2 Statistical Process Control Key pointsA process may in control (capable) or out of control (not capable).All processes have some natural variation3. The data used for SPC are the mean and standard deviation of the process (found by sampling).4. The central limit theorem is the underlying theory.5. SPC only applies if the process samples show a bell curve distribution.
3 Process Control(a) In statistical control and capable of producing within control limitsFrequencyLower control limitUpper control limit(b) In statistical control but not capable of producing within control limitsThis slide helps introduce different process outputs.It can also be used to illustrate natural and assignable variation.(c) Out of control(weight, length, speed, etc.)SizeFigure S6.2
4 Statistical Process Control A process control system is set up to monitor a process and signal when assignable (or real) causes of variation are present.(This means there is a cause for the variation that can be fixed)
5 Statistical Process Control (SPC) Measures the variability found in every process, fromNatural or common causesSpecial or assignable causesSPC gives a statistical signal when assignable causes are presentQuality control work is to detect and eliminate assignable causes of variationPoints which might be emphasized include:- Statistical process control measures the performance of a process, it does not help to identify a particular specimen produced as being “good” or “bad,” in or out of tolerance.- Statistical process control requires the collection and analysis of data - therefore it is not helpful when total production consists of a small number of units- While statistical process control can not help identify a “good” or “bad” unit, it can enable one to decide whether or not to accept an entire production lot. If a sample of a production lot contains more than a specified number of defective items, statistical process control can give us a basis for rejecting the entire lot. The issue of rejecting a lot which was actually good can be raised here, but is probably better left to later.
6 Natural Variations Also called common causes Affect virtually all production processesExpected amount of variationOutput measures follow a probability distributionFor any distribution there is a measure of central tendency and dispersionIf the distribution of outputs falls within acceptable limits, the process is said to be “in control”
7 Natural variation Data range UCL mean LCL Time / number of samples Resulting normal distributionUCLmeanLCLTime / number of samples
9 Control Charts for Variables Characteristics that can take any real valueMay be in whole or in fractional numbersContinuous random variablesx-chart tracks changes in the central tendency (mean)R-chart indicates a change in spread (range)Once the categories are outlined, students may be asked to provide examples of items for which variable or attribute inspection might be appropriate. They might also be asked to provide examples of products for which both characteristics might be important at different stages of the production process.These two charts must be used together
10 Mean and Range Charts (a) These sampling distributions result in the charts below(Sampling mean is shifting upward, but range is consistent)x-chart(x-chart detects shift in central tendency)UCLLCLR-chart(R-chart does not detect change in mean)UCLLCLFigure S6.5
11 Mean and Range Charts (b) These sampling distributions result in the charts below(Sampling mean is constant, but dispersion is increasing)x-chart(x-chart indicates no change in central tendency)UCLLCLR-chart(R-chart detects increase in dispersion)UCLLCLFigure S6.5
12 Patterns in Control Charts Upper control limitTargetLower control limitAsk the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.Normal behavior. Process is “in control.”Figure S6.7
13 Patterns in Control Charts Upper control limitTargetLower control limitAsk the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.One plot out above (or below). Investigate for cause. Process is “out of control.”Figure S6.7
14 Patterns in Control Charts Upper control limitTargetLower control limitAsk the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.Trends in either direction, 5 plots. Investigate for cause of progressive change.Figure S6.7
15 Patterns in Control Charts Upper control limitTargetLower control limitAsk the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.Two plots very near lower (or upper) control. Investigate for cause.Figure S6.7
16 Patterns in Control Charts Upper control limitTargetLower control limitAsk the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.Run of 5 above (or below) central line. Investigate for cause.Figure S6.7
17 Patterns in Control Charts Upper control limitTargetLower control limitAsk the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.Erratic behavior. Investigate.Figure S6.7
19 Each square represents one sample of five boxes of cereal SamplesTo measure the process, we take output samples and analyze the sample statistics following these stepsEach square represents one sample of five boxes of cereal(a) Samples of the product, (boxes of cereal) taken off the filling machine line, vary in weightFrequencyWeight#Figure S6.1
20 The solid line represents the distribution SamplesTo measure the process, we take output samples and analyze the sample statistics following these stepsThe solid line represents the distributionFrequencyWeight(b) After enough samples are taken from a stable process, they form a pattern called a distributionFigure S6.1
21 SamplesTo measure the process, we take output samples and analyze the sample statistics following these steps(c) There are many types of distributions, including the normal (bell-shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shapeFigure S6.1WeightCentral tendencyVariationShapeFrequency
22 SamplesTo measure the process, we take samples and analyze the sample statistics following these steps(d) If only natural causes of variation are present, the output of a process forms a stable distribution that is predictablePredictionWeightTimeFrequencyFigure S6.1
23 SamplesTo measure the process, we take samples and analyze the sample statistics following these stepsPrediction?WeightTimeFrequency(e) If assignable causes are present, the process output is not stable and is not predicableFigure S6.1
25 Central Limit Theorem (theoretical basis for SPC) In any population, the distribution of means of samples from the population will follow a normal curveThe mean of the sampling distribution will be the same as the population mean mThe standard deviation of the sampling distribution ( ) will equal the population standard deviation (s ) divided by the square root of the sample size, nThis slide introduces the difference between “natural” and “assignable” causes.The next several slides expand the discussion and introduce some of the statistical issues.
26 Sampling Distribution Figure S6.4Sampling distribution of meansProcess distribution of means= m(mean)It may be useful to spend some time explicitly discussing the difference between the sampling distribution of the means and the mean of the process population.
27 Population and Sampling Distributions Population distributionsBetaNormalUniformDistribution of sample meansStandard deviation of the sample meansMean of sample means =| | | | | | |99.73% of allfall within ±95.45% fall within ±Figure S6.3
28 Central Limit Theorem (theoretical basis for SPC) So for any population (process) that we have sampled, and taken the means of the samples, we will create a normal distribution.This allows us to use properties of the normal distribution to understand our samples (and therefore the population (process)).The key feature to use is the standard deviation of a normal population
29 Central Limit TheoremAs the properties of normal distributions are known we can say that a process with only natural variation will have results that show68.26% of the process inside ± 1 std dev of the mean95.45% of the process inside ± 2 std dev of the mean99.73 % of the process inside ± 3 std dev of the mean(Turn it around)If a result is outside the ± 3 std dev limit we can be 99.73% sure that there is a special cause at work – and we have a problem.
30 Central Limit Theorem (as sure as 1+1 = 2) Z = 1Z = 2Z = 3( z = the number of std dev allowed. (see table s6.2, page 229)
31 Key PointsSampling any population using small samples and taking the mean of each sample.The population of ‘means-of-samples’ will be a bell curve.This allows us to use central limit theorem rules.Which is what the SPC charts are based on using mean and std deviation to define the process.
33 Setting Chart Limits For x-Charts when we know std dev(s) Where = mean of the sample means or a target value set for the processz = number of normal standard deviationssx = standard deviation of the sample meanss = population (process) standard deviationn = sample size
34 Setting Control Limits Randomly select and weigh nine (n = 9) boxes each hourAverage weight in the first sampleWEIGHT OF SAMPLEHOUR(AVG. OF 9 BOXES)116.1516.5916.3216.8616.41014.8315.5715.21114.2481217.3
35 Setting Control Limits Average mean of 12 samples
36 Setting Control Limits Average mean of 12 samples
37 Setting Control Limits Control Chart for samples of 9 boxesVariation due to assignable causesOut of controlSample number| | | | | | | | | | | |17 = UCL15 = LCL16 = MeanVariation due to natural causesOut of control
38 Setting Chart Limits For x-Charts when we don’t know std dev(s) where average range of the samplesA2 = control chart factor found in Table S6.1= mean of the sample means
39 Control Chart Factors TABLE S6.1 Factors for Computing Control Chart Limits (3 sigma)SAMPLE SIZE,nMEAN FACTOR,A2UPPER RANGE,D4LOWER RANGE,D321.8803.26831.0232.5744.7292.2825.5772.1156.4832.0047.4191.9240.0768.3731.8640.1369.3371.8160.18410.3081.7770.22312.2661.7160.284
40 Setting Control Limits Super Cola ExampleLabeled as “net weight 12 ounces”Process average = 12 ouncesAverage range = .25 ounceSample size = 5From Table S6.1UCL =Mean = 12LCL =
44 Setting Control Limits Average range = 5.3 poundsSample size = 5From Table S6.1 D4 = 2.115, D3 = 0UCL = 11.2Mean = 5.3LCL = 0
45 Steps In Creating Control Charts Collect 20 to 25 samples, often of n = 4 or n = 5 observations each, from a stable process and compute the mean and range of eachCompute the overall means ( and ), set appropriate control limits, usually at the 99.73% level, and calculate the preliminary upper and lower control limitsIf the process is not currently stable and in control, use the desired mean, m, instead of to calculate limits.
46 Steps In Creating Control Charts Graph the sample means and ranges on their respective control charts and determine whether they fall outside the acceptable limitsInvestigate points or patterns that indicate the process is out of control – try to assign causes for the variation, address the causes, and then resume the processCollect additional samples and, if necessary, revalidate the control limits using the new data