Process Capability Analysis in Minitab_Manual

CONTENTSINDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE 14 I I I I I I I I I Process Capability Process Capability Overview, 14-2 Capability Analysis (Normal Distribution), 14-6 Capability Analysis (Between/Within), 14-14 Capability Analysis (Weibull Distribution), 14-19 Capability Sixpack (Normal Distribution), 14-24 Capability Sixpack (Between/Within), 14-30 Capability Sixpack (Weibull Distribution), 14-34 Capability Analysis (Binomial), 14-37 Capability Analysis (Poisson), 14-41 MINITAB User’s Guide 2 14-1 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Overview Process Capability Overview Once a process is in statistical control, that is producing consistently, you probably then want to determine if it is capable, that is meeting specification limits and producing “good” parts. You determine capability by comparing the width of the process variation with the width of the specification limits. The process needs to be in control before you assess its capability; if it is not, then you will get incorrect estimates of process capability. You can assess process capability graphically by drawing capability histograms and capability plots. These graphics help you assess the distribution of your data and verify that the process is in control. You can also calculate capability indices, which are ratios of the specification tolerance to the natural process variation. Capability indices, or statistics, are a simple way of assessing process capability. Because they are unitless, you can use capability statistics to compare the capability of one process to another. Choosing a capability command MINITAB provides a number of different capability analysis commands from which you can choose depending on the the nature of data and its distribution. You can perform capability analyses for: I I I normal or Weibull probability models (for measurement data) normal data that might have a strong source of between-subgroup variation binomial or Poisson probability models (for attributes or count data) If your data are badly skewed, you can use the Box-Cox transformation or use a Weibull probability model—see Non-normal data on page 14-6. Note It is essential to choose the correct distribution when conducting a capability analysis. For example, MINITAB provides capability analyses based on both normal and Weibull probability models. The commands that use a normal probability model provide a more complete set of statistics, but your data must approximate the normal distribution for the statistics to be appropriate for the data. For example, Capability Analysis (Normal) estimates expected parts per million out-of-spec using the normal probability model. Interpretation of these statistics rests on two assumptions: that the data are from a stable process, and that they follow an approximately normal distribution. Similarly, Capability Analysis (Weibull) calculates parts per million out-of-spec using a Weibull distribution. In both cases, the validity of the statistics depends on the validity of the assumed distribution. If the data are badly skewed, probabilities based on a normal distribution could give rather poor estimates of the actual out-of-spec probabilities. In that case, it is better to either transfom the data to make the normal distribution a more appropriate model, or choose a different probability model for the data. With MINITAB, you can use the 14-2 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Process Capability Overview Box-Cox power transformation or a Weibull probability model. Non-normal data on page 14-6 compares these two methods. If you suspect that there may be a strong between-subgroup source of variation in your process, use Capability Analysis (Between/Within) or Capability Sixpack (Between/ Within). Subgroup data may have, in addition to random error within subgroups, random variation between subgroups. Understanding both sources of subgroup variation may provide you with a more realistic estimate of the potential capability of a process. Capability Analysis (Between/Within) and Capability Sixpack (Between/Within) computes both within and between standard deviations and then pools them to calculate the total standard deviation. MINITAB also provides capability analyses for attributes (count) data, based on the binomial and Poisson probability models. For example, products may be compared against a standard and classified as defective or not (use Capability Analysis (Binomial)). You can also classify products based on the number of defects (use Capability Analysis (Poisson)). MINITAB’s capability commands I Capability Analysis (Normal) draws a capability histogram of the individual measurements overlaid with normal curves based on the process mean and standard deviation. This graph helps you make a visual assessment of the assumption of normality. The report also includes a table of process capability statistics, including both within and overall statistics. Capability Analysis (Between/Within) draws a capability histogram of the individual measurements overlaid with normal curves, which helps you make a visual assessment of the assumption of normality. Use this analysis for subgroup data in which there is a strong between-subgroup source of variation, in addition to the within-subgroup variation. The report also includes a table of between/within and overall process capability statistics. Capability Analysis (Weibull) draws a capability histogram of the individual measurements overlaid with a Weibull curve based on the process shape and scale. This graph helps you make a visual assessment of the assumption that your data follow a Weibull distribution. The report also includes a table of overall process capability statistics. Capability Sixpack (Normal) combines the following charts into a single display, along with a subset of the capability statistics: – an X (or Individuals), R or S (or Moving Range), and run chart, which can be used to verify that the process is in a state of control – a capability histogram and normal probability plot, which can be used to verify that the data are normally distributed – a capability plot, which displays the process variability compared to the specifications 14-3 I I I MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE CONTENTS Chapter 14 I INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Overview Capability Sixpack (Between/Within) is appropriate for subgroup data in which there is a strong between-subgroup source of variation. Capability Sixpack (Between/ Within) combines the following charts into a single display, along with a subset of the capability statistics: – an Individuals Chart, Moving Range Chart, and R Chart or S Chart, which can be used to verify that the process is in a state of control – a capability histogram and normal probability plot, which can be used to verify that the data are normally distributed – a capability plot, which displays the process variability compared to specifications Capability Sixpack (Weibull) combines the following charts into a single display, along with a subset of the capability statistics: – an X (or Individuals), R (or Moving Range), and run chart, which can be used to verify that the process is in a state of control – a capability histogram and Weibull probability plot, which can be used to verify that the data come from a Weibull distribution – a capability plot, which displays the process variability compared to the specifications I Although the Capability Sixpack commands give you fewer statistics than the Capability Analysis commands, the array of charts can be used to verify that the process is in control and that the data follow the chosen distribution. Note Capability statistics are simple to use, but they have distributional properties that are not fully understood. In general, it is not good practice to rely on a single capability statistic to characterize a process. See [2], [4], [5], [6], [9], [10], and [11] for a discussion. I Capability Analysis (Binomial) is appropriate when your data consists of the number of defectives out of the total number of parts sampled. The report draws a P chart, which helps you verify that the process is in a state of control. The report also includes a chart of cumulative %defectives, histogram of %defectives, and defective rate plot. Capability Analysis (Poisson) is appropriate when your data take the form of the number of defects per item. The report draws a U chart, which helps you to verify that the process is in a state of control. The report also includes a chart of the cumulative mean DPU (defects per unit), histogram of DPU, and a defect rate plot. I Capability statistics Process capability statistics are numerical measures of process capability—that is, they measure how capable a process is of meeting specifications. These statistics are simple and unitless, so you can use them to compare the capability of different processes. Capability statistics are basically a ratio between the allowable process spread (the width of the specification limits) and the actual process spread (6σ). Some of the statistics take into account the process mean or target. 14-4 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE PPL—associated with overall variation I Capability Analysis (Between/Within) and Capability Sixpack (Between/Within) I I Capability Analysis (Weibull) and Capability Sixpack (Weibull) I For more information. and few believe that a value less than 1 is acceptable. You can calculate Cpm by entering a target in the Options subdialog box. Cpk. Ppk.33 to be a minimum acceptable value for the process capability statistics. PPL—associated with overall variation Cp. Ppk. CPL. taking into account the process mean relative to the midpoint between specifications: minimum [(USL − µ) / 3σ. but falls on or between them ratio of the tolerance (the width of the specification limits) to the actual spread. A value less than 1 indicates that your process variation is wider than the specification tolerance. PPU. Ppk. since Cpm measures process mean relative to the target rather than the midpoint between specifications. Many practitioners consider 1. CPU.LSL / 3σ Note If the process target is not the midpoint between specifications. Capability statistics on page 14-21. CPL. you may prefer to use Cpm in place of Cpk. and Capability statistics on page 14-26.µ / 3σ µ . PPL—associated with overall variation Pp. PPU. and Cpm (if you specify a target)—associated with within and between variation Pp. See [9] for a discussion. (µ − LSL) / 3σ] CPU or PPU CPL or PPL the process only has an upper specification limit the process only has a lower specification limit USL . PPU. and Cpm (if you specify a target)—associated with within variation Pp. Here are some guidelines for how the statistics are used: This statistic… Cp or Pp is used when… the process is centered between the specification limits Definition ratio of the tolerance (the width of the specification limits) to the actual spread (the process tolerance): (USL − LSL) / 6σ Cpk or Ppk the process is not centered between the specification limits. see Capability statistics on page 14-9. CPU. Cpk. MINITAB User’s Guide 2 14-5 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE .CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Process Capability Overview Process capability command Capability Analysis (Normal) and Capability Sixpack (Normal) Capability statistics I Cp. such as the process mean. The report also includes statistics of the process data. since it provides estimates of both overall and within process capability. Capability Sixpack (Normal). the observed performance. use Capability Analysis (Weibull) and Capability Sixpack (Weibull). and capability statistics Calculates both within and overall process parameters and capability statistics Draws a normal curve over the histogram to help you determine whether the transformation made the data “more normal” Weibull model Uses actual data units for the histogram.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Analysis (Normal Distribution) Non-normal data When you have non-normal data. and the process specifications. use Capability Analysis (Normal). it is probably better to choose the normal model. the target (if you enter one). I To transform the data. Capability Analysis (Between/Within). I This table summarizes the differences between the methods. the within and overall standard deviation. within and overall standard deviations). Capability Analysis (Normal Distribution) Use Capability Analysis (Normal) to produce a process capability report when your data are from a normal distribution or when you have Box-Cox transformed data. process parameters (shape and scale). If both models fit the data about the same. target. See Box-Cox Transformation for Non-Normal Data on page 12-6. Normal model with Box-Cox transformation Uses transformed data for the histogram. process parameters (mean. The two normal curves are generated using the process mean and within standard deviation and the process mean and overall standard deviation. and a complete table of overall and within capability statistics. you can either transfom the data in such a way that the normal distribution is a more appropriate model. and the expected within and overall 14-6 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . or choose a Weibull probability model for the data. The report includes a capability histogram overlaid with two normal curves. or Capability Sixpack (Between/Within) with the optional Box-Cox power transformation. and capability statistics Calculates only overall process parameters and capability statistics Draws a Weibull curve over the histogram to help you determine whether the data fit the Weibull distribution Which method is better? The only way to answer that question is to see which model fits the data better. To use a Weibull probability model. specification limits. then set up a second column of subgroup indicators. To use the Box-Cox transformation. whether the process is centered on the target. you must have two or more observations in at least one subgroup in order to estimate the process standard deviation. MINITAB omits it from the calculations. see Data on page 12-3. then click OK. respectively. choose Subgroups across rows of. enter a lower and/or upper specification limit. 2 Do one of the following: I When subgroups or individual observations are in one column. I 3 In Lower spec or Upper spec. h To perform a capability analysis (normal probability model) 1 Choose Stat ® Quality Tools ® Capability Analysis (Normal). If your data are very skewed. see the discussion under Non-normal data on page 14-6. Subgroup data can be structured in one column. When subgroups are in rows. and whether it is capable of consistently meeting the process specifications. enter the data column in Single column. data must be positive. A model which assumes the data are from a normal distribution suits most process data. For examples. enter a subgroup size of 1. If you have data in subgroups. enter the data in a single column. 4 If you like. The report can be used to visually assess whether the data are normally distributed. If an observation is missing. When you have subgroups of unequal size. enter a subgroup size or column of subgroup indicators. and enter the columns containing the rows in the box. use any of the options listed below. For individual observations. or in rows across several columns. MINITAB User’s Guide 2 14-7 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . Data You can use individual observations or data in subgroups. In Subgroup size.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Analysis (Normal Distribution) performance. Individual observations should be structured in one column. You must enter at least one of them. the expected % out of spec is set to 0 for “boundaries. entering 12 says to use an interval 12 standard deviations wide. enter a process target. The default is to display capability statistics. As a result. The default is parts per million. Estimate subdialog box I estimate the process standard deviation (σ) various ways—see Estimating the process variation on page 14-10. display benchmark Z scores instead of capability statistics. If the observed % out-of-spec comes up nonzero. this is an obvious indicator of incorrect data. MINITAB User’s Guide 2 I I I I I I I I 14-8 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE .” meaning measurements cannot fall outside those limits. six on either side of the process mean. calculate the capability statistics using an interval other than six standard deviations wide (three on either side of the process mean) by entering a sigma tolerance. MINITAB calculates Cpm in addition to the standard capability statistics. MINITAB still calculates the observed % out-of-spec. Options subdialog box I use the Box-Cox power transformation when you have very skewed data—see Use the Box-Cox power transformation for non-normal data on page 12-68. then USL (upper specification limits) and LSL (lower specification limit) will be replaced by UB (upper boundary) and LB (lower boundary) on the analysis.” If you choose boundaries. Note I enter historical values for µ (the process mean) and σ (the process potential standard deviation) if you have known process parameters or estimates from past data. The default is to perform both. expected “within” performance. enter a minimum and/or maximum scale to appear on the capability histogram. or nominal specification. If you do not specify a value for µ or σ. When you define the upper and lower specification limits as boundaries. display observed performance. MINITAB estimates them from the data. perform only the within-subgroup analysis or only the overall analysis. The default is to display the graph. display the capability analysis graph or not.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Analysis (Normal Distribution) Options Capability Analysis (Normal) dialog box I define the upper and lower specification limits as “boundaries. For example. and expected “overall” performance in percents or parts per million. replace the default graph title with your own title. CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Analysis (Normal Distribution) Storage subdialog box I store your choice of statistics in worksheet columns. MINITAB estimates σoverall considering the variation for the whole study. CPU. and PPL represent the overall capability of the process. Overall capability depicts how the process is actually performing relative to the specification limits. The large curve represents overall variation—the variation for the whole study. Pp. Ppk. MINITAB calculates the capability statistics associated with within variation (Cp. Capability statistics When you use the normal distribution model for the capability analysis. Minitab estimates σwithin considering the variation within subgroups. Cpk. PPL). the within variation estimate is based on a moving range. Ppk PPU. Note When your subgroup size is one. Cpk. PPU. Each small curve represents within (or potential) variation. To interpret these statistics. or variation for one subgroup (one moment in time). or it may indicate sources of variation not estimated by within capability. so that adjacent observations are effectively treated as subgroups. Within capability depicts how the process could perform relative to the specification limits. MINITAB User’s Guide 2 14-9 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . and CPL) and with overall variation (Pp. To calculate these. see Capability statistics on page 14-4. and CPL represents the potential capability of your process—what your process would be capable of if the process did not have shifts and drifts in the subgroup means. CPU. The statistics available for storage depend on the options you have chosen in the Capability Analysis (Normal) dialog box and subdialog boxes. Cp. but not the shift and drift between subgroups. if shifts and drifts could be eliminated. A substantial difference between overall and within variation may indicate that the process is out of control. When calculating these statistics. MINITAB User’s Guide 2 I 14-10 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . Ppk. – the median of the moving range—choose Median moving range. Both Capability Analysis (Normal) and Capability Sixpack (Normal) calculate within (within-subgroup) and overall variation. which are listed below. MINITAB provides several options. To change the length of the moving range from 2. sigma (σ). For a discussion of the relative merits of these methods. and PPL. h To specify a method for estimating σwithin 1 In the Capability Analysis (Normal) or Capability Sixpack (Normal) main dialog box. To calculate σoverall. check Use moving range of length and enter a number in the box. To not use an unbiasing constant in the estimation. The capability statistics associated with the within variation are Cp. MINITAB uses the standard deviation of all of the data. To change the length of the moving range from 2. – the pooled standard deviation (the default)—choose Pooled standard deviation. To calculate σwithin. – the square root of MSSD (mean of the squared successive differences)—choose Square root of MSSD. see [1]. uncheck Use unbiasing constants. to base the estimate on: – the average of the moving range (the default)—choose Average moving range. 2 Do one of the following: I For subgroup sizes greater than one. For individual observations (subgroup size is one). CPU. and CPL. Cpk. PPU. To not use an unbiasing constant in the estimation. The statistics associated with the overall variation are Pp. click Estimate. uncheck Use unbiasing constants. To not use an unbiasing constant in the estimation. – the average of the subgroup standard deviations—choose Sbar. check Use moving range of length and enter a number in the box. uncheck Use unbiasing constants.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Analysis (Normal Distribution) Estimating the process variation An important step in a capability analysis with normal data is estimating the process variation using the standard deviation. to base the estimate on: – the average of the subgroup ranges—choose Rbar. You decide to run a capability study to see whether Supplier 1 alone is capable of meeting your engineering specifications. One of the parts. Upon examination of the inventory records. In Target (adds Cpm to table). the number of poor quality assemblies has dropped significantly.MTW.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Analysis (Normal Distribution) 3 Click OK. After dropping Supplier 2. enter 5. enter Supp1. 3 In Single column. 1 Open the worksheet CAMSHAFT. In Subgroup size. 2 Choose Stat ® Quality Tools ® Capability Analysis (Normal). 5 Click Options. must be 600 mm +2 mm long to meet engineering specifications. There has been a chronic problem with camshaft lengths being out of specification—a problem which has caused poor-fitting assemblies down the production line and high scrap and rework rates. In Upper spec. enter 598. An X and R chart showed you that Supplier 2’s camshaft production was out of control. e Example of a capability analysis (normal probability model) Suppose you work at an automobile manufacturer in a department that assembles engines. you discovered that there were two suppliers for the camshafts. 4 In Lower spec. Graph window output MINITAB User’s Guide 2 14-11 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . but the problems have not completely disappeared. so you decided to stop accepting production runs from them until they get their production under control. enter 602. enter 600. a camshaft. Click OK in each dialog box. And the left tail of the distribution falls outside the lower specification limits. you measure warping in ten tiles each working day for ten days. 2 Choose Stat ® Control Charts ® Box-Cox Transformation. as shown by the histogram overlaid with a normal curve. The Cpk index indicates whether the process will produce units within the tolerance limits. Click OK. A histogram shows that your data do not follow a normal distribution. your data should approximately follow a normal distribution.MTW.06. e Example of a capability analysis with a Box-Cox transformation Suppose you work for a company that manufactures floor tiles and are concerned about warping in the tiles. indicating that they need to improve their process by reducing variability and centering the process around the target. Then you will do the capability analysis. But you can see that the process mean (599. so you decide to use the Box-Cox power transformation to try to make the data “more normal.90.” First you need to find the optimal lambda (λ) value for the transformation. and therefore. enter Warping. 14-12 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . the PPM < LSL—the number of parts per million whose characteristic of interest is less than the lower spec—is 3621. your own.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Analysis (Normal Distribution) Interpreting the results If you want to interpret the process capability statistics. This means you will sometimes see camshafts that do not meet the lower specification of 598 mm. Since Supplier 1 is currently your best supplier. The Cpk index for Supplier 1 is only 0. you will work with them to improve their process. Likewise. To ensure production quality. This requirement appears to have been fulfilled.55) falls short of the target (600). type 10. In Subgroup size. 3 In Single column. 1 Open the worksheet TILES. performing the Box-Cox transformation with that value. This means that approximately 3621 out of a million camshafts do not meet the lower specification of 598 mm. but practically speaking. as marked by vertical lines on the graph.5 is a reasonable choice because it falls within the 95% confidence interval. In our example. Choose Lambda = 0. using λ = 0. enter Warping.5. In Subgroup size. 0. enter 10. such as the square root (a lambda of 0. 2 In Single column.449.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Analysis (Normal Distribution) Graph window output The best estimate of lambda is 0. So you will run the Capability Analysis with a Box-Cox transformation. you may want a lambda value that corresponds to an intuitive transformation. 4 Click Options. 5 Check Box-Cox power transformation (W = Y**Lambda). Click OK in each dialog box. Graph window output MINITAB User’s Guide 2 14-13 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . 1 Choose Stat ® Quality Tools ® Capability Analysis (Normal). enter 8. 3 In Upper spec.5).5 (square root). if you enter one. Both statistics are 0. to see how the fit compares—see Example of a capability analysis (Weibull probability model) on page 14-22. Ideally. all subgroups should be the same size.76.33. You can also see on the histogram that some of the process data fall beyond the upper spec limit. To use the Box-Cox transformation. random error within subgroups may not be the only source of variation to consider. If your subgroups are not all the same size. the overall process variation is due to both the between-subgroup variation and the within-subgroup variation. The normal curves are generated using the process mean and overall standard deviation and the process mean and total standard deviation. When you collect data in subgroups. due to missing data or unequal subgroup sizes. Capability Analysis (Between/Within) Use Capability Analysis (Between/Within) to produce a process capability report using both between-subgroup and within-subgroup variation. or in rows across several columns. Subgroup data can be structured in one column. Capability Analysis (Between/Within) computes standard deviations within subgroups and between subgroups.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Analysis (Between/Within) Interpreting the results As you can see from the normal curve overlaying the histogram. The report includes a capability histogram overlaid with two normal curves. target. Because you only entered an upper specification limit. the capability statistics printed are CPU and Cpk. data must be positive. These will be combined (pooled) to compute the total standard deviation. Under these conditions. You decide to perform a capability analysis with this data using a Weibull model. The report also includes statistics of the process data. 14-14 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . Data You can use data in subgroups. so your process does not appear to be capable. below the guideline of 1. only subgroups of the majority size are used for estimating the between-subgroup variation. The total standard deviation will be used to calculate the capability statistics. the Box-Cox transformation “normalized” the data. such as the process mean. and a complete table of overall and total (between and within) capability statistics. with two or more observations. or you may specify historical standard deviations. total (between and within) and overall standard deviation. Now the process capability statistics are appropriate for this data. There may also be random error between subgroups. and observed and expected performance. such as Cp and Cpk. 2 Do one of the following: I When subgroups are in one column. I 3 In Lower spec or Upper spec. enter a subgroup size or column of subgroup indicators. use any of the options listed below. When you define the upper and lower specification limits as boundaries. enter a lower and/or upper specification limit. 4 If you like. the expected % out of spec is set to 0 for a boundary. In Subgroup size.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Analysis (Between/Within) h To perform a capability analysis (between/within) 1 Choose Stat ® Quality Tools ® Capability Analysis (Between/Within). You must enter at least one of them. If you choose a boundary. MINITAB User’s Guide 2 14-15 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . respectively. MINITAB does not calculate capability statistics for that side. When subgroups are in rows. If the observed % out-of-spec comes up nonzero. choose Subgroups across rows of. Note I enter historical values for µ (the process mean) and σ within subgroups and/or σ between subgroups if you have known process parameters or estimates from past data. enter the data column in Single column.” meaning measurements cannot fall outside those limits. MINITAB still calculates the observed % out-of-spec. Options Capability Analysis (Between/Within) dialog box I define the upper and lower specification limits as “boundaries. If you do not specify a value for µ or σ. then click OK. this is an obvious indicator of incorrect data. MINITAB estimates them from the data. and enter the columns containing the rows in the box. As a result. Pp. and PPL) and between/within capability statistics (Cp. σtotal is used to calculate the capability statistics. six on either side of the process mean. display observed performance. To interpret these statistics. or nominal specifications. or the overall analysis only. The default is to perform both. Ppk. Minitab estimates σwithin and σbetween and pools them to estimate σtotal. To calculate these. Then. MINITAB calculates both overall capability statistics (Pp. When calculating these statistics. entering 12 says to use an interval 12 standard deviations wide. calculate the capability statistics using an interval other than six standard deviations wide (three on either side of the process mean) by entering a sigma tolerance. expected “between/within” performance. MINITAB calculates Cpm in addition to the standard capability statistics. The statistics available for storage depend on the options you have chosen in the Capability Analysis (Between/ Within) dialog box and subdialog boxes. MINITAB estimates σoverall considering the variation for the whole study. enter a minimum and/or maximum scale to appear on the capability histogram. Cpk. see Capability statistics on page 14-4. display the capability analysis graph or not. PPU. PPU. CPU. 14-16 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . I I I I I I I Storage subdialog box I store your choice of statistics in worksheet columns. The default is parts per million. and expected “overall” performance in percents or parts per million. Cpk. enter a process target. CPU. perform the between/within subgroup analysis only.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Analysis (Between/Within) Estimate subdialog box I estimate the within and between standard deviations ( σ) various ways—see Estimating the process variation on page 14-17. and PPL represent the overall capability of the process. Capability statistics When you use Capability Analysis (Between/Within). and CPL represents the potential capability of your process—what your process would be capable of if the process did not have shifts and drifts in the subgroup means. Ppk. Cp. For example. Options subdialog box I use the Box-Cox power transformation when you have very skewed data—see Use the Box-Cox power transformation for non-normal data on page 12-68. and CPL). The default is to display the graph. replace the default graph title with your own title. the pooled standard deviation (the default)—choose Pooled standard deviation. and CPL. the average of the subgroup standard deviations—choose Sbar. check Use moving range of length and enter a number in the box. Both Capability Analysis (Between/ Within) and Capability Sixpack (Between/Within) calculate within. see Help. CPU. PPU. MINITAB uses the standard deviation of all of the data. For the formulas used to estimate the process standard deviations (σ). between. The capability statistics associated with total variation are Cp. MINITAB User’s Guide 2 14-17 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . MINITAB provides several options. Ppk. see [1]. total (between/within). To calculate σtotal. To calculate σwithin and σbetween. and PPL. For a discussion of the relative merits of these methods. uncheck Use unbiasing constants. To calculate σoverall. choose one of the following: I the average of the moving range (the default)—choose Average moving range. 2 To change the method for estimating σwithin. uncheck Use unbiasing constants. Cpk. sigma (σ). click Estimate. To not use an unbiasing constant in the estimation.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Analysis (Between/Within) Estimating the process variation An important step in a capability analysis with normal data is estimating the process variation using the standard deviation. To change the length of the moving range from 2. I 3 To change the method for estimating σbetween. choose one of the following: I I the average of the subgroup ranges—choose Rbar. MINITAB pools σwithin and σbetween. and overall variation. which are listed below. To not use an unbiasing constant in the estimation. The statistics associated with overall variation are Pp. h To specify methods for estimating σwithin and σbetween 1 In the Capability Analysis (Between/Within) or Capability Sixpack (Between/Within) main dialog box. 94.21. This analysis tells you that your process is fairly capable. the square root of MSSD (mean of the squared successive differences)—choose Square root of MSSD. You take three samples from 25 consecutive rolls and measure coating thickness.MTW. but could be improved. 1 Open the worksheet BWCAPA. The Cpk index is only 1. Graph window output Interpreting results You can see that the process mean (49. You are concerned that the paper is being coated with the correct thickness of film and that the coating is applied evenly throughout the roll.8829) falls close to the target of 50. I 4 Click OK. 4 In Lower spec. enter Roll. enter 53. e Example of a capability analysis (between/within) Suppose you are interested in the capability of a process that coats rolls of paper with a thin film. To change the length of the moving range from 2. check Use moving range of length and enter a number in the box. enter 47. enter Coating. Click OK.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Analysis (Between/Within) I the median of the moving range—choose Median moving range. This means that approximately 194 out of a million coatings will not meet the specification limits. indicating that the process is fairly capable. The thickness must be 50 ±3 to meet engineering specifications. In Upper spec. In Subgroup size. 3 In Single column. uncheck Use unbiasing constants. To not use an unbiasing constant in the estimation. The Cpk index indicates whether the process will produce units within the tolerance limits. 14-18 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . The PPM Total for Expected “Between/Within” Performance is 193. 2 Choose Stat ® Quality Tools ® Capability Analysis (Between/Within). Data You can enter your data in a single column or in multiple columns if you have arranged subgroups across rows. When using the Weibull model. use Capability Analysis (Normal Distribution) on page 14-6 with the optional Box-Cox power transformation. Because the Weibull capability analysis does not calculate within capability statistics. the actual overall capability. If you have data that do not follow a normal distribution. and the observed and expected overall performance. Cp and Cpk. target (if you enter one). Ppk. Data must be positive. see Data on page 12-3.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Analysis (Weibull Distribution) Capability Analysis (Weibull Distribution) Use the Capability Analysis (Weibull) command to produce a process capability report when your data are from a Weibull distribution. Pp. If an observation is missing. and whether the process is capable of meeting the specifications consistently. whether the data follow a Weibull distribution. MINITAB User’s Guide 2 14-19 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . The report also includes statistics of the process data. shape. MINITAB omits it from the calculations. PPU. The report can be used to visually assess the distribution of the process relative to the target. and PPL. rather than mean and variance estimates as in the normal case. For examples. The Weibull curve is generated from the process shape and scale. and you want to calculate the within capability statistics. MINITAB calculates the overall capability statistics. scale. such as the mean. The calculations are based on maximum likelihood estimates of the shape and scale parameters for the Weibull distribution. see Non-normal data on page 14-6. The report includes a capability histogram overlaid with a Weibull curve and a table of overall capability statistics. and process specifications. For a comparison of the methods used for non-normal data. MINITAB does not used subgroups in calculations. when calculating the expected % out-of-spec. MINITAB still calculates the observed % out-of-spec. and enter the columns containing the rows in the box.” meaning that it is impossible for a measurement to fall outside that limit. enter a lower and/or upper specification limit.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Analysis (Weibull Distribution) h To perform a capability analysis (Weibull probability model) 1 Choose Stat ® Quality Tools ® Capability Analysis (Weibull). These limits must be positive numbers. Options Capability Analysis (Weibull) dialog box I define the upper and lower specification limits as “boundaries. though the lower spec can be 0. this is an obvious indicator of incorrect data. 2 Do one of the following: I When subgroups or individual observations are in one column. choose Single column and enter the column containing the data. then click OK. When you define the upper or lower specification limits as boundaries. As a result. When subgroups are in rows. Note 14-20 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . 4 If you like. I 3 In Lower spec or Upper spec. You must enter at least one of them. choose Subgroups across rows of. use any of the options listed below. MINITAB sets this value to 0 for a boundary. respectively. If the observed % out-of-spec comes up nonzero. gives an exponential distribution. calculate the capability statistics using an interval other than six standard deviations wide (three on either side of the process mean) by entering a sigma tolerance. especially the shape. six on either side of the process mean. a β = 2 gives a Rayleigh distribution. enter a process target or nominal specification. Its defining parameters are the shape (β) and scale (δ). MINITAB only calculates the overall capability statistics. MINITAB estimates σoverall considering the variation for the whole study. The appearance of the distribution varies widely. To interpret these statistics. Pp. see Capability statistics on page 14-4. including such distributions as the exponential and Rayleigh. and PPL.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Analysis (Weibull Distribution) Options subdialog box I enter historical values for the Weibull shape and scale parameters—see Weibull family of distributions on page 14-21. and PPL represent the overall capability of the process. PPU. they also define the probabilities used to calculate the capability statistics. I I I Capability statistics When you use the Weibull model for the capability analysis. If you do not enter historical values. MINITAB calculates Cpm in addition to the standard capability statistics. can have large effects on the associated probabilities. Caution Because the shape and scale parameters define the properties of the Weibull distribution. MINITAB User’s Guide 2 14-21 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . MINITAB obtains maximum likelihood estimates from the data. for instance. rather than mean and variance estimates as in the normal case. If you enter “known” values for the parameters. If you like. Weibull family of distributions The Weibull distribution is actually a family of distributions. Ppk. Ppk. The calculations are based on maximum likelihood estimates of the shape and scale parameters for the Weibull distribution. keep in mind that small changes in the parameters. you can enter historical values for the shape and scale. PPU. When calculating these statistics. For example. Pp. depending on the size of β. A β = 1. entering 12 says to use an interval 12 standard deviations wide. replace the default graph title with your own title. choose Historical value.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Analysis (Weibull Distribution) h To enter historical values for the shape and scale parameters 1 In the Capability Analysis (Weibull) or Capability Sixpack (Weibull) main dialog box. and enter a positive value for the scale. 1 Open the worksheet TILES. Click OK. 2 Choose Stat ® Quality Tools ® Capability Analysis (Weibull). e Example of a capability analysis (Weibull probability model) Suppose you work for a company that manufactures floor tiles. 2 Under Shape parameter. 4 In Upper spec.MTW. Click OK. A histogram of the data showed that it did not come from a normal distribution—see Example of a capability analysis with a Box-Cox transformation on page 14-12. 14-22 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . type 8. you measured warping in ten tiles each working day for ten days. So you decide to perform a capability analysis based on a Weibull probability model. and are concerned about warping in the tiles. enter Warping. To ensure production quality. and enter a positive value in the box 3 In Scale parameter. choose one of the following: I I I 1 (Exponential) 2 (Rayleigh) Historical value. click Options. 3 In Single column. Thus. MINITAB User’s Guide 2 14-23 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE .33.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Analysis (Weibull Distribution) Graph window output Interpreting the results The capability histogram does not show evidence of any serious discrepancies between the assumed model and the data.77. see Example of a capability analysis with a Box-Cox transformation on page 14-12. This means you will sometimes see warping higher than the upper specification of 8 mm. the PPM > USL—the number of parts per million above the upper spec—is 20000.00. Likewise. below the guideline of 1.000 out of a million tiles will warp more than the upper specification of 8 mm. But you can see that the right tail of the distribution falls over the upper specification limit. The Ppk and PPU indices tell you whether the process will produce tiles within the tolerance limits. your process does not appear to be capable. Both indices are 0. To see the same data analyzed with Capability Analysis (Normal). This means that 20. then set up a second column of subgroup indicators. you must have two or more observations in at least one subgroup in order to estimate the process standard deviation. If a single observation in the subgroup is missing. MINITAB omits it from the calculations of the statistics for that subgroup. Subgroup data can be structured in one column. If you have data in subgroups. Individual observations should be structured in one column. Combined with the capability statistics. Data You can enter individual observations or data in subgroups. or in rows across several columns. The histogram and normal probability plot can be used to verify that the data are normally distributed. when this variation is proportional to the mean). For examples. there is a gap in the chart where the statistic for that subgroup would have been plotted.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Sixpack (Normal Distribution) Capability Sixpack (Normal Distribution) Use the Capability Sixpack (Normal) command to assess process capability in a glance when your data are from the normal distribution or you have Box-Cox transformed data. If your data are either very skewed or the within-subgroup variation is not constant (for example. Ppk. and σoverall The X . this information can help you assess whether your process is in control and the product meets specifications. Lastly. 14-24 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . see Data on page 12-3. the capability plot gives a graphical view of the process variability compared to the specifications. If an entire subgroup is missing. To use the Box-Cox transformation. Cpk. and run charts can be used to verify that the process is in a state of control. When you have subgroups of unequal size. data must be positive. Cpm (if you enter a target). see the discussion under Non-normal data on page 14-6. and σwithin. enter the subgroups in a single column. Capability Sixpack combines the following information into a single display: I I I I I I I an X chart (or Individuals chart for individual observations) an R chart or S chart (or MR chart for individual observations) a run chart of the last 25 subgroups (or last 25 observations) a histogram of the process data a normal probability plot a process capability plot within and overall capability statistics: Cp. A model that assumes the data are from a normal distribution suits most process data. Pp. Such an omission may cause the control chart limits and the center line to have different values for that subgroup. Subgroups need not be the same size. R. enter a lower and/or upper specification limit. You must enter at least one of them. respectively. use Defining Tests for Special Causes on page 12-5. To adjust the sensitivity of the tests. MINITAB estimates them from the data. I 3 In Lower spec or Upper spec. use any of the options listed below. 2 Do one of the following: I When subgroups or individual observations are in one column. Tests subdialog box I do your choice of eight tests for special causes—see Do tests for special causes on page 12-64. enter a subgroup size of 1. If you do not specify a value for µ or σ. When subgroups are in rows.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Sixpack (Normal Distribution) h To make a capability sixpack (normal probability model) 1 Choose Stat ® Quality Tools ® Capability Sixpack (Normal). enter a subgroup size or column of subgroup indicators. Options Capability Sixpack (Normal) dialog box I enter your own value for µ (the process mean) and σ (the process potential standard deviation) if you have known process parameters or estimates from past data. 4 If you like. MINITAB User’s Guide 2 14-25 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . choose Subgroups across rows of. then click OK. enter the data column in Single column. For individual observations. and enter the columns containing the rows in the box. In Subgroup size. replace the default graph title with your own title. Minitab estimates σwithin considering the variation within subgroups. Pp. Cpk. Ppk. Cpk. see Capability statistics on page 14-4. MINITAB calculates Cpm in addition to the standard capability statistics. Cp. I Note When you estimate σ using the average of subgroup ranges (Rbar). The default estimate of σ is based on a pooled standard deviation. 14-26 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . MINITAB displays an S chart. and PPL represent the overall capability of the process. but not the shift and drift between subgroups. For example. Cp. I I I Options subdialog box I use the Box-Cox power transformation when you have very skewed data—see Use the Box-Cox power transformation for non-normal data on page 12-68. MINITAB displays an S chart. enter the process target or nominal specification. and CPL represents the potential capability of your process—what your process would be capable of if the process did not have shifts and drifts in the subgroup means. When you estimate σ using the average of subgroup standard deviations (Sbar). entering 12 says to use an interval 12 standard deviations wide. and σwithin. MINITAB estimates σoverall considering the variation for the whole study. MINITAB displays an R chart. CPU. and σoverall. When calculating these statistics. When you estimate σ using the pooled standard deviation and your subgroup size is less than ten. To interpret these statistics. calculate the capability statistics using an interval other than six standard deviations wide (three on either side of the process mean) by entering a sigma tolerance.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Sixpack (Normal Distribution) Estimate subdialog box I estimate the process standard deviation (σ) various ways—see Estimating the process variation on page 14-10. I I I I Capability statistics Capability Sixpack (Normal) displays both the within and overall capability statistics. The default is 25. and Pp. six on either side of the process mean. change the number of subgroups or observations to display in the run chart. When you estimate σ using the pooled standard deviation and your subgroup size is ten or greater. To calculate these. PPU. Ppk. Cpm (if you specify a target). MINITAB displays an R chart. One of the parts. Graph window output Interpreting the results On both the X chart and the R chart. 1 Open the worksheet CAMSHAFT. again. Click OK. implying a stable process. Upon examination of the inventory records.MTW. the number of poor quality assemblies have dropped significantly. An X and R chart showed you that Supplier 2’s camshaft production was out of control. but the problems have not completely disappeared.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Sixpack (Normal Distribution) e Example of a capability sixpack (normal probability model) Suppose you work at an automobile manufacturer in a department that assembles engines. In Subgroup size. After dropping Supplier 2. 2 Choose Stat ® Quality Tools ® Capability Sixpack (Normal). Yours do not. the points are randomly distributed between the control limits. enter 5. There has been a chronic problem with camshaft lengths being out of specification—a problem which has caused poor-fitting assemblies down the production line and high scrap and rework rates. so you decided to stop accepting production runs from them until they get their production under control. implying a stable process. 3 In Single column. must be 600 mm ±2 mm long to meet engineering specifications. enter 602. enter 598. enter Supp1. In Upper spec. you discovered that there were two suppliers for the camshafts. 4 In Lower spec. It is also important to compare points on the R chart with those on the X chart to see if the points follow each other. MINITAB User’s Guide 2 14-27 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . a camshaft. You decide to run a capability sixpack to see whether Supplier 1 alone is capable of meeting your engineering specifications. MTW. So you will run the capability sixpack using a Box-Cox transformation on the data. the values of Cp (1. But from the capability plot. If you want to interpret the process capability statistics.5 (square root). indicating that Supplier 1 needs to improve their process. 4 In Upper spec. This means you will sometimes see camshafts that do not meet the lower specification of 598 mm. To ensure production quality. Also. From previous analyses. e Example of a capability sixpack with a Box-Cox tranformation Suppose you work for a company that manufactures floor tiles.33. Click OK in each dialog box. 1 Open the worksheet TILES. 6 Check Box-Cox power transformation (W = Y**Lambda).90) are below the guideline of 1. the data approximately follow the normal curve. 14-28 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . 3 In Single column. you measure warping in ten tiles each working day for ten days. and that a Box-Cox transformation using a lambda value of 0.16) and Cpk (0. 5 Click Options. On the capability histogram. with no trends or shifts— also indicating process stability. 2 Choose Stat ® Quality Tools ® Capability Sixpack (Normal). type 10. and are concerned about warping in the tiles.” For details.5 makes the data “more normal.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Sixpack (Normal Distribution) The points on the run chart make a random horizontal scatter. Choose Lambda = 0. the points approximately follow a straight line. These patterns indicate that the data are normally distributed. enter Warping. On the normal probability plot. your data should approximately follow a normal distribution. type 8. you can see that the process tolerance falls below the lower specification limit. see Example of a capability analysis with a Box-Cox transformation on page 14-12. In Subgroup size. you found that the tile data do not come from a normal distribution. 76) and Ppk (0. the data follow the normal curve. so your process does not appear to be capable. however. Now the process capability statistics are appropriate for this data. the points approximately follow a straight line. Also.75) fall below the guideline of 1. the points are randomly distributed between the control limits. And the values of Cpk (0. implying a stable process. on the normal probability plot. These patterns indicate that the Box-Cox transformation “normalized” the data. with no trends or shifts— also indicating process stability.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Sixpack (Normal Distribution) Graph window output Interpreting the results On both the X chart and the R chart. shows that the process is not meeting specifications. implying a stable process. The points on the run chart make a random horizontal scatter. MINITAB User’s Guide 2 14-29 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE .33. As you can see from the capability histogram. It is also important to compare points on the R chart with those on the X chart for the same data to see if the points follow each other. The capability plot. Yours do not—again. and R or S charts can verify whether or not the process is in control. Cpm (if you specify a target). Cp. Ppk. Pp. The Individuals. and σtotal. Control limits for the Individuals and Moving Range charts are based on the majority subgroup size. The histogram and normal probability plot can verify whether or not the data are normally distributed. To use the Box-Cox transformation. with two or more observations per subgroup. A model that assumes that the data are from a normal distribution suits most process data. Data You can enter data in subgroups. If your subgroups are not all the same size. σbetween. and σoverall. all subgroups should be the same size. data must be positive. see the discussion under Non-normal data on page 14-6. Combined with the capability statistics. If your data are either very skewed or the within subgroup variation is not constant (for example. this information can help you assess whether your process is in control and the product meets specifications.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Sixpack (Between/Within) Capability Sixpack (Between/Within) Use the Capability Sixpack (Between/Within) command when you suspect that you may have both between-subgroup and within-subgroup variation. 14-30 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . the capability plot gives a graphical view of the process variability compared to specifications. Lastly. due to missing data or unequal sample sizes. only subgroups of the majority size are used for estimating the between-subgroup variation. Subgroup data can be structured in one column or in rows across several columns. Cpk. σwithin. Ideally. Moving Range. when the variation is proportional to the mean). Capability Sixpack (Between/Within) allows you to assess process capability at a glance and combines the following information into a single display: I I I I I I I an Individuals chart a Moving Range chart an R chart or S chart a histogram of the process data a normal probability plot a process capability plot between/within and overall capability statistics. respectively. 4 If you like. Tests subdialog box I do your choice of the eight tests for special causes—see Do tests for special causes on page 12-64. I 3 In Lower spec or Upper spec. To adjust the sensitivity of the tests.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Sixpack (Between/Within) h To make a capability sixpack (between/within) 1 Choose Stat ® Quality Tools ® Capability Sixpack (Between/Within). MINITAB User’s Guide 2 14-31 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . MINITAB estimates them from the data. choose Subgroups across rows of. enter the data column in Single column. In Subgroup size. enter a lower and/or upper specification limit. Options Capability Sixpack (Between/Within) dialog box I enter a historical value for µ (the process mean) and/or σ (within-subgroup and/or between-subgroup standard deviations) if you have known process parameters or estimates from past data. You must enter at least one of them. enter a subgroup size or column of subgroup indicators. If you do not specify a value for µ or σ. use any of the options listed below. 2 Do one of the following: I When subgroups are in one column. and enter the columns containing the rows in the box. then click OK. use Defining Tests for Special Causes on page 12-5. When subgroups are in rows. For example. enter the process target or nominal specification. MINITAB displays an S chart. MINITAB calculates both overall capability statistics (Pp. entering 12 says to use an interval 12 standard deviations wide. I I I Options subdialog box I use the Box-Cox power transformation when you have very skewed data—see Non-normal data on page 14-6.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Sixpack (Between/Within) Estimate subdialog box I estimate the process standard deviation (σ) various ways—see Estimating the process variation on page 14-17. replace the default graph title with your own title. When you estimate σ using the average of subgroup standard deviations (Sbar).MTW. When you estimate σ using the pooled standard deviation and your subgroup size is ten or greater. you use MINITAB to conduct a Capability Sixpack (Between/Within). 2 Select Stat ® Quality Tools ® Capability Sixpack (Between/Within). Ppk. I I I Capability statistics When you use Capability Analysis (Between/Within). and CPL). 14-32 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . You take three samples from 25 consecutive rolls and measure coating thickness. e Example of a capability sixpack (between/within) Suppose you are interested in the capability of a process that coats rolls of paper with a thin film. To interpret these statistics. 1 Open the worksheet BWCAPA. see Capability statistics on page 14-4. PPU. MINITAB displays an S chart. When you estimate σ using the pooled standard deviation and your subgroup size is less than ten. Cpk. CPU. You are concerned that the paper is being coated with the correct thickness of film and that the coating is applied evenly throughout the roll. calculate the capability statistics using an interval other than six standard deviations wide (three on either side of the process mean) by entering a sigma tolerance. I Note When you estimate σ using the average of subgroup ranges (Rbar). MINITAB displays an R chart. and PPL) and between/within capability statistics (Cp. six on either side of the process mean. The thickness must be 50 ±3 to meet engineering specifications. Because you are interested in determining whether or not the coating is even throughout a roll. MINITAB calculates Cpm in addition to the standard capability statistics. MINITAB displays an R chart. 5 Click Tests. thereby implying that your process is in control. on the normal probability plot. enter Roll. the points approximately follow a straight line.14) fall just below the guideline of 1. No points failed the eight tests for special causes. The values of Cpk (1. The points on the Individuals and Moving Range chart do not appear to follow each other. The capability plot shows that the process is meeting specifications. enter 47. Click OK in each dialog box. enter Coating. 4 In Lower spec. In Subgroup size.33. Choose Perform all eight tests.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Sixpack (Between/Within) 3 In Single column. This criteria appears to have been met. the data approximately follow the normal curve. MINITAB User’s Guide 2 14-33 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . In the capability histogram. Graph window output Interpreting results If you want to interpret the process capability statistics.21) and Ppk (1. enter 53. In Upper spec. again indicating a stable process. so your process could use some improvement. Also. your data need to come from a normal distribution. and run charts can be used to verify that the process is in a state of control. Data You can enter individual observations or data in subgroups. The histogram and Weibull probability plot can be used to verify that the data approximate a Weibull distribution. rather than mean and variance estimates as in the normal case. see Capability Analysis (Normal Distribution) on page 14-6 with the optional Box-Cox power transformation. Individual observations should be structured in one column. σwithin). see Use the Box-Cox power transformation for non-normal data on page 12-68. If you have data that do not follow a normal distribution. Pp and Ppk. entering Lambda = 0(natural log). you can use the Capability Sixpack (Weibull) command to assess process capability in a glance. see Non-normal data on page 14-6. Ppk. Subgroup data can be structured in one column or in rows across several columns. Capability Sixpack (Weibull) combines the following information into a single display: I I I I I I I an X chart (or I chart for individual observations) an R chart (or MR chart for individual observations) a run chart of the last 25 subgroups (or last 25 observations) a histogram of the process data a Weibull probability plot a process capability plot overall capability statistics Pp. shape (β). Combined with the capability statistics. then set up a second column of subgroup indicators. R. MINITAB only calculates the overall capability statistics. and scale (δ) The X. Lastly. your data must follow a normal distribution. If the Weibull distribution fits your data well. use the control chart command with the optional Box-Cox transformation. the capability plot gives a graphical view of the process variability compared to the specifications. For examples. The calculations are based on maximum likelihood estimates of the shape and scale parameters for the Weibull distribution. see Data on page 12-3. and you want to calculate the within statistics (Cp. a lognormal distribution would probably also provide a good fit. For more details. For a comparison of the methods used for non-normal data.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Sixpack (Weibull Distribution) Capability Sixpack (Weibull Distribution) When a Weibull distribution is a good approximation of the distribution of your process data. To transform your data. 14-34 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . When using the Weibull model. Tip To make a control chart that you can interpret properly. Cpk. When you have subgroups of unequal size. this information can help you assess whether your process is in control and can produce output that consistently meets the specifications. enter the subgroups in a single column. MINITAB obtains maximum likelihood estimates from the data. In Subgroup size. though the lower spec can be 0. Caution MINITAB User’s Guide 2 14-35 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . For individual observations. 4 If you like. enter the data column in Single column. use any of the options listed below. When subgroups are in rows. enter a subgroup size or column of subgroup indicators. If an entire subgroup is missing. MINITAB omits it from the calculations of the statistics for that subgroup. can have large effects on the associated probabilities. then click OK. enter a lower and/or upper specification limit. When you enter “known” values for the parameters. 2 Do one of the following: I When subgroups or individual observations are in one column. keep in mind that small changes in the parameters. These limits must be positive numbers. If you do not enter values. Options Options subdialog box I enter your own value for the Weibull shape and scale parameters—see Weibull family of distributions on page 14-21. This may cause the control chart limits and the center line to have different values for that subgroup. choose Subgroups across rows of. there is a gap in the chart where the statistic for that subgroup would have been plotted. If a single observation in the subgroup is missing. enter a subgroup size of 1.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Sixpack (Weibull Distribution) Data must be positive. h To make a capability sixpack (Weibull probability model) 1 Choose Stat ® Quality Tools ® Capability Sixpack (Weibull). I 3 In Lower spec or Upper spec. and enter the columns containing the rows in the box. You must enter at least one of them. especially the shape. you measured warping in ten tiles each working day for ten days. type 10. and are concerned about warping in the tiles. see Capability statistics on page 14-4. 2 Choose Stat ® Quality Tools ® Capability Sixpack (Weibull). rather than mean and variance estimates as in the normal case. For example. A histogram of the data revealed that it did not come from a normal distribution—see Example of a capability analysis with a Box-Cox transformation on page 14-12. So you decide to make a capability sixpack based on a Weibull probability model.MTW. replace the default graph title with your own title. Click OK.CONTENTS Chapter 14 I INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Sixpack (Weibull Distribution) change the number of subgroups or observations to display in the run chart. entering 12 says to use an interval 12 standard deviations wide. To ensure production quality. 14-36 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . six on either side of the process mean. The default is 25. In Subgroup size. enter Warping. type 8. calculate the capability statistics using an interval other than six standard deviations wide (three on either side of the process mean) by entering a sigma tolerance. e Example of a capability sixpack (Weibull probability model) Suppose you work for a company that manufactures floor tiles. 3 In Single column. I I Capability statistics Capability Sixpack (Weibull) displays the overall capability statistics. For information on interpreting these statistics. 1 Open the worksheet TILES. These calculations are based on maximum likelihood estimates of the shape and scale parameters for the Weibull distribution. Pp and Ppk. 4 In Upper spec. you could record the number of people who call in sick on a particular day and the number of people scheduled to work each day.77) falls below the guideline of 1. see Example of a capability sixpack with a Box-Cox tranformation on page 14-28. Or. And the value of Ppk (0. on the Weibull probability plot. The capability plot. so your process does not appear to be capable. however.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Analysis (Binomial) Graph window output Interpreting the results The capability histogram does not show evidence of any serious discrepancies between the assumed model and the data. you might have a pass/fail gage that determines whether an item is defective or not. To see the same data analyzed with Capability Sixpack (Normal). Also. shows that the process is not meeting specifications. You could then record the total number of parts inspected and the number failed by the gage. For example. the points approximately follow a straight line. MINITAB User’s Guide 2 14-37 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . Binomial distributions are usually associated with recording the number of defective items out of the total number sampled. Capability Analysis (Binomial) Use Capability Analysis (Binomial) to produce a process capability report when your data are from a binomial distribution.33. CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Analysis (Binomial) Use Capability Analysis (Binomial) if your data meet the following conditions: I I I I each item is the result of identical conditions each item can result in one or two possible outcomes (success/failure. On any given data. The other plots and charts simply exclude the missing observations. Suppose you have collected data on the number of parts inspected and the number of parts that failed inspection. When subgroup sizes are unequal. go/no-go) the probability of a success (or failure) is constant for each item the outcomes of the items are independent of each other Capability Analysis (Binomial) produces a process capability report that includes the following: I I P chart—verifies that the process is in a state of control Chart of cumulative %defective—verifies that you have collected data from enough samples to have a stable estimate of %defective Histogram of %defective—displays the overall distribution of the %defectives from the samples collected Defective rate plot—verifies that the %defective is not influenced by the number of items sampled I I Data Use data from a binomial distribution. 14-38 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . enter subgroup size in another column: Failed Inspected 11 1003 12 968 9 897 13 1293 9 989 15 1423 Missing data If an observation is missing. If the total number inspected varies. you must also enter a corresponding column of subgroup sizes. there is a gap in the P chart where that subgroup would have been plotted. Each entry in the worksheet column should contain the number of defectives for a subgroup. both numbers may vary. Enter the number that failed inspection in one column. In general. h To perform a capability analysis (binomial probability model) 1 Choose Stat ® Quality Tools ® Capability Analysis (Binomial). this could be caused by fatigued inspectors. the control limits are a function of the subgroup size. I MINITAB User’s Guide 2 14-39 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . if you tend to have a smaller %defective when more items are sampled. This value must be between 0 and 1. The subgroup size has no bearing on the other charts because they only display the %defective. 2 In Defectives.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Analysis (Binomial) Unequal subgroup sizes In the P chart. enter the column containing the number of defectives. For example. When you do have unequal subgroup sizes. a common problem. the plot of %defective versus sample size will permit you to verify that there is no relationship between the two. Options Capability Analysis (binomial) dialog box I enter a historical value for the proportion of defectives. enter the sample size value in Constant size. When your sample sizes vary. 4 If you like. enter the column containing sample sizes in Use sizes in. the control limits are further from the center line for smaller subgroups than they are for larger ones. 3 Do one of the following: I I When your sample size is constant. use any of the options listed below. then click OK. enter a value for the % defective target. how capable it is of answering incoming calls. replace the default graph title with your own title. Options subdialog box I I choose a color scheme for printing. Click OK. use Defining Tests for Special Causes on page 12-5. Graph window output Interpreting results The P chart indicates that there is one point out of control.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Analysis (Binomial) Tests subdialog box I perform your choice of the four tests for special causes—see Do tests for special causes on page 13-15. 3 In Defectives. 1 Open the worksheet BPCAPA. 14-40 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE .75. This process could use a lot of improvement. You also record the total number of incoming calls. e Example of capability analysis (binomial probability model) Suppose you are responsible for evaluating the responsiveness of your telephone sales department. You record the number of calls that were not answered (a defective) by sales representatives due to unavailability each day for 20 days. The rate of defectives does not appear to be affected by sample size. enter Unavailable. The process Z is around 0. enter Calls.MTW. 2 Choose Stat ® Quality Tools ® Capability Analysis (Binomial). The chart of cumulative %defect shows that the estimate of the overall defective rate appears to be settling down around 22%. To adjust the sensitivity of the tests. but more data may need to be collected to verify this. that is. which is very poor. 4 In Use sizes in. you may also record the size of each surface sampled. Poisson data is usually associated with the number of defects observed in an item. When subgroup sizes are unequal. Or. The report includes the following: I U chart—verifies that the process was in a state of control at the time the report was generated Chart of cumulative mean DPU (defects per unit)—verifies that you have collected data from enough samples to have a stable estimate of the mean Histogram of DPU—displays the overall distribution of the defects per unit from the samples collected Defect plot rate—verifies that DPU is not influenced by the size of the items sampled I I I Data Each entry in the worksheet column should contain the number of or defects for a subgroup. where the item occupies a specified amount of time or specified space. you may want to record the number of breaks in a piece of wire. you may want to record the number of scratches on the surface of the appliance. so you may also keep track of the size. For any given unit. you must also enter a corresponding column of subgroup sizes. Suppose you have collected data on the number of defects per unit and the size of the unit.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Analysis (Poisson) Capability Analysis (Poisson) Use Capability Analysis (Poisson) to produce a process capability report when your data are from a Poisson distribution. Since the sizes of the surface may be different. Enter the number of defects in one column. say in square inches. if you manufacture appliances. The size of the item may vary. If the lengths of the wire vary. if you manufacture electrical wiring. you will have to record the size of each piece sampled. If the unit size varies. both numbers may vary. enter unit size in another column: Failed Inspected 3 89 4 94 7 121 2 43 11 142 6 103 MINITAB User’s Guide 2 14-41 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . For example. Use Capability Analysis (Poisson) when your data meet the following conditions: I I the rate of defects per unit of space or time is the same for each item the number of defects observed in the items are independent of each other Capability Analysis (Poisson) produces a process capability report for data from a Poisson distribution. use any of the options listed below. then click OK. if you tend to have a smaller DPU when more items are sampled. 2 In Defects. When you do have unequal subgroup sizes. the plot of defects per unit (DPU) versus sample size will permit you to verify that there is no relationship between the two. 14-42 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . h To perform a capability analysis (Poisson distribution model) 1 Choose Stat ® Quality Tools ® Capability Analysis (Poisson). In general.CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Capability Analysis (Poisson) Missing data If an observation is missing. The subgroup size has no bearing on the other charts. When your unit sizes vary. enter the column containing unit sizes in Use sizes in. enter the unit size value in Constant size. The other plots and charts simply exclude the missing observation(s). enter the column containing the number of defects. there is a gap in the U chart where the subgroup would have been plotted. the control limits are a function of the subgroup size. For example. because they only display the DPU. 3 Do one of the following: I I When your unit size is constant. the control limits are further from the centerline for smaller subgroups than they are for larger ones. 4 If you like. a common problem. this could be caused by fatigued inspectors. Unequal subgroup sizes In the U chart. I Tests subdialog box I perform the four tests for special causes—see Do tests for special causes on page 13-15. Options subdialog box I I choose to use a full color. replace the default graph title with your own title. use Defining Tests for Special Causes on page 12-5. 4 In Uses sizes in. or black and white color scheme for printing. To adjust the sensitivity of the tests. enter Weak Spots. You take random lengths of electrical wiring and test them for weak spots in their insulation by subjecting them to a test voltage. 1 Open the worksheet BPCAPA. Click OK. 3 In Defects.MTW. enter a target DPU (defects per unit) for the process. MINITAB estimates it from the data. 2 Choose Stat ® Quality Tools ® Capability Analysis (Poisson). enter Lengths.CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability Capability Analysis (Poisson) Options Capability Analysis (Poisson) dialog box I enter historical values for µ (the process mean) if you have known process parameters or estimates from past data. You record the number of weak spots and the length of each piece of wire (in feet). e Example of capability analysis (Poisson probability distribution) Suppose you work for a wire manufacturer and are concerned about the effectiveness of the wire insulation process. If you do not specify a value for µ. partial color. MINITAB User’s Guide 2 14-43 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE References Graph window output Interpreting results The U Chart indicates that there are three points out of control. References [1] L. Dearborn.0265.W. “A New Measure of Process Capability: Cpm.223–229.196–210. signifying that enough samples were collected to have a good estimate of the mean DPU. “The Use and Abuse of Cpk. 20.108. Owen. The rate of DPU does not appear to be affected by the lengths of the wire. “Bootstrap Lower Confidence Limits for Capability Indices. Wasserman (1992). pp. Continuing Process Control and Process Capability Improvement. pp. [5] B. 109. Franklin and G. Gunter (1989). Part 2. [6] B. D. “Lower Confidence Limits on Process Capability Indices. 24. Spiring (1988). October. “The Use and Abuse of Cpk. 14-44 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE .S. pp. S. July. 22. and F. May. pp.” Journal of Quality Technology. [4] L. 22.A. Gunter (1989). July. Cheng.162–175. pp. Chou. 80.” Journal of Quality Technology. [3] Ford Motor Company (1983).79. 22.” Quality Progress.A. Michigan. Chan. Part 3.” Quality Progress.” Journal of Quality Technology.K. [2] Y. March. S. Borrego (1990). The chart of cumulative mean DPU (defects per unit) has settled down around the value 0. Ford Motor Company. Indiana. [12] T.” Journal of Quality Technology. 18.S.188–195. John Wiley & Sons.227–228. [8] V. “Reducing Variability: A New Approach to Quality.M. Western Electric Corporation. Ryan (1989). Kotz. 216–231. 24. and N. “Confidence Bounds for Capability Indices. Kane (1986). [14] H. October. Statistical Quality Control Handbook. July.L.P. Sullivan (1984). 24. Indianapolis. [11] R. [9] R.” Quality Progress. Statistical Methods for Quality Improvement. Modern Methods for Quality Control and Improvement. [15] Western Electric (1956). Wadsworth. 24. Johnson (1992).L.15– 21. MINITAB User’s Guide 2 14-45 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE . John Wiley & Sons. pp. 41–52.H. Godfrey (1986). pp. pp. Rodriguez (1992).E. Stephens.N.B.” Tappi. and A. Pearn. “How to Estimate Percentage of Product Failing Specifications.72. K.CONTENTS References INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE Process Capability [7] A. “Recent Developments in Process Capability Analysis. “Distributional and Inferential Properties of Process Capability Indices.” Journal of Quality Technology. October. Kushler and P.” Journal of Quality Technology. [13] L.” Journal of Quality Technology. pp. Jaehn (1989). October. S. 1984. “Process Capability Indices. pp. [10] W.H. Hurley (1992).P.176–187. pp. CONTENTS Chapter 14 INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE References 14-46 MINITAB User’s Guide 2 CONTENTS INDEX MEET MTB UGUIDE 1 UGUIDE 2 SC QREF HOW TO USE .