Normal Plotting for Process CharacterizationProcess Capability Measures and Their EstimationProcess CapabilityCpCpkCaveatsIE 361 Module 17Process Capability Analysis: Part 1Reading: Sections 5.1, 5.2 Statistical Quality Assurance Methods forEngineersProf. Steve Vardeman and Prof. Max MorrisIow a State UniversityVarde man and Morris (Iowa State University) IE 361 Module 17 1 / 23Normal Plotting and QuantilesIf (by virtue of process monitoring and wise intervention) one is willing tosay that a data set represents a stable process, it may be used tocharacterize process output. Section 5.1 of SQAME discusses severalgraphical techniques for summarizing a sample and therefore representingthe process that stands behind it. Here we emphasize one of these, socalled "normal plotting," a tool for investigating the extent to which adata set (and thus the process that produced it) can be described using anormal distribution.Normal plots are made using so called quantiles. The p quantile (or100  pth percentile) of a distribution is a number such that a fraction pof the distribution lies to the left and a fraction 1  p lies to the right. Ifone scores at the .8 quantile (80th percentile) on an exam, 80% of thosetaking the exam had lower marks and 20% had higher marks. Or, since95% of the standard normal distribution is to the left of 1.645, 1.645 is the.95 quantile of that distribution. We will use the notation Q(p) to standfor the p quantile of any distribution.Varde man and Morris (Iowa State University) IE 3 61 Module 17 2 / 23Normal PlotsFor a data set consisting of n values x1 x2     xn(xiis the ithsmallest data value), we’ll adopt the convention that xiis thep = (i  .5)/n quantile of the data set, that isQd atai  .5n= xiFor Qz(p)the standard normal quantile function, a normal plot is thenmade by plotting ordered pairsQd atai  .5n, Qzi  .5ni.e.xi, Qzi  .5nVarde man and Morris (Iowa State University) IE 3 61 Module 17 3 / 23Normal Plots (Operational Details and Interpretation)Standard normal quantiles Qz(p)can be found by locating values of p inthe body of a typical cumulative normal probability table and then readingcorresponding quantiles from the table’s margin. And statistical packageslike JMP provide "inverse cumulative probability" functions and "normalplotting" functions that can be used to automate this.This plot allows comparison of data quantiles and (standard) normal ones.A "straight line" normal plot indicates that a data set has the same shapeas the normal distributions, and suggests that the process that standsbehind the data set can be modeled as producing normally distributedobservations. (Section 5.1 of SQAME has a careful discussion ofinterpretation of such Q-Q plots for those who need a review of this Stat231 material.)Varde man and Morris (Iowa State University) IE 3 61 Module 17 4 / 23Normal Plots (Example 17-1)Table 5.7 of SQAME contains measured "tongue thickness" for n = 20steel levers. Below is a normal plot for those data. It shows the largestthickness is much too large to "…t" with the other observations. It wouldneed to be pulled substantially "back to the left" to make the plot "linear."Important departure from a "normal"/Gaussian shape is indicated.Figure: Normal Plot of the Data of Table 5.7 (Vertical Axis is Linear in NormalQuantile, But is Marked as Cumulative Probability)Varde man and Morris (Iowa State University) IE 3 61 Module 17 5 / 23Normal PlottingImportance -Judging Adequacy of a Normal ModelProbability plotting is important for several reasons. First, it helps onejudge how much faith to place in calculations based on a normaldistribution, and suggests in what ways the calculations might tend to bewrong. For example, the normal plot for the tongue thicknesses suggeststhat if the mechanism that operated to produce the single very large valueis truly "part of the process," using a normal distribution to describemanufactured thickness will likely underpredict the frequency of large datavalues.Varde man and Morris (Iowa State University) IE 3 61 Module 17 6 / 23Normal PlottingImportance -Parameter EstimationProbability plotting is also sometimes helpful in providing graphicalestimates of distribution parameters. For example, if one makes a normalplot of an exactly normal distribution, the slope of the plot is thereciprocal of σ and the horizontal intercept is µ. That suggests that for areal data set whose normal plot is fairly linear,1the horizontal intercept of an approximating line is a sensible estimateof the mean of the process generating the data, and2the reciprocal of the slope is a sensible estimate of the standarddeviation of the process generating the data.Varde man and Morris (Iowa State University) IE 3 61 Module 17 7 / 23Normal Plotting and Capability AnalysisThe facts that (for bell-shaped data sets) normal plotting provides asimple way of approximating a standard deviation and that 6σ is oftenused as a measure of the intrinsic spread of measurements generated by aprocess, together lead to the common practice of basing processcapability analyses on normal plotting. The …gure on panel 9 shows avery common type of industrial form that essentially facilitates the makingof a normal plot by removing the necessity of evaluating the standardnormal quantiles Qz(p). (On the special vertical scale one may simply usethe plotting position p rather than Qz(p), as would be required whenusing regular graph paper.) After plotting a data set and drawing in anapproximating straight line, 6σ can be read o¤ the plot as the di¤erence inhorizontal coordinates for points on the line at the "+3σ" and "3σ"vertical levels (i.e., with p = .0013 and p = .9987).Varde man and Morris (Iowa State University) IE 3 61 Module 17 8 / 23A Capability Analysis FormForms like the one below encourage plotting of process data and allowpeople to easily estimate and develop intuition about "process spread."Figure: A "Capability Analysis Sheet" (That is Essentially a Piece of NormalProbability Paper) (Page 211 SQAME )Varde man and Morris (Iowa State University) IE 3 61 Module 17 9 / 23Process Capability MeasuresGraphical methods provide a visual representation of the pattern ofvariation associated with a process. Often, in addition to these tools, it isconvenient to have some numerical "capability index" summary measuresto quote. We discuss the "process capability" and the two