moving ange Chart - For Variables Data - For ttribute Data Other Control Chart types: X-Bar and S Chart u Chart Median X and Chart p Chart c Chart np Chart CONTOL CHT VIWGPH 3 Control Chart Decision Tree re you charting attribute data? NO Data are variables data Is sample size equal to 1? YS Use Xm chart for variables data YS NO Use Xm chart for attribute data For sample size between 2 and 15, use X-Bar and Chart CONTOL CHT VIWGPH 4 CONTOL CHT 5

6 What are the elements of a Control Chart? ach Control Chart actually consists of two graphs, an upper and a lower, which are described below under plotting areas Control Chart is made up of eight elements The first three are identified in Viewgraphs 5; the other five in Viewgraph 6 1 Title The title briefly describes the information which is displayed 2 Legend This is information on how and when the data were collected 3 Data Collection Section The counts or measurements are recorded in the data collection section of the Control Chart prior to being graphed 4 Plotting reas Control Chart has two areas an upper graph and a lower graph where the data is plotted a The upper graph plots either the individual values, in the case of an Individual X and Moving ange chart, or the average (mean value) of the sample or subgroup in the case of an X-Bar and chart b The lower graph plots the moving range for Individual X and Moving ange charts, or the range of values found in the subgroups for X-Bar and charts 5 Vertical or Y-xis This axis reflects the magnitude of the data collected The Y-axis shows the scale of the measurement for variables data, or the count (frequency) or percentage of occurrence of an event for attribute data 6 Horizontal or X-xis This axis displays the chronological order in which the data were collected 7 Control Limits Control limits are set at a distance of 3 sigma above and 3 sigma below the centerline [ef 6, pp 60-61] They indicate variation from the centerline and are calculated by using the actual values plotted on the Control Chart graphs 8 Centerline This line is drawn at the average or mean value of all the plotted data The upper and lower graphs each have a separate centerline 6 CONTOL CHT

7 lements of a Control Chart 1 2 Title: Legend: Date M S U M N T S verage ange CONTOL CHT VIWGPH 5 lements of a Control Chart 5 4a b CONTOL CHT VIWGPH 6 CONTOL CHT 7

8 What are the steps for calculating and plotting an X-Bar and Control Chart for Variables Data? The X-Bar (arithmetic mean) and (range) Control Chart is used with variables data when subgroup or sample size is between 2 and 15 The steps for constructing this type of Control Chart are: Step 1 - Determine the data to be collected Decide what questions about the process you plan to answer efer to the Data Collection module for information on how this is done Step 2 - Collect and enter the data by subgroup subgroup is made up of variables data that represent a characteristic of a product produced by a process The sample size relates to how large the subgroups are nter the individual subgroup measurements in time sequence in the portion of the data collection section of the Control Chart labeled MSUMNTS (Viewgraph 7) STP 3 - Calculate and enter the average for each subgroup Use the formula below to calculate the average (mean) for each subgroup and enter it on the line labeled verage in the data collection section (Viewgraph 8) Where: x x i n x x 1 x 2 x 3 x n n The average of the measurements within each subgroup The individual measurements within a subgroup The number of measurements within a subgroup verage xample Subgroup X X X X X verage: X = = = CONTOL CHT

9 Constructing an X-Bar & Chart Step 2 - Collect and enter data by subgroup Title: Legend: Date 1 Feb 2 Feb 3 Feb 4 Feb 5 Feb 6 Feb 7 Feb 8 Feb 9 Feb M S U M N T S verage ange nter data by subgroup in time sequence V G CONTOL CHT VIWGPH 7 Constructing an X-Bar & Chart Step 3 - Calculate and enter subgroup averages Title: Legend: Date 1 Feb 2 Feb 3 Feb 4 Feb 5 Feb 6 Feb 7 Feb 8 Feb 9 Feb M S U M N T S verage ange V G nter the average for each subgroup CONTOL CHT VIWGPH 8 CONTOL CHT 9

10 Step 4 - Calculate and enter the range for each subgroup Use the following formula to calculate the range () for each subgroup nter the range for each subgroup on the line labeled ange in the data collection section (Viewgraph 9) NG (Largest Value in each Subgroup) (Smallest Value in each Subgroup) ange xample Subgroup X X X X X verage: ange: The rest of the steps are listed in Viewgraph 10 Step 5 - Calculate the grand mean of the subgroup s average The grand mean of the subgroup s average (X-Bar) becomes the centerline for the upper plot Where: x x k x x 1 x 2 x 3 x k k The grand mean of all the individual subgroup averages The average for each subgroup The number of subgroups Grand Mean xample x CONTOL CHT

11 Constructing an X-Bar & Chart Step 4 - Calculate and enter subgroup ranges Title: Legend: Date 1 Feb 2 Feb 3 Feb 4 Feb 5 Feb 6 Feb 7 Feb 8 Feb 9 Feb M S U M N T S V G verage ange nter the range for each subgroup CONTOL CHT VIWGPH 9 Constructing an X-Bar & Chart Step 5 - Calculate grand mean Step 6 - Calculate average of subgroup ranges Step 7 - Calculate UCL and LCL for subgroup averages Step 8 - Calculate UCL for ranges Step 9 - Select scales and plot Step 10 - Document the chart CONTOL CHT VIWGPH 10 CONTOL CHT 11

12 Step 6 - Calculate the average of the subgroup ranges The average of all subgroups becomes the centerline for the lower plotting area Where: i k k k The individual range for each subgroup The average of the ranges for all subgroups The number of subgroups verage of anges xample Step 7 - Calculate the upper control limit (UCL) and lower control limit (LCL) for the averages of the subgroups t this point, your chart will look like a un Chart Now, however, the uniqueness of the Control Chart becomes evident as you calculate the control limits Control limits define the parameters for determining whether a process is in statistical control To find the X-Bar control limits, use the following formula: UCL X X 2 LCL X X 2 NOT: Constants, based on the subgroup size (n), are used in determining control limits for variables charts You can learn more about constants in Tools and Methods for the Improvement of Quality [ef 3] Use the following constants ( ) in the computation [ef 3, Table 8]: 2 n n n CONTOL CHT

13 Upper and Lower Control Limits xample UCL X X 2 (1540) (0577)(18) LCL X X 2 (1540) (0577)(18) Step 8 - Calculate the upper control limit for the ranges When the subgroup or sample size (n) is less than 7, there is no lower control limit To find the upper control limit for the ranges, use the formula: UCL D 4 LCL D 3 (for subgroups 7) Use the following constants (D ) in the computation [ef 3, Table 8]: 4 n D n D n D xample UCL D 4 (2114)(18) CONTOL CHT 13

14 Step 9 - Select the scales and plot the control limits, centerline, and data points, in each plotting area The scales must be determined before the data points and centerline can be plotted Once the upper and lower control limits have been computed, the easiest way to select the scales is to have the current data take up approximately 60 percent of the vertical (Y) axis The scales for both the upper and lower plotting areas should allow for future high or low out-ofcontrol data points Plot each subgroup average as an individual data point in the upper plotting area Plot individual range data points in the lower plotting area (Viewgraph 11) Step 10 - Provide the appropriate documentation ach Control Chart should be labeled with who, what, when, where, why, and how information to describe where the data originated, when it was collected, who collected it, any identifiable equipment or work groups, sample size, and all the other things necessary for understanding and interpreting it It is important that the legend include all of the information that clarifies what the data describe When should we use an Individual X and Moving ange (Xm) Control Chart? You can use Individual X and Moving ange (Xm) Control Charts to assess both variables and attribute data Xm charts reflect data that do not lend themselves to forming subgroups with more than one measurement You might want to use this type of Control Chart if, for example, a process repeats itself infrequently, or it appears to operate differently at different times If that is the case, grouping the data might mask the effects of such differences You can avoid this problem by using an Xm chart whenever there is no rational basis for grouping the data What conditions must we satisfy to use an Xm Control Chart for attribute data? The only condition that needs to be checked before using the Xm Control Chart is that the average count per sample IS GT THN ON There is no variation within a subgroup since each subgroup has a sample size of 1, and the difference between successive subgroups is used as a measure of variation This difference is called a moving range There is a corresponding moving range value for each of the individual X values except the very first value 14 CONTOL CHT

15 Constructing an X-Bar & Chart Step 9 - Select scales and plot V G N G CONTOL CHT VIWGPH 11 CONTOL CHT 15

16 What are the steps for calculating and plotting an Xm Control Chart? Step 1 - Determine the data to be collected Decide what questions about the process you plan to answer Step 2 - Collect and enter the individual measurements nter the individual measurements in time sequence on the line labeled Individual X in the data collection section of the Control Chart (Viewgraph 12) These measurements will be plotted as individual data points in the upper plotting area STP 3 - Calculate and enter the moving ranges Use the following formula to calculate the moving ranges between successive data entries nter them on the line labeled Moving in the data collection section The moving range data will be plotted as individual data points in the lower plotting area (Viewgraph 13) m i X i 1 X i Where: X i X i 1 Is an individual value Is the next sequential value following X i Note: The brackets ( ) refer to the absolute value of the numbers contained inside the bracket In this formula, the difference is always a positive number xample 16 CONTOL CHT

17 Constructing an Xm Chart Step 2 - Collect and enter individual measurements Title: Legend: Date 1 pr 2 pr 3 pr 4 pr 5 pr 6 pr 7 pr 8 pr 9 pr 10pr Individual X Moving I N D I V I D U L nter individual measurements in time sequence X CONTOL CHT VIWGPH 12 Constructing an Xm Chart Step 3 - Calculate and enter moving ranges Title: Legend: Date 1 pr 2 pr 3 pr 4 pr 5 pr 6 pr 7 pr 8 pr 9 pr 10pr Individual X Moving I N D I V I D U L nter the moving ranges X CONTOL CHT VIWGPH 13 CONTOL CHT 17

18 Steps 4 through 9 are outlined in Viewgraph 14 Step 4 - Calculate the overall average of the individual data points The average of the Individual-X data becomes the centerline for the upper plot Where: x x i k x x 1 x 2 x 3 x n k The average of the individual measurements n individual measurement The number of subgroups of one xample X Step 5 - Calculate the average of the moving ranges The average of all moving ranges becomes the centerline for the lower plotting area Where: m m n k m m 1 m 2 m 3 m n k 1 The average of all the Individual Moving anges The Individual Moving ange measurements The number of subgroups of one verage of Moving anges xample m (10 1) Step 6 - Calculate the upper and lower control limits for the individual X values The calculation will compute the upper and lower control limits for the upper plotting area To find these control limits, use the formula: UCL x X (266)m UCL x X (266)m NOT: Formulas in Steps 6 and 7 are based on a two-point moving range 18 CONTOL CHT

19 Constructing an Xm Chart Step 4 - Calculate average of data points Step 5 - Calculate average of moving ranges Step 6 - Calculate UCL and LCL for individual X Step 7 - Calculate UCL for ranges Step 8 - Select scales and plot Step 9 - Document the chart CONTOL CHT VIWGPH 14 CONTOL CHT 19

20 You should use the constant equal to 266 in both formulas when you compute the upper and lower control limits for the Individual X values Upper and Lower Control Limits xample UCL X X (266)m (169) (266)(32) 2541 LCL X X (266)m (169) (266)(32) 839 Step 7 - Calculate the upper control limit for the ranges This calculation will compute the upper control limit for the lower plotting area There is no lower control limit To find the upper control limit for the ranges, use the formula: UCL m UCL m (3268)m NON You should use the constant equal to 3268 in the formula when you compute the upper control limit for the moving range data xample UCL m (3268)m (3268)(32) 1045 Step 8 - Select the scales and plot the data points and centerline in each plotting area Before the data points and centerline can be plotted, the scales must first be determined Once the upper and lower control limits have been computed, the easiest way to select the scales is to have the current spread of the control limits take up approximately 60 percent of the vertical (Y) axis The scales for both the upper and lower plotting areas should allow for future high or low out-of-control data points Plot each Individual X value as an individual data point in the upper plotting area Plot moving range values in the lower plotting area (Viewgraph 15) Step 9 - Provide the appropriate documentation ach Control Chart should be labeled with who, what, when, where, why, and how information to describe where the data originated, when it was collected, who collected it, any identifiable equipment or work groups, sample size, and all the other things necessary for understanding and interpreting it It is important that the legend include all of the information that clarifies what the data describe 20 CONTOL CHT

21 I N D I V I D U L X Constructing an Xm Chart Step 8 - Select scales and plot N G CONTOL CHT VIWGPH 15 CONTOL CHT 21

22 NOT: If you are working with attribute data, continue through steps 10, 11, 12a, and 12b (Viewgraph 16) Step 10 - Check for inflated control limits You should analyze your Xm Control Chart for inflated control limits When either of the following conditions exists (Viewgraph 17), the control limits are said to be inflated, and you must recalculate them: If any point is outside of the upper control limit for the moving range (UCL ) m If two-thirds or more of the moving range values are below the average of the moving ranges computed in Step 5 Step 11 - If the control limits are inflated, calculate 3144 times the median moving range For example, if the median moving range is equal to 6, then (3144)(6) = The centerline for the lower plotting area is now the median of all the values (vice the mean) when they are listed from smallest to largest eview the discussion of median and centerline in the un Chart module for further clarification When there is an odd number of values, the median is the middle value When there is an even number of values, average the two middle values to obtain the median xample VN ODD (Total = 12) MDIN = 65 (Total = 11) MDIN = CONTOL CHT

23 Constructing an Xm Chart Step 10 - Check for inflated control limits Step 11 - If inflated, calculate 3144 times median m Step 12a - Do not recompute if 3144 times median m is greater than 266 times average of moving ranges Step 12b - Otherwise, recompute all control limits and centerlines CONTOL CHT VIWGPH 16 Constructing an Xm Chart Step 10 - Check for inflated control limits UCL m ny point is above upper control limit of moving range m 0 2/3 or more of data points are below average of moving range (13 of 19 data points = 68%) CONTOL CHT VIWGPH 17 CONTOL CHT 23

24 Step 12a - Do not compute new limits if the product of 3144 times the median moving range value is greater than the product of 266 times the average of the moving ranges Step 12b - ecompute all of the control limits and centerlines for both the upper and lower plotting areas if the product of 3144 times the median moving range value is less than the product of 266 times the average of the moving range The new limits will be based on the formulas found in Viewgraph 18 These new limits must be redrawn on the corresponding charts before you look for signals of special causes The old control limits and centerlines are ignored in any further assessment of the data 24 CONTOL CHT

25 Step 12b - Constructing an Xm Chart Upper Plot UCL X = X + (3144) (Median Moving ange) LCL X = X - (3144) (Median Moving ange) Centerline X = X Lower Plot UCL m = (3865) (Median Moving ange) LCL m = None Centerline m = Median Moving ange CONTOL CHT VIWGPH 18 CONTOL CHT 25

26 What do we need to know to interpret Control Charts? Process stability is reflected in the relatively constant variation exhibited in Control Charts Basically, the data fall within a band bounded by the control limits If a process is stable, the likelihood of a point falling outside this band is so small that such an occurrence is taken as a signal of a special cause of variation In other words, something abnormal is occurring within your process However, even though all the points fall inside the control limits, special cause variation may be at work The presence of unusual patterns can be evidence that your process is not in statistical control Such patterns are more likely to occur when one or more special causes is present Control Charts are based on control limits which are 3 standard deviations (3 sigma) away from the centerline You should resist the urge to narrow these limits in the hope of identifying special causes earlier xperience has shown that limits based on less than plus and minus 3 sigma may lead to false assumptions about special causes operating in a process [ef 6, p 82] In other words, using control limits which are less than 3 sigma from the centerline may trigger a hunt for special causes when the process is already stable The three standard deviations are sometimes identified by zones ach zone s dividing line is exactly one-third the distance from the centerline to either the upper control limit or the lower control limit (Viewgraph 19) Zone is defined as the area between 2 and 3 standard deviations from the centerline on both the plus and minus sides of the centerline Zone B is defined as the area between 1 and 2 standard deviations from the centerline on both sides of the centerline Zone C is defined as the area between the centerline and 1 standard deviation from the centerline, on both sides of the centerline There are two basic sets of rules for interpreting Control Charts: ules for interpreting X-Bar and Control Charts similar, but separate, set of rules for interpreting Xm Control Charts NOT: These rules should not be confused with the rules for interpreting un Charts 26 CONTOL CHT

27 Control Chart Zones UCL ZON Centerline LCL ZON B ZON C ZON C ZON B ZON 1/3 distance from Centerline to Control Limits CONTOL CHT VIWGPH 19 CONTOL CHT 27

28 What are the rules for interpreting X-Bar and Charts? When a special cause is affecting the data, the nonrandom patterns displayed in a Control Chart will be fairly obvious The key to these rules is recognizing that they serve as a signal for when to investigate what occurred in the process When you are interpreting X-Bar and Control Charts, you should apply the following set of rules: UL 1 (Viewgraph 20): Whenever a single point falls outside the 3 sigma control limits, a lack of control is indicated Since the probability of this happening is rather small, it is very likely not due to chance UL 2 (Viewgraph 21): Whenever at least 2 out of 3 successive values fall on the same side of the centerline and more than 2 sigma units away from the centerline (in Zone or beyond), a lack of control is indicated Note that the third point can be on either side of the centerline 28 CONTOL CHT

29 ule 1 - Interpreting X-Bar & Charts Out of Limits UCL Centerline LCL ZON ZON B ZON C ZON C ZON B ZON CONTOL CHT VIWGPH 1 ule 2 - Interpreting X-Bar & Charts UCL ZON ZON B Centerline ZON C ZON C ZON B LCL 2 out of 3 successive values in Zone ZON CONTOL CHT VIWGPH 1 CONTOL CHT 29

30 UL 3 (Viewgraph 22): Whenever at least 4 out of 5 successive values fall on the same side of the centerline and more than one sigma unit away from the centerline (in Zones or B or beyond), a lack of control is indicated Note that the fifth point can be on either side of the centerline UL 4 (Viewgraph 23): Whenever at least 8 successive values fall on the same side of the centerline, a lack of control is indicated What are the rules for interpreting Xm Control Charts? To interpret Xm Control Charts, you have to apply a set of rules for interpreting the X part of the chart, and a further single rule for the m part of the chart ULS FO INTPTING TH X-POTION of Xm Control Charts: pply the four rules discussed above, XCPT apply them only to the upper plotting area graph UL FO INTPTING TH m POTION of Xm Control Charts for attribute data: ule 1 is the only rule used to assess signals of special causes in the lower plotting area graph Therefore, you don t need to identify the zones on the moving range portion of an Xm chart 30 CONTOL CHT

31 ule 3 - Interpreting X-Bar & Charts UCL Centerline LCL 4 out of 5 successive values in Zones & B ZON ZON B ZON C ZON C ZON B ZON CONTOL CHT VIWGPH 1 ule 4 - Interpreting X-Bar & Charts UCL ZON ZON B Centerline LCL 8 successive values on same side of Centerline ZON C ZON C ZON B ZON CONTOL CHT VIWGPH 1 CONTOL CHT 31

32 When should we change the control limits? There are only three situations in which it is appropriate to change the control limits: When removing out-of-control data points When a special cause has been identified and removed while you are working to achieve process stability, you may want to delete the data points affected by special causes and use the remaining data to compute new control limits When replacing trial limits When a process has just started up, or has changed, you may want to calculate control limits using only the limited data available These limits are usually called trial control limits You can calculate new limits every time you add new data Once you have 20 or 30 groups of 4 or 5 measurements without a signal, you can use the limits to monitor future performance You don t need to recalculate the limits again unless fundamental changes are made to the process When there are changes in the process When there are indications that your process has changed, it is necessary to recompute the control limits based on data collected since the change occurred Some examples of such changes are the application of new or modified procedures, the use of different machines, the overhaul of existing machines, and the introduction of new suppliers of critical input materials How can we practice what we've learned? The exercises on the following pages will help you practice constructing and interpreting both X-Bar and and Xm Control Charts nswer keys follow the exercises so that you can check your answers, your calculations, and the graphs you create for the upper and lower plotting areas 32 CONTOL CHT

33 XCIS 1: team collected the variables data recorded in the table below X X X X Use these data to answer the following questions and plot a Control Chart: 1 What type of Control Chart would you use with these data? 2 Why? 3 What are the values of X-Bar for each subgroup? 4 What are the values of the ranges for each subgroup? 5 What is the grand mean for the X-Bar data? 6 What is the average of the range values? 7 Compute the values for the upper and lower control limits for both the upper and lower plotting areas 8 Plot the Control Chart 9 re there any signals of special cause variation? If so, what rule did you apply to identify the signal? CONTOL CHT 33

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