Presentation on theme: "Control Charts for Variables"— Presentation transcript:
1Control Charts for Variables EBB 341 Quality Control
2Variation There is no two natural items in any category are the same. Variation may be quite large or very small.If variation very small, it may appear that items are identical, but precision instruments will show differences.
33 Categories of variation Within-piece variationOne portion of surface is rougher than another portion.Apiece-to-piece variationVariation among pieces produced at the same time.Time-to-time variationService given early would be different from that given later in the day.
4Source of variation Equipment Material Environment Operator Tool wear, machine vibration, …MaterialRaw material qualityEnvironmentTemperature, pressure, humadityOperatorOperator performs- physical & emotional
5Control Chart Viewpoint Variation due toCommon or chance causesAssignable causesControl chart may be used to discover “assignable causes”
6Some Terms Run chart - without any upper/lower limits Specification/tolerance limits - not statisticalControl limits - statistical
7Control chart functions Control charts are powerful aids to understanding the performance of a process over time.InputPROCESSOutputWhat’s causing variability?
8Control charts identify variation Chance causes - “common cause”inherent to the process or random and not controllableif only common cause present, the process is considered stable or “in control”Assignable causes - “special cause”variation due to outside influencesif present, the process is “out of control”
9Control charts help us learn more about processes Separate common and special causes of variationDetermine whether a process is in a state of statistical control or out-of-controlEstimate the process parameters (mean, variation) and assess the performance of a process or its capability
10Control charts to monitor processes To monitor output, we use a control chartwe check things like the mean, range, standard deviationTo monitor a process, we typically use two control chartsmean (or some other central tendency measure)variation (typically using range or standard deviation)
11Types of Data Variable data Product characteristic that can be measuredLength, size, weight, height, time, velocityAttribute dataProduct characteristic evaluated with a discrete choiceGood/bad, yes/no
12Control chart for variables Variables are the measurable characteristics of a product or service.Measurement data is taken and arrayed on charts.
13Control charts for variables X-bar chartIn this chart the sample means are plotted in order to control the mean value of a variable (e.g., size of piston rings, strength of materials, etc.).R chartIn this chart, the sample ranges are plotted in order to control the variability of a variable.S chartIn this chart, the sample standard deviations are plotted in order to control the variability of a variable.S2 chartIn this chart, the sample variances are plotted in order to control the variability of a variable.
14X-bar and R chartsThe X- bar chart is developed from the average of each subgroup data.used to detect changes in the mean between subgroups.The R- chart is developed from the ranges of each subgroup dataused to detect changes in variation within subgroups
15Control chart components Centerlineshows where the process average is centered or the central tendency of the dataUpper control limit (UCL) and Lower control limit (LCL)describes the process spread
16The Control Chart Method X bar Control Chart:UCL = XDmean + A2 x RmeanLCL = XDmean - A2 x RmeanCL = XDmean R Control Chart:UCL = D4 x RmeanLCL = D3 x RmeanCL = Rmean Capability Study:PCR = (USL - LSL)/(6s); where s = Rmean /d2
17Control Chart Examples 4/11/2017Control Chart ExamplesUCLNominalVariationsLCLAnd finally, this chart shows a process with two points actually outside the control limits, an easy indicator to detect but not the only one.These rules, there are a few more, are commonly referred to as the Western Electric Rules and can be found in any advanced quality reference.Sample numberStatistics30
19Define the problem Select a quality characteristic to be measured Use other quality tools to help determine the general problem that’s occurring and the process that’s suspected of causing it.Select a quality characteristic to be measuredIdentify a characteristic to study - for example, part length or any other variable affecting performance.
20Choose a subgroup size to be sampled Choose homogeneous subgroupsHomogeneous subgroups are produced under the same conditions, by the same machine, the same operator, the same mold, at approximately the same time.Try to maximize chance to detect differences between subgroups, while minimizing chance for difference with a group.
21Collect the dataGenerally, collect subgroups (100 total samples) before calculating the control limits.Each time a subgroup of sample size n is taken, an average is calculated for the subgroup and plotted on the control chart.
22Determine trial centerline The centerline should be the population mean, Since it is unknown, we use X Double bar, or the grand average of the subgroup averages.
23Determine trial control limits - Xbar chart The normal curve displays the distribution of the sample averages.A control chart is a time-dependent pictorial representation of a normal curve.Processes that are considered under control will have 99.73% of their graphed averages fall within 6.
25Determining an alternative value for the standard deviation
26Determine trial control limits - R chart The range chart shows the spread or dispersion of the individual samples within the subgroup.If the product shows a wide spread, then the individuals within the subgroup are not similar to each other.Equal averages can be deceiving.Calculated similar to x-bar charts;Use D3 and D4 (appendix 2)
27Example: Control Charts for Variable Data 4/11/2017Example: Control Charts for Variable DataSlip Ring Diameter (cm)Sample X RExample4.3Statistics
28Calculation From Table above: Sigma X-bar = 50.09 Sigma R = 1.15 Thus;X-Double bar = 50.09/10 = cmR-bar = 1.15/10 = cmNote: The control limits are only preliminary with 10 samples.It is desirable to have at least 25 samples.
40Revised CL & Control Limits Calculation based on discarding subgroup 4 & 20 (X-bar chart) and subgroup 18 for R chart:= ( )/(25-2)= 6.40 mm= ( )/25 - 1= = 0.08 mm
41New Control Limits New value: Using standard value, CL & 3 control limit obtained using formula:
42From Table B:A = for a subgroup size of 4,d2 = 2.059, D1 = 0, and D2 = 4.698Calculation results:
43Trial Control Limits & Revised Control Limit Revised control limitsUCL = 6.46CL = 6.40LCL = 6.34UCL = 0.18CL = 0.08LCL = 0
44Revise the chartsIn certain cases, control limits are revised because:out-of-control points were included in the calculation of the control limits.the process is in-control but the within subgroup variation significantly improves.
45Revising the charts Interpret the original charts Isolate the causes Take corrective actionRevise the chartOnly remove points for which you can determine an assignable cause
46Process in ControlWhen a process is in control, there occurs a natural pattern of variation.Natural pattern has:About 34% of the plotted point in an imaginary band between 1s on both side CL.About 13.5% in an imaginary band between 1s and 2s on both side CL.About 2.5% of the plotted point in an imaginary band between 2s and 3s on both side CL.
47The Normal Distribution -3 -2 -1 +1 +2 +3 Mean 68.26% 95.44% 4/11/2017-3 -2 -1 +2 +3Mean68.26%95.44%99.74%The NormalDistribution = Standard deviationLSLUSL-3+3CLAnd at +/- 3 sigma, the most common choice of confidence/control limits in quality control application, the area is 99.97%.Statistics20
4834.13% of data lie between and 1 above the mean (). 34.13% between and 1 below the mean.Approximately two-thirds (68.28 %) within 1 of the mean.13.59% of the data lie between one and two standard deviationsFinally, almost all of the data (99.74%) are within 3 of the mean.
49Normal Distribution Review Define the 3-sigma limits for sample means as follows:What is the probability that the sample means will lie outside 3-sigma limits?Note that the 3-sigma limits for sample means are different from natural tolerances which are at
504/11/2017Common CausesThe first set of slides presents the various aspects of common and assignable causes. This slide and the two following show a normal process distribution and how it allows for expected variability which is termed common causes.Statistics2
51Process Out of ControlThe term out of control is a change in the process due to an assignable cause.When a point (subgroup value) falls outside its control limits, the process is out of control.
52Assignable Causes (a) Mean Average Grams 4/11/2017 The new distribution will look as shown in this slide.GramsStatistics7
53Assignable Causes (b) Spread Average Grams 4/11/2017 The new distribution has a much greater spread (higher standard deviation).GramsStatistics9
54Assignable Causes (c) Shape Average Grams 4/11/2017 A skewed (non-normal) distribution will result in a different pattern of variability.GramsStatistics11
55Control Charts Assignable causes likely UCL Nominal LCL 1 2 3 Samples 4/11/2017Control ChartsAssignable causes likelyUCLNominalHowever, the third sample plots outside the original distribution, indicating the likely presence of an assignable cause.LCLSamplesStatistics24
56Control Chart Examples 4/11/2017Control Chart ExamplesUCLNominalVariationsLCLAnd finally, this chart shows a process with two points actually outside the control limits, an easy indicator to detect but not the only one.These rules, there are a few more, are commonly referred to as the Western Electric Rules and can be found in any advanced quality reference.Sample numberStatistics30
57Control Limits and Errors 4/11/2017Control Limits and ErrorsType I error:Probability of searching fora cause when none exists(a) Three-sigma limitsUCLProcessaverageAs long as the area encompassed by the control limits is less than 100% of the area under the distribution, there will be a probability of a Type I error. A Type I error occurs when it is concluded that a process is out of control when in fact pure randomness is present. Given +/- 3 sigma, the probability of a Type I error is = , a very small probability.LCLStatistics32
58Control Limits and Errors 4/11/2017Control Limits and ErrorsType I error:Probability of searching fora cause when none exists(b) Two-sigma limitsUCLProcessaverageWhen the control limits are changed to +/- 2 sigma, the probability of a Type I error goes up considerably, from to = While this is still a small probability, it is a change that should be carefully considered. In practice, this would mean more samples would be identified inappropriately as out-of-control. Even if they were subsequently ‘Okd’, there would be increased costs due to many more cycles through the four step improvement process shown previously.LCLStatistics33
59Control Limits and Errors 4/11/2017Control Limits and ErrorsType II error:Probability of concludingthat nothing has changed(a) Three-sigma limitsUCLShift in processaverageProcessaverageReturning to the 3 sigma limits, we can see the probability of making a Type II error, in this case failing to detect a shift in the process mean.LCLStatistics34
60Control Limits and Errors 4/11/2017Control Limits and ErrorsType II error:Probability of concludingthat nothing has changed(b) Two-sigma limitsUCLShift in processaverageProcessaverageBy reducing the control limits to +/- 2 sigma, we see the probability of failing to detect the shift has been reduced.LCLStatistics35
61Achieve the purposeOur goal is to decrease the variation inherent in a process over time.As we improve the process, the spread of the data will continue to decrease.Quality improves!!
63Examine the processA process is considered to be stable and in a state of control, or under control, when the performance of the process falls within the statistically calculated control limits and exhibits only chance, or common causes.
64Consequences of misinterpreting the process Blaming people for problems that they cannot controlSpending time and money looking for problems that do not existSpending time and money on unnecessary process adjustmentsTaking action where no action is warrantedAsking for worker-related improvements when process improvements are needed first
65Process variationWhen a system is subject to only chance causes of variation, 99.74% of the measurements will fall within 6 standard deviationsIf 1000 subgroups are measured, 997 will fall within the six sigma limits.-3 -2 -1 +2 +3Mean68.26%95.44%99.74%
66Chart zonesBased on our knowledge of the normal curve, a control chart exhibits a state of control when:Two thirds of all points are near the center value.The points appear to float back and forth across the centerline.The points are balanced on both sides of the centerline.No points beyond the control limits.No patterns or trends.