Process Capability Platform Overview
The Process Capability platform provides the tools needed to measure the compliance of a process to given specifications. By default, JMP shows a Goal Plot, Capability Box Plots, and a Capability Index Plot for the variables that you fit with normal distributions. Capability indices for nonnormal variables are plotted on the Capability Index Plot. You can add normalized box plots, summary reports, and individual detail reports for the variables in your analysis.
You can supply specification limits in several ways:
• in the data table, using a column property
• by requesting the Spec Limits Dialog in the launch window
• by loading the limits from a specification limits data table
You can specify two-sided, one-sided, or asymmetric specification limits.
Note: The Process Capability platform expands significantly on the Capability analyses that are available through Analyze > Distribution and through Analyze > Quality and Process > Control Chart.
Capability Indices
A capability index is a ratio that relates the ability of a process to produce product that meets specification limits. The index relates estimates of the mean and standard deviation of the quality characteristic to the specification limits. Within estimates of capability are based on an estimate of the standard deviation constructed from within-subgroup variation. Overall estimates of capability use an estimate of standard deviation constructed from all of the process data. See “Capability Indices for Normal Distributions” and “Variation Statistics”.
Estimates of the mean or standard deviation are well-defined only if the processes related to centering or spread are stable. Therefore, interpretation of within capability indices requires that process spread is stable. Interpretation of overall capability indices requires that both process centering and spread are stable.
Capability indices constructed from small samples can be highly variable. The Process Capability platform provides confidence intervals for most capability indices. Use these to determine the range of potential values for your quality characteristic’s actual capability.
Note: 
When confidence intervals are not provided (for example, for nonnormal distributions) you can use the Simulate feature to construct confidence intervals. For an example, see “Simulation of Confidence Limits for a Nonnormal Process Ppk”.
When confidence intervals are not provided (for example, for nonnormal distributions) you can use the Simulate feature to construct confidence intervals. For an example, see “Simulation of Confidence Limits for a Nonnormal Process Ppk”.Guidelines for values of capability indices can be found in Montgomery (2013). The minimum recommended value is 1.33. Six Sigma initiatives aim for much higher capability levels that correspond to extremely low rates of defective parts per million.
Capability Indices for Nonnormal Processes
The Process Capability platform constructs capability indices for process measurements with the following distributions: Normal, gamma, Johnson, lognormal, and Weibull. A Best Fit option determines the best fit among these distributions and provides capability indices for this fit. The platform also provides a Nonparametric fit option that gives nonparametric estimates of capability.
For the nonnormal methods, estimates are constructed using two approaches: the ISO/Quantile method (Percentiles) and the Bothe/Z-scores method (Z-Score). For details about these methods, see “Capability Indices for Nonnormal Distributions: Percentile and Z-Score Methods”.
Note: Process Capability analysis for individual responses is accessible through Analyze > Quality and Process > Control Chart Builder. However, nonnormal distributions are available only in the Process Capability platform.
Overall and Within Estimates of Sigma
Most capability indices in the Process Capability platform can be computed based on estimates of the overall (long-term) variation and the within-subgroup (short-term) variation. If the process is stable, these two measures of variation should yield similar results since the overall and within subgroup variation should be similar. The normalized box plots and summary tables can be calculated using either the overall or the within-subgroup variation. See “Additional Examples of the Process Capability Platform” for examples of capability indices computed for stable and unstable processes.
You can specify subgroups for estimating within-subgroup variation in the launch window. You can specify a column that defines subgroups or you can select a constant subgroup size. For each of these methods, you can choose to estimate the process variation using the average of the unbiased standard deviations or using the average of the ranges. If you do not specify subgroups, the Process Capability platform constructs a within-subgroup estimate of the process variation using a moving range of subgroups of size two. Finally, you can specify a historical sigma to be used as an estimate of the process standard deviation.
Capability Index Notation
The Process Capability platform provides two sets of capability indices. See “Capability Indices for Normal Distributions” for details about the calculation of the capability indices.
• Cpk, Cpl, Cpu, Cp, and Cpm. These indices are based on a within-subgroup (short-term) estimate of the process standard deviation.
• Ppk, Ppl, Ppu, Pp, and Cpm. These indices are based on an overall (long-term) estimate of the process standard deviation. Note that the process standard deviation does not exist if the process is not stable. See Montgomery (2013).
The Process Capability platform uses the appropriate AIAG notation for capability indices: Ppk labeling denotes an index constructed from an overall variation estimate and Cpk denotes an index constructed from a within-subgroup variation estimate.
Note: The AIAG (Ppk) Labeling platform preference is selected by default. You can change the reporting to use Cp notation only by deselecting this preference under Process Capability.
For more information about process capability analysis, see Montgomery (2013) and Wheeler (2004).
Example of the Process Capability Platform with Normal Variables
This example uses the Semiconductor Capability.jmp sample data table. The variables represent standard measurements that a semiconductor manufacturer might make on a wafer as it is being processed. Specification limits for the variables have been entered in the data table through the Column Properties > Spec Limits property.
1. Select Help > Sample Data Library and open Semiconductor Capability.jmp.
2. Select Analyze > Quality and Process > Process Capability.
3. Select PNP1, PNP2, NPN2, PNP3, IVP1, PNP4, NPN3, and IVP2, and click Y, Process.
4. Click OK.
5. Select Label Overall Sigma Points from the Goal Plot red triangle menu.
Figure 10.2 Example Results for Semiconductor Capability.jmp
The Goal Plot in Figure 10.2 shows the spec-normalized mean shift on the x-axis and the spec-normalized standard deviation on the y-axis for each variable. The triangular region defined by the red lines in the bottom center of the plot is the goal triangle. It defines a region of capability index values. You can adjust the goal triangle using the Ppk slider below the plot. When the slider is set to 1, note that PNP1, PNP3, IVP1, and IVP2 are outside of the goal triangle and possibly out of specification.
The Capability Box Plots report shows a box plot for each variable in the analysis. The values for each column are centered by their target value and scaled by the specification range. In this example, all process variables have both upper and lower specification limits, and these are symmetric about the target value. It follows that the solid green line shows where the target should be and the dashed lines represent the specification limits.
It appears that the majority of points for IVP1 are above its upper specification limit (USL), and the majority of points for IVP2 are less than its target. PNP2 seems to be on target with all data values inside the specification limits.
The Capability Index Plot plots the Ppk values for each variable. Four variables come from very capable processes, with Ppk values of 2 or more. Four variables have Ppk values below 1.
Example of the Process Capability Platform with Nonnormal Variables
The Process Measurements.jmp data table contains measurements made on seven different processes used to construct a product. For each process, specification limits are saved as column properties. You begin by examining the distributions of your process data. You see that the distributions are not normal. Then you use the nonnormal capability features of the Process Capability platform to compute capability indices.
View the Distributions
1. Select Help > Sample Data Library and open Process Measurements.jmp.
2. Select Analyze > Distribution.
3. Select all seven columns from the Select Columns list and click Y, Columns.
4. Check the box next to Histograms Only.
5. Click OK.
For most processes, the histograms show evidence that the theoretical distribution of measurements is skewed and does not follow a normal distribution. Therefore, for each process, you find the best fitting distributions among all of the available parametric distributions.
Perform a Capability Analysis
1. Select Analyze > Quality and Process > Process Capability.
2. Select all seven columns from the Columns list and click Y, Process.
3. Select all seven columns in the Y, Process list.
4. Open the Distribution Options panel and select Best Fit from the Distribution list.
5. Click Set Process Distribution.
The suffix &Dist(Best Fit) is added to each variable name in the Y, Process list. The Best Fit option specifies that the best-fitting parametric distribution should be fit to each variable. The available parametric distributions are normal, gamma, Johnson, lognormal, and Weibull. See Figure 10.3.
6. Open the Nonnormal Distribution Options outline. Note that the Nonnormal Capability Indices Method is set to Percentiles, the Johnson Distribution Fitting Method is set to Quantile Matching, and the Distribution Comparison Criterion is set to AICc.
Figure 10.3 Completed Launch Window
The Quantile Matching method is the default method used for fitting Johnson distributions because of its stability and speed as compared to Maximum Likelihood. Note that Maximum Likelihood is used in the Distribution platform.
7. Click OK.
8. Select Label Overall Sigma Points from the Goal Plot red triangle menu.
9. Select Label Overall Sigma Points from the Capability Index Plot red triangle menu.
Figure 10.4 Initial Report with Variables Labeled
Note: Click on a label in the plot and drag it to make the plot more interpretable. Click on the right side frame of the Capability Index Plot and drag it to the right to make the labels easier to distinguish.
The Goal Plot shows only one point and it corresponds to Process 7. The Capability Box Plots report shows a single box plot for Process 7. This is because the best fit for Process 7 is a normal distribution.
10. Beneath the Capability Index Plot, set the Ppk value to 2.
The Capability Index Plot shows Ppk values for all seven processes. Only two processes, Process 2 and Process 7, have capability values that exceed 2. Note that the best fitting nonnormal distributions are shown in parentheses to the right of the variable names in the Capability Index Plot. The best fitting distribution for Process 7 is not shown because it is a normal distribution.
11. Select Individual Detail Reports from the Process Capability red triangle menu.
Because you requested Best Fit in the launch window, the Compare Distributions option has been selected from each distribution’s red triangle menu.
12. Scroll to the report entitled Process 4(Lognormal) Capability.
Figure 10.5 Individual Detail Report for Process 4
The title of the report for Process 4 indicates that the capability calculations are based on a lognormal fit. All of the check boxes in the Compare Distributions report, except the one for Nonparametric, are checked, indicating that these five distributions are fit. (This is because you requested a Best Fit in the launch window.) The button that is selected in the Selected column indicates that the Lognormal distribution is the distribution that is used in the remainder of the Process 4(Lognormal) Capability report to estimate capability and nonconformance.
The Compare Distributions report enables you to compare the five distributional fits. The Histogram - Compare Distributions report gives a visual assessment of the fit and the Comparison Details report shows fit statistics for the selected distributions. Both the plot and the fit statistics indicate that the lognormal distribution gives the best fit among the selected distributions.
The Individual Detail Report information that is shown by default includes a histogram showing the estimated best-fit distribution, a summary of the process information, capability indices based on an overall estimate of sigma, parameter estimates for the fitted lognormal distribution, and observed and expected nonconformance levels.
Launch the Process Capability Platform
Launch the Process Capability Platform by selecting Analyze > Quality and Process > Process Capability. In Figure 10.6, which uses the Semiconductor Capability.jmp data table, all outlines and panels have been opened.
Figure 10.6 Process Capability Launch Window
The Process Capability launch window contains the following outlines and options:
After you click OK in the launch window, the Spec Limits window appears unless one of the following occurs:
• All of the columns contain specification limits.
• You selected No (skip columns with no spec limits) on the launch window.
The Spec Limits window also appears if you select Yes on the launch window. Otherwise, the Process Capability report window appears.
Process Selection
Select the process variables to include in the capability analysis.
Y, Process
Assigns the variables that you want to analyze.
Notes:
• The Transform menu is not available for the Select Column list in the Process Capability launch window. Right-click a column heading in the data table and select New Formula Column to create a transform column for use in Process Capability. See the Enter and Edit Data chapter in the Using JMP book for more information about creating new formula columns.
• Reference columns for virtually joined tables are not available in the Process Capability platform.
Process Subgrouping
This group of options enables you to assign each variable in the Y, Process list a subgroup ID column or a constant subgroup size.
Create Subgroups Using an ID Column
1. Select a variable or variables in the Y, Process list.
2. Select Subgroup ID Column from the Subgroup with options.
3. Select a subgroup ID column in the Select Columns list.
4. Click Nest Subgroup ID Column.
The subgroup ID column appears in brackets to the right of the variable names in the Y, Process list.
Create Subgroups Using a Constant Subgroup Size
1. Select a variable or variables in the Y, Process list.
2. Select Constant Subgroup Size from the Subgroup with options.
3. Enter the subgroup size next to Set Constant Subgroup Size.
4. Click Subgroup by Size.
The subgroup size appears in brackets to the right of the variable names in the Y, Process list.
Nest Subgroup ID Column
Available when you select Subgroup ID Column. Assigns a column that you select from the Select Columns list to define the subgroups for the selected Y, Process columns.
Subgroup by Size
Available when you select Constant Subgroup Size. Assigns the subgroup size that you specify in the Set Constant Subgroup Size box to define the subgroups for the selected Y, Process columns.
Set Constant Subgroup Size
Available when you select Constant Subgroup Size. Specify the constant subgroup size for the selected Y, Process columns. You need to assign this value using Subgroup by Size.
Within-Subgroup Variation Statistic
(Available when Process Subgrouping is used. Specifies if the within-subgroup estimate of standard deviation is calculated using standard deviations or ranges.
Historical Information
Use this outline to assign historically accepted values of the standard deviation to variables in the Y, Process list.
1. Select a variable or variables in the Y, Process list.
2. Enter a value next to Set Historical Sigma.
3. Select Use Historical Sigma to assign that value to the selected variables.
The specified value appears in parentheses in the expression “&Sigma()” to the right of the variable names in the Y, Process list.
Note: If you set a historical sigma, then subgroup assignments for the selected process variable are no longer relevant and are removed.
Distribution Options
Unless otherwise specified, all Y, Process variables are analyzed using the assumption that they follow a normal distribution. Use the Distribution Options outline to assign other distributions or calculation methods to variables in the Y, Process list and to specify options related to nonnormal calculations.
• The available distributions are the Normal, gamma, Johnson, lognormal, and Weibull distributions. Except for Johnson distributions, maximum likelihood estimation is used to fit distributions. See “Johnson Distribution Fit Method”.
• The Best Fit option determines the best fit among the available distributions and applies this fit.
• The Nonparametric option fits a distribution using kernel density estimation.
For more options related to nonnormal fits, see “Nonnormal Distribution Options”.
Specify a Distribution
1. Select a variable or variables in the Y, Process list.
2. Select a distribution from the Distribution list.
3. Select Set Process Distribution to assign that distribution to the selected variables.
The specified distribution appears in parentheses in the expression “&Dist()” to the right of the variable names in the Y, Process list.
Note: If you select a distribution other than Normal, you cannot assign a Subgroup ID column or a Historical Sigma. These selections are not supported by the methods used to calculate nonnormal capability indices. See “Capability Indices for Nonnormal Distributions: Percentile and Z-Score Methods”.
Nonnormal Distribution Options
Nonnormal Capability Indices Method
Specifies the method used to compute capability indices for nonnormal distributions. See “Capability Indices for Nonnormal Distributions: Percentile and Z-Score Methods”.
Johnson Distribution Fit Method
Specifies the method used to find the best-fitting Johnson distribution. Before estimating the parameters, the best-fitting family of distributions is determined from among the Johnson Su, Sb, and Sl families. The procedure described in Slifker and Shapiro (1980) is used to find the best-fitting family.
Quantile Matching
The default method. It is more stable and faster than Maximum Likelihood. Quantile Matching Parameter estimates, assuming the best-fitting family, are obtained using a quantile-matching approach. See Slifker and Shapiro (1980).
Maximum Likelihood
Parameters for the best-fitting family are determined using maximum likelihood.
Distribution Comparison Criterion
(Available when a Best Fit Distribution is selected.) Specify the criterion that you want to use in determining a Best Fit. This criterion also determines the ordering of distributions in the Comparison Details report. See “Order by Comparison Criterion”.
Other Specifications
By
Produces a separate report for each level of the By variable. If more than one By variable is assigned, a separate report is produced for each possible combination of the levels of the By variables.
Specify Alpha Level
Specifies the significance level for confidence limits.
Show Spec Limits Dialog
Specifies how to handle columns that do not have specification limits.
Note: It is good practice to ensure that specification limits for all process variables are specified as Spec Limits column properties or to load specification limits from a Limits Data table (see “Limits Data Table”). Otherwise, you can specify limits interactively in the Spec Limits window that appears after you click OK in the launch window (unless you select No (skip columns with no spec limits) on the launch window).
Entering Specification Limits
The lower specification limit (LSL), upper specification limit (USL), and target define the lower bound, upper bound, and target value for a quality process.
There are several ways to enter specification limits:
• Enter limits in the Spec Limits window after selecting columns in the launch window. See “Spec Limits Window”.
• Import limits from a JMP data table (known as a Limits Table). See “Limits Data Table”.
• Enter limits as Spec Limits column properties in the data table. See “Spec Limits Column Property”.
• If you are creating a Process Capability report by running a JSL script, enter limits in the script. See “The Process Capability Report”.
Only one specification limit is required for a selected column. If only the USL is specified, the box plots and Goal Plot point are colored blue. If only the LSL is specified, the box plots and Goal Plot point are colored red.
Spec Limits Window
After you click OK on the launch window, the Spec Limits window appears if any of the columns do not contain limits and you did not select No (skip columns with no spec limits) on the launch window. The Spec Limits window also appears if you select Yes on the launch window. Figure 10.7 shows the Spec Limits window for the Cities.jmp sample data table after selecting OZONE, CO, SO2, and NO as process variables in the launch window. Enter the known specification limits and click OK to view the Process Capability report.
Figure 10.7 Spec Limits Window for Cities.jmp
Limits Data Table
You can also specify a limits data table with the Load spec limits from data table option from the Spec Limits window. Click the Select Data Table button and then select the appropriate data table that contains the specification limits for the analysis. After you select the appropriate limits table, the values populate the window. Click OK to view the Process Capability report.
A limits data table can be in two different formats: tall or wide. A tall limits data table has one column for the responses and the limits key words are the other columns. A wide limits data table has a column for each response with one column to label the limits keys. Either of these formats can be read using the Load spec limits from data table option.
• A tall table contains four columns and has one row for each process. The first column has a character data type and contains the names of the columns analyzed in the Process Capability platform. The other three columns need to be named, _LSL, _USL, and _Target.
Figure 10.8 Example of a Tall Specification Limits Table
• A wide table contains three rows and one column for each column analyzed in the Process Capability platform plus a _LimitsKey column. In the _LimitsKey column, the three rows need to contain the identifiers _LSL, _USL, and _Target.
Figure 10.9 Example of a Wide Specification Limits Table
The easiest way to create a limits data table is to save results computed by the Process Capability platform. The Save Spec Limits option in the Process Capability red triangle menu automatically saves limits from the sample values. After entering or loading the specification limits, you can do the following:
• Select Save Spec Limits as Column Properties to save the limits to the columns in the data table.
• Select Save Spec Limits to New Table to save the limits to a new tall specification limits data table. If you have selected at least one nonnormal distribution, a column called Distribution that contains the specified distributions is also added to the limits data table.
For more information, see “Process Capability Platform Options”.
Spec Limits Column Property
When you perform a capability analysis, you can use Column Properties > Spec Limits to save specification limits as a column property. The Spec Limits property applies only to numeric columns.
Some processes have one-sided specifications. Some have no target. You can enter any of these that apply: a lower specification limit, an upper specification limit, or a target value.
Figure 10.10 displays the Spec Limits section of the Column Properties window for OZONE in the sample data table Cities.jmp.
Figure 10.10 Spec Limits Section of the Column Properties Window
Tip: Saving specification limits as a column property ensures consistency when you repeat an analysis.
The Process Capability Report
By default, the Process Capability platform provides the following reports:
• “Goal Plot” (provided only if at least one variable is fit with a normal distribution and shows only points for variables fit with normal distributions)
• “Capability Box Plots” (provided only if at least one variable is fit with a normal distribution and shows only box plots for variables fit with normal distributions)
Figure 10.2 shows an example of a default Process Capability report.
Using the Process Capability red triangle menu, you can add individual detail reports, normalized box plots, and summary reports. The red triangle menu also has options for identifying out-of-spec values in your data table, creating a summary data table, changing the display order of analyzed columns, and saving out spec limits. These options are described in “Process Capability Platform Options”.
You can change the default report at File > Preferences > Platforms > Process Capability. You can also make changes to the appearance of reports produced by options by selecting the relevant Process Capability topic at File > Preferences > Platforms.
Goal Plot
The Goal Plot shows, for each variable, the spec-normalized mean shift on the x-axis, and the spec-normalized standard deviation on the y-axis. It is useful for getting a quick, summary view of how the variables are conforming to specification limits. By default, the Goal Plot shows only those points for each column that are calculated using the overall sigma. Hold your cursor over each point to view the variable name and the sigma method used to calculate the point. See “Goal Plot” for details about the calculation of the coordinates for the Goal Plot.
Note: Process variables with distributions other than Normal are not plotted on the Goal Plot.
Goal Plot Points
Points on the Goal Plot correspond to columns, not rows. Selecting a point in the Goal Plot selects the corresponding column in the data table.
The points on the Goal Plot are also linked to the rows of the Goal Plot Summary Table, where each row corresponds to a column. You can select a point in the Goal Plot, right-click, and apply row states. These row states are applied to the rows of the Goal Plot Summary Table. Row states that you apply in the Goal Plot Summary Table are reflected in the Goal Plot. To see this table, select Make Goal Plot Summary Table from the Process Capability red triangle menu. See “Make Goal Plot Summary Table”.
Tip: If you hide a point in the Goal Plot, you can show the point again by changing the corresponding row state in the Goal Plot Summary Table.
Goal Plot Triangle
The goal plot triangle appears in the center of the bottom of the Goal Plot. The slider beneath the plot enables you to adjust the size of goal triangle in the plot.
By default, the Ppk slider and the value beneath it are set to Ppk = 1. This approximates a non-conformance rate of 0.0027, if the distribution is normal. The goal triangle represents the Ppk shown in the box. To change the Ppk value, move the slider or enter a number in the box.
JMP gives the Goal Plot in terms of Ppk values by default. You can change this preference at File > Preferences > Platforms > Process Capability. When the AIAG (Ppk) Labeling preference is unchecked, all of the Ppk labeling is changed to Cpk labeling, including the label of the slider under the goal plot.
Goal Plot Options
The Goal Plot red triangle menu has the following options:
Show Within Sigma Points
Shows or hides the points calculated using the within sigma estimate.
Show Overall Sigma Points
Shows or hides the points calculated using the overall sigma estimate.
Shade Levels
Shows or hides the Ppk level shading. See Figure 10.11. When you select Shade Levels, shaded areas appear in the plot. The shaded areas are described as follows, with p representing the value shown in the box beneath Ppk:
‒ Points in the red area have Ppk < p.
‒ Points in the yellow area have p < Ppk < 2p.
‒ Points in the green area have 2p < Ppk.
Label Within Sigma Points
Shows or hides labels for points calculated using the within sigma estimate.
Label Overall Sigma Points
Shows or hides labels for points calculated using the overall sigma estimate.
Defect Rate Contour
Shows or hides a contour representing a specified defect rate.
Figure 10.11 shows the Goal Plot for the entire data set for the Semiconductor Capability.jmp sample data table after selecting Shade Levels and Show Within Sigma Points from the Goal Plot red triangle menu.
Figure 10.11 Goal Plot
One-Sided or Missing Specification Limits
When there is only one specification limit for a column, markers and colors are used in the following ways:
• If only the upper specification limit (USL) is specified, the point on the Goal Plot is represented by a right-pointing triangle and is colored blue.
• If only the lower specification limit (LSL) is specified, the point on the Goal Plot is represented by a left-pointing triangle and is colored red.
• If at least one process has only an upper specification limit, the right half of the goal triangle is blue.
• If at least one process has only a lower specification limit, the left half of the goal triangle is red.
Processes with only an upper specification limit are represented by blue and should be compared to the blue (right) side of the goal triangle. Processes with only a lower specification limit are represented by red and should be compared to the red (left) side of the goal triangle. For details about how the coordinates of points are calculated, see “Goal Plot”.
Capability Box Plots
The Capability Box Plots show a box plot for each variable selected in the analysis. The values for each column are centered by their target value and scaled by the difference between the specification limits. If the target is not centered between the specification limits, the values are scaled by twice the minimum difference between the target and specification limits. For each process column Yj (see “Notation for Goal Plots and Capability Box Plots” for a description of the notation):
For a process with a one-sided specification, see “One-Sided or Missing Specification Limits”. For the situation where no target is specified, see “Capability Box Plots for Processes with Missing Targets”.
Note: Process variables with distributions other than Normal are not plotted on the Capability Box Plot.
Figure 10.11 shows a Capability Box Plots report for eight variables in the Semiconductor Capability.jmp sample data table.
Figure 10.12 Capability Box Plot
The plot displays dotted green lines drawn at ±0.5.
• For a process with a target that is centered between its specification limits, the dotted green lines represent the standardized specification limits.
• For a process with a target that is not centered between its specification limits, one of the dotted green lines represents the standardized specification limit for the limit closer to the target. The other dotted green line represents the same distance in the opposite direction.
This plot is useful for comparing variables with respect to their specification limits. For example, in Figure 10.12, the majority of points for IVP1 are above its USL, and the majority of its points for IVP2 are less than its target. PNP2 seems to be on target with all data points in the specification limits.
One-Sided or Missing Specification Limits
When there is only one specification limit for a column, colors are used in the following ways:
• If only the upper specification limit (USL) is specified, the box plot is colored blue.
• If only the lower specification limit (LSL) is specified, the box plot is colored red.
• If at least one process has only an upper specification limit, the dotted line at 0.5 is blue.
• If at least one process has only a lower specification limit, the dotted line at -0.5 is red.
Suppose that only the lower specification limit is specified and that the process target is specified. The capability box plot is based on the following values for the transformed observations. See “Notation for Goal Plots and Capability Box Plots” for a description of the notation:
Suppose that only the upper specification limit is specified and that the process target is specified. The capability box plot is based on the following values for the transformed observations:
For details about how missing targets are handled with one-sided specification limits, see “Single Specification Limit and No Target”.
Capability Index Plot
The Capability Index Plot shows Ppk values for all variables that you entered as Y, Process.
• Each variable name appears on the horizontal axis. If you fit a nonnormal distribution, the fitted distribution name appears in the plot as a parenthetical suffix to the variable name.
• The vertical axis shows Ppk values.
• A horizontal line is placed at the Ppk value specified by the slider beneath the plot.
Figure 10.13 shows a Capability Index Plot report for the Process Measurements.jmp sample data table. Seven of the variables are fit with nonnormal distributions. Process 7 is fit with a normal distribution. Points have been labeled using the Label Overall Sigma Points option that is available in the Capability Index Plot red triangle menu.
Figure 10.13 Capability Index Plot with Nonnormal Distributions
Capability Index Plot Options
The Capability Index Plot red triangle menu has the following options:
Show Within Sigma Points
Shows or hides the points calculated using the within sigma estimate.
Show Overall Sigma Points
Shows or hides the points calculated using the overall sigma estimate.
Shade Levels
Shows or hides the Ppk level shading. When you select Shade Levels, shaded areas appear in the plot. The shaded areas are described as follows, with p representing the value shown in the box beneath Ppk:
‒ Points in the red area have Ppk < p.
‒ Points in the yellow area have p < Ppk < 2p.
‒ Points in the green area have 2p < Ppk.
Label Within Sigma Points
Shows or hides labels for points calculated using the within sigma estimate.
Label Overall Sigma Points
Shows or hides labels for points calculated using the overall sigma estimate.
Process Capability Platform Options
The Process Capability red triangle menu contains the following options:
Individual Detail Reports
Shows or hides individual detail reports for each variable in the analysis. See “Individual Detail Reports” for more information.
Goal Plot
Shows or hides a goal plot for the data. The Goal Plot shows the spec-normalized mean shift on the x-axis and the spec-normalized standard deviation on the y-axis for each variable. See “Goal Plot” for more information. (Only variables for which you specify normal distributions are shown on the plot.)
Capability Box Plots
Shows or hides a capability box plot for each variable in the analysis. The values for each column are centered by their target value and scaled by twice the minimum difference between the target value and the specification limits. See “Capability Box Plots” for more information. (Box plots are shown only for variables for which you specify normal distributions.)
Normalized Box Plots
Provides two options for plots that show normalized box plots for each process variable. Each column is standardized by subtracting its mean and dividing by an estimate of the column’s standard deviation. The box plot is constructed using quantiles for the standardized values. See “Normalized Box Plots” for more information. (Normalized box plots are shown only for variables for which you specify normal distributions.)
Within Sigma Normalized Box Plots
Shows or hides a plot called Within Sigma Normalized Box Plots. The box plots are constructed using the within-subgroup estimate of standard deviation.
Overall Sigma Normalized Box Plots
Shows or hides a plot called Overall Sigma Normalized Box Plots. The box plots are constructed using the overall estimate of standard deviation.
Capability Index Plot
Shows overall Ppk values for all variables that you entered as Y, Process. See “Capability Index Plot”.
Summary Reports
Provides two options for summary reports of capability indices. See “Summary Reports” for more information.
Within Sigma Summary Report
Shows or hides a summary report of capability indices calculated using the within-subgroup estimate of standard deviation. (Results are available only for variables with specified normal distributions.)
Overall Sigma Summary Report
Shows or hides a summary report of capability indices calculated using the overall estimate of standard deviation.
Action Options
The following red triangle menu options perform actions:
Out of Spec Values
Provides options for the cells in the data table containing values that are out of spec.
Select Out of Spec Values
Selects all rows and columns in the data table that contain at least one value that does not fall within the specification limits.
Color Out of Spec Values
Colors the cells in the data table that correspond to values that are out of spec. The cell is colored blue if the value is above the USL and red if the value is below the LSL.
Tip: To remove colors in specific cells, select all cells of interest. Right-click in one of the cells and select Clear Color. To remove colors in all cells, deselect Color Out of Spec Values.
Make Goal Plot Summary Table
Creates a summary table for the points plotted in the Goal Plot. This table includes the variable’s name, its spec-normalized mean shift, and its spec-normalized standard deviation. Each variable has two rows in this table: one for each sigma type (within and overall). See “Make Goal Plot Summary Table” for more information.
Order By
Reorders the box plots, summary reports, and individual detail reports. You can reorder by Initial Order, Reverse Initial Order, Within Sigma Cpk Ascending, Within Sigma Cpk Descending, Overall Sigma Ppk Ascending, or Overall Sigma Ppk Descending. The options that order by Within Sigma reorder plot elements only for variables with specified normal distributions.
Save Spec Limits
Provides options for saving specification limits.
Save Spec Limits as Column Properties
Saves the specification limits to a column property for each variable in the analysis. If no spec limit column property is present, the column property is created. If a spec limit column property is present, the values in the column property are overwritten. See “Spec Limits Column Property” for more information.
Save Spec Limits to New Table
Saves the specification limits to a limits data table in tall format. See “Limits Data Table” for more information.
Save Distributions as Column Properties
Saves the distribution used in calculating capability as a Process Capability Distribution column property. See the Column Info Window chapter in the Using JMP book.
If a column contains the Distribution property specifying a nonnormal distribution and no Process Capability Distribution property, then the Process Capability platform applies a nonnormal fit. The Process Capability platform uses the distribution specified in the Distribution column property, or a Johnson fit if that distribution is not supported in Process Capability. If a column contains the Process Capability Distribution property, then the Process Capability platform uses the distribution specified in the Process Capability Distribution column property.
Note: If you want to use a specific distribution in the Process Capability platform, save it as a Process Capability Distribution column property.
Relaunch Dialog
Opens the platform launch window and recalls the settings used to create the report.
See the JMP Reports chapter in the Using JMP book for more information about the following options:
Local Data Filter
Shows or hides the local data filter that enables you to filter the data used in a specific report.
Redo
Contains options that enable you to repeat or relaunch the analysis. In platforms that support the feature, the Automatic Recalc option immediately reflects the changes that you make to the data table in the corresponding report window.
Save Script
Contains options that enable you to save a script that reproduces the report to several destinations.
Save By-Group Script
Contains options that enable you to save a script that reproduces the platform report for all levels of a By variable to several destinations. Available only when a By variable is specified in the launch window.
Individual Detail Reports
The Individual Detail Reports option displays a capability report for each variable in the analysis.
Normal Distributions
Figure 10.14 shows the Individual Detail Report for PNP1 from the Semiconductor Capability.jmp sample data table as described in “Example of the Process Capability Platform with Normal Variables”.
Figure 10.14 Individual Detail Report
The Individual Details report for a variable with a normal distribution shows a histogram, process summary details, and capability and nonconformance statistics. The histogram shows the distribution of the values, the lower and upper specification limits and the process target (if they are specified), and one or two curves showing the assumed distribution. The histogram in Figure 10.14 shows two normal curves, one based on the overall estimate of standard deviation and the other based on the within-subgroup estimate.
When you fit your process with a normal distribution, the Process Summary includes the Stability Ratio, which is a measure of stability of the process. The stability ratio is defined as follows:
(Overall Sigma/Within Sigma)2
A stable process has stability ratio near one. Higher values indicate less stability.
Nonnormal Distributions
Note: Capability indices based on within-subgroup variation and stability ratios are not available for processes for which you have specified nonnormal distributions.
Figure 10.15 shows the Individual Detail Report for Process 1 from the Process Measurements.jmp sample data table as described in “Example of the Process Capability Platform with Nonnormal Variables”.
Figure 10.15 Individual Detail Report for Process 1
The report opens with a note summarizing the Nonnormal Distribution Options that you selected in the launch window.
The Individual Details report for a variable with a nonnormal distribution shows a histogram, process summary details, and capability and nonconformance statistics. The histogram shows the distribution of the values, the lower and upper specification limits and the process target (if they are specified). A curve showing the fitted distribution is superimposed on the histogram. If you selected a Nonparametric distribution, the curve shown in the histogram is the nonparametric density.
The report also shows a Parameter Estimates report if you selected a nonnormal parametric distribution or a Nonparametric Density report if you selected a Nonparametric fit. See “Parameter Estimates” and “Nonparametric Density”.
Individual Detail Report Options
The outline title for each variable in the Individual Detail Reports section is of the form <Variable Name> Capability. However, if you request nonnormal capability, the relevant distribution name is shown parenthetically in the outline title.
Each Capability report has a red triangle menu with the following options:
Compare Distributions
Shows or hides the control panel for comparing distributions for the process. See “Compare Distributions”.
Process Summary
Shows or hides the summary statistics for the variable, including the overall sigma estimate, and, if you have specified a normal distribution, the within sigma estimate and the stability ratio.
Histogram
Shows or hides the histogram of the values of the variable. The histogram report includes a red triangle menu that controls the following features of the histogram:
Show Spec Limits
Shows or hides vertical red lines on the histogram at the specification limits for the process.
Show Target
Shows or hides a vertical green line on the histogram at the process target.
Show Within Sigma Density
Shows or hides an approximating normal density function on the histogram with mean given by the sample mean and standard deviation given by the within estimate of sigma.
Show Overall Sigma Density
Shows or hides an approximating normal density function on the histogram with mean given by the sample mean and standard deviation given by the overall estimate of sigma.
Show Count Axis
Shows or hides an additional axis to the right of the histogram plot showing the count of observations.
Show Density Axis
Shows or hides an additional axis to the right of the histogram plot showing the proportion of observations.
Capability Indices
Controls display of the following capability index reports:
Within Sigma Capability
(Available when distribution is Normal.) Shows or hides capability indices (and confidence intervals) based on the within (short-term) sigma.
Within Sigma Z Benchmark
(Available when distribution is Normal.) Shows or hides Z benchmark indices based on the within (short-term) sigma.
Overall Sigma Capability
Shows or hides capability indices (and confidence intervals) based on the overall (long-term) sigma.
Overall Sigma Z Benchmark
(Available when distribution is Normal.) Shows or hides Z benchmark indices based on the overall (long-term) sigma.
Nonconformance
Shows or hides the observed and expected percentages of observations below the LSL, above the USL, and outside of the specification limits. The Nonconformance table contains hidden columns for observed and expected PPM and counts.
Parameter Estimates
(Available when a distribution other than Normal or Nonparametric is selected.) Shows or hides the Parameter Estimates report, which gives estimates for the parameters of the selected distribution.
The estimates for all except the Johnson family distributions are obtained using maximum likelihood. For details about Johnson family fits, see “Johnson Distribution Fit Method”.
The parameters and probability density functions for the normal, gamma, Johnson, lognormal, and Weibull distributions are described in “Capability Indices for Nonnormal Distributions: Percentile and Z-Score Methods”. These are the same parameterizations used in the Distribution platform, with the exception that Process Capability does not support threshold parameters. See the Distributions chapter in the Basic Analysis book.
Nonparametric Density
(Available when Nonparametric is selected as the distribution.) Shows or hides the Nonparametric Density report, which gives the kernel bandwidth used in fitting the nonparametric distribution. The kernel bandwidth is given by the following, where n is the number of observations and S is the uncorrected sample standard deviation:
Compare Distributions
The Compare Distributions report enables you to compare and apply various distributional fits. Note the following:
• Your selected distribution is indicated in the Selected column.
• The report initially shows fit statistics for your Selected distribution and other fitted distributions in the Comparison Details report. If you selected Best Fit, the Comparison Details report initially shows statistics for all parametric fits.
• Check the distributions in the Distribution list that you want to compare.
‒ The probability density function for the best fitting distribution in each family that you select is superimposed on the histogram in the Histogram - Compare Distributions report.
‒ If the distribution is parametric, a row for that family containing fit results is added to the Comparison Details report.
‒ If Nonparametric is checked in the Distribution list, the Nonparametric Density report, showing the best fitting kernel bandwidth, is added to the Compare Distributions report. See “Nonparametric Density”.
‒ You can change your selected distribution by selecting its radio button under Selected. The capability report is updated to show results for the selected distribution.
Figure 10.16 shows the Compare Distributions report for Process 1 in the Process Measurements.jmp sample data table. The Selected distribution, which is Lognormal, is being compared to a Normal distribution. The Comparison Details report shows fit statistics for both distributions.
You can obtain probability plots by selecting the Probability Plots option from the Compare Distributions red triangle menu. The points in the probability plot for the normal distribution in Figure 10.16 do not follow the line closely. This indicates a poor fit.
Figure 10.16 Compare Distributions with Probability Plot for Normal
Compare Distributions Options
The following options are available in the Compare Distributions red triangle menu.
Comparison Details
For each distribution, gives AICc, BIC, and -2Loglikelihood values. See the Statistical Details appendix in the Fitting Linear Models book. (Not available for a Nonparametric fit.)
Comparison Histogram
Shows or hides the Histogram report.
Probability Plots
Shows or hides a report that displays probability plots for each parametric distribution that you fit. See Figure 10.16. An observation’s horizontal coordinate is its mid-point adjusted Kaplan-Meier estimate. An observation’s vertical coordinate is the value of the quantile of the fitted distribution for the observation’s rank. For the normal distribution, the overall estimate of sigma is used in determining the fitted distribution.
Note: When it is not possible to calculate the quantiles for a probability scale for the gamma distribution, the plot shows a linear scale.
The red triangle menus associated with each Probability Plot contain the following options.
Simultaneous Empirical Confidence Limits
Shows or hides confidence limits that have a simultaneous 95% confidence level of containing the true probability function, given that the data come from the selected parametric family. These limits have the same estimated precision at all points. Use them to determine whether the selected parametric distribution fits the data well. See Nair (1984) and Meeker and Escobar (1998).
Caution: The simultaneous empirical confidence limits are not affected by the selection of Alpha Level in the Process Capability launch window.
Simultaneous Empirical Confidence Limits Shading
Shows or hides shading of the region between the Simultaneous Empirical Confidence Limits.
Parametric Fit Line
Shows or hides the line that shows the predicted probabilities for the observations based on the fitted distribution.
Parametric Fit Confidence Limits Shading
Shows or hides shading of the region between parametric fit confidence intervals. The parametric fit confidence limits have confidence level (1 - Alpha), where Alpha is the value that you specify in the launch window. (Available only when the parametric fit confidence limits are meaningful and when it is possible to calculate them.)
When possible, the intervals are computed by expressing the parametric distribution F as a location-scale family, so that F(y) = G(z), where z = (y - μ)/σ. The approximate standard error of the fitted location-scale component at a point is computed using the delta method. Using the standard error estimate, a Wald confidence interval for z is computed for each point. The confidence interval for the cumulative distribution function F is obtained by transforming the Wald interval using G. Note that, in some cases, special accommodations are required to provide appropriate intervals near the endpoints of the interval of process measurements.
Order by Comparison Criterion
Orders the distributions in the Comparison Details report according to the criterion that you select. The default ordering is by AICc, unless you selected another criterion in the Distribution Comparison Criterion panel in the launch window.
Normalized Box Plots
The Within Sigma Normalized Box Plots and Overall Sigma Normalized Box Plots options show or hide box plots that have been normalized using the within sigma and overall sigma, respectively. When drawing normalized box plots, JMP standardizes each column by subtracting the mean and dividing by the standard deviation. The box plots are formed for each column using these standardized values.
Figure 10.17 Within Sigma Normalized Box Plot
Figure 10.17 shows the Within Sigma Normalized Box Plot for a selection of the process variables in the Semiconductor Capability.jmp sample data table using wafer as a subgroup variable.
The green vertical lines represent the specification limits for each variable normalized by the mean and standard deviation of each variable. The gray dotted vertical lines are drawn at ±0.5, since the data is standardized to a standard deviation of 1.
Summary Reports
The Within Sigma Capability Summary Report and Overall Sigma Capability Summary Report options show or hide a table that contains the following statistics for each variable: LSL, Target, USL, Sample Mean, Sigma, Cpk, Cpl, Cpu, Cp, Cpm, and Nonconformance statistics. These statistics are calculated using the within sigma and overall sigma, respectively. Figure 10.18 shows a subset of columns for both summary reports as described in “Example of the Process Capability Platform with Normal Variables”. The following optional columns are available for this report:
• Confidence intervals for Cpk, Cpl, Cpu, CP, and Cpm
• Expected and observed PPM statistics (outside, below LSL, above USL)
• Sample standard deviation
• The sample size (N), the minimum, and the maximum.
To reveal these optional columns, right-click on the report and select the column names from the Columns submenu.
Note that the report (based on overall sigma) shows the overall capability indices Ppk, Ppl, Ppu, and Pp instead of the within capability indices Cpk, Cpl, Cpu, and Cp. The labeling of the overall capability indices depends on the setting of the AIAG (Ppk) Labeling preference.
Figure 10.18 Within Sigma and Overall Sigma Capability Summary Reports
Make Goal Plot Summary Table
The Make Goal Plot Summary Table option produces a summary data table that includes each variable’s name, its spec-normalized mean shift (Mean Shift Standardized to Spec), and its spec-normalized standard deviation (Std Dev Standardized to Spec). For each variable, there is a row for each of the two sigma types (Within and Overall).
Note: If a variable is fit with a distribution other than normal, the name of the fitted distribution is appended parenthetically to the variable name. The Mean Shift Standardized to Spec and Std Dev Standardized to Spec values are not provided for nonnormal variables.
The points in the Goal Plot are linked to the rows in the Goal Plot Summary Table. If you apply row states to a point in the Goal Plot, you can change the corresponding row states in the Goal Plot Summary Table. Conversely, if you apply row states in the Goal Plot Summary Table, they are reflected on the Goal Plot.
Figure 10.19 shows the Goal Plot Summary Table for the Semiconductor Capability.jmp sample data table as described in “Example of the Process Capability Platform with Normal Variables”.
Figure 10.19 Summary Table
Additional Examples of the Process Capability Platform
Process Capability for a Stable Process
In this example, you verify the assumptions that enable you to estimate PPM defective rates based on a capability analysis. You access Process Capability through Control Chart Builder and then directly. The data consist of 22 subgroups of size five. There are six missing readings, with three in each of two consecutive subgroups.
Process Capability through Control Chart Builder
You can use Control Chart Builder to check process stability and the normality assumption for your process characteristic. You can also obtain Process Capability information directly within Control Chart Builder.
1. Select Help > Sample Data Library and open Quality Control/Clips2.jmp.
2. Right-click the Gap column and select Column Info.
3. Select the Spec Limits column property.
4. Select Show as graph reference lines and click OK.
5. Select Analyze > Quality and Process > Control Chart Builder.
6. Drag Date to the Subgroup zone.
7. Drag Gap to the Y zone.
Figure 10.20 XBar and R Control Chart for Gap
The control chart indicates that Gap is stable over time. Because Gap has the Spec Limits column property, a Process Capability Analysis report appears to the right of the control chart.
Figure 10.21 Histogram in Process Capability Analysis for Gap
The histogram and fitted normal blue curve suggest that the distribution of Gap is approximately normal. Although the process is stable, the distribution of Gap is shifted to the right of the specification range.
The Process Summary report shows the specification limits that are saved to the Spec Limits column property. It also shows that the estimate of sigma calculated from within-subgroup variation (Within Sigma) does not differ greatly from the overall estimate given by the sample standard deviation (Overall Sigma). Consequently, the Stability Ratio is near one (0.958966). This is expected because the process is stable.
8. Right-click in the body of the Nonconformance report and select Expected Within PPM from the Columns submenu.
Figure 10.22 Capability Indices and Nonconformance Report
The Cpk value calculated using subgroup variation is 0.966, indicating that the process is not very capable. The Cpl value suggests good performance, but this is because the process is shifted away from the lower specification limit. Defective parts generally result from large values of Gap.
Note that the confidence interval for Cpk is wide; it ranges from 0.805 to 1.128. This occurs even though there are 104 observations. Capability indices are surprisingly variable, due to the fact that they are ratios. It is easy to reach incorrect conclusions based on the point estimate of a capability index.
The estimates of out-of-specification product given in the Nonconformance report provide a direct measure of process performance. The PPM values in the Nonconformance report indicate that Gap hardly ever falls below the lower specification limit (1.4 parts per million). However, the number of parts for which Gap falls above the upper specification limit is 1869.0 parts per million.
For an uncentered process, the Cp value indicates potential capability if the process were adjusted to be centered. If this process were adjusted to be centered at the target value of 14.8, then its capability would be 1.264, with a confidence interval from 1.071 to 1.457.
Process Capability Platform
Now that you have verified stability and normality for Gap, you can obtain additional information in the Process Capability platform.
1. Select Analyze > Quality and Process > Process Capability.
2. Select Gap and click Y, Process.
3. Open the Process Subgrouping outline.
4. Select Date in the Select Columns list and Gap in the Roles list.
5. Click Nest Subgroup ID Column.
By default, the Within-Subgroup Variation Statistic selection is set to Average of Unbiased Standard Deviations. In the Control Chart Builder example (“Process Capability through Control Chart Builder”), subgroup ranges were used.
6. Click OK.
Figure 10.23 Goal Plot and Box Plot for Gap
The Goal Plot shows the Ppk index for Gap as being essentially equal to 1. The box plot shows that most values fall within specifications, but the preponderance of data values are shifted to the right within the specification range.
7. From the Process Capability red triangle menu, select Individual Detail Reports.
The report is the one obtained using Control Chart Builder, except that the Within Sigma is based on average standard deviations rather than average ranges. See “Histogram in Process Capability Analysis for Gap” and “Capability Indices and Nonconformance Report”.
Process Capability for an Unstable Process
The following example shows a case where the overall variation differs from the within variation because the process is not stable. It uses the Coating.jmp data table from the Quality Control folder of Sample Data (taken from the ASTM Manual on Presentation of Data and Control Chart Analysis). The process variable of interest is the Weight column, grouped into subgroups by the Sample column.
Process Capability Platform
1. Select Help > Sample Data Library and open Quality Control/Coating.jmp.
2. Select Analyze > Quality and Process > Process Capability.
3. Select Weight and click Y, Process.
4. Open the Process Subgrouping outline.
5. Select Sample in the Select Columns list on the left.
6. Select Weight in the Cast Selected Columns into Roles list on the right.
7. Click Nest Subgroup ID Column.
8. Click OK.
9. Enter 16 for LSL, 20 for Target, and 24 for USL in the Spec Limits window.
10. Click OK.
11. Select Show Within Sigma Points from the Goal Plot red triangle menu.
12. Select Individual Detail Reports from the Process Capability red triangle menu.
Figure 10.24 Process Capability Report for Coating.jmp Data
Figure 10.24 shows the resulting Process Capability report. The Goal Plot shows two points that represent the mean shift and standard deviation standardized to the specification limits. The point labeled Overall Sigma is calculated using the overall sample standard deviation. The point labeled Within Sigma is calculated using a within-subgroup estimate of the standard deviation.
The point calculated using Overall Sigma is outside the goal triangle corresponding to a Ppk of 1. This indicates that the variable Weight results in non-conforming product.
However, the point calculated using Within Sigma is inside the goal triangle. This indicates that, if the process were stable, Weight values would have a high probability of falling within the specification limits.
Control Chart to Assess Stability
Use Control Chart Builder to determine whether the Weight measurements are stable.
1. Select Help > Sample Data Library and open Quality Control/Coating.jmp.
1. Select Analyze > Quality and Process > Control Chart Builder.
2. Drag Sample to the Subgroup zone.
3. Drag Weight to the Y zone.
Figure 10.25 XBar and R Chart for Weight
The control chart indicates that the Weight measurements are unstable. The process is affected by special causes and is unpredictable. This makes the interpretation of capability indices and nonconformance estimates highly questionable. Even estimates based on Overall Sigma are questionable, because the process is not predictable.
The histogram in Figure 10.24 shows the distribution of the Weight values with normal density curves using both sigma estimates superimposed over the histogram. The normal curve that uses the Overall Sigma estimate is flatter and wider than the normal curve that uses the Within Sigma estimate. This normal curve is more dispersed because the estimate of Overall Sigma is inflated by the special causes that make the process unstable. If the process were stable, the narrower normal curve would reflect process behavior.
You can also compare the Cpk estimate (1.142) to the Ppk estimate (0.814). The fact that Ppk is much smaller than Cpk is additional evidence that this is an unpredictable process. The Cpk estimate is a forecast of the capability that you would achieve by bringing the process to a stable state.
Note: The Individual Detail Reports Cutoff preference determines whether the Individual Reports appear by default. If the preference is enabled, the Individual Reports appear by default if the number of process variables is less than or equal to the number specified in the preference. You can change this preference in Preferences > Platforms > Process Capability.
Simulation of Confidence Limits for a Nonnormal Process Ppk
In this example, you first perform a capability analysis for the three nonnormal variables in Tablet Measurements.jmp. You then use Simulate to find confidence limits for the nonconformance percentage for the variable Purity.
Nonnormal Capability Analysis
If you prefer not to follow the steps below, you can obtain the results in this section by running the Process Capability table script in Tablet Measurements.jmp.
1. Select Help > Sample Data Library and open Tablet Measurements.jmp.
2. Select Analyze > Quality and Process > Process Capability.
3. Select Weight, Thickness, and Purity and click Y, Process.
4. Select Weight, Thickness, and Purity in the Cast Selected Columns into Roles list on the right.
5. Open the Distribution Options outline.
6. From the Distribution list, select Best Fit.
7. Click Set Process Distribution.
The &Dist(Best Fit) suffix is added to each column name in the list on the right.
8. Click OK.
A Capability Index Plot appears, showing the Ppk values. Note that only the Thickness variable appears above the line that denotes Ppk = 1. Purity is nearly on the line. Although the number of measurements, 250, seems large, the estimated Ppk value is still quite variable. For this reason, you construct a confidence interval for the true Purity Ppk value.
Note: Because a Goal Plot is not shown, you can conclude that a normal distribution fit was not the best fit for any of the three variables.
9. Select Individual Detail Reports from the Process Capability red triangle menu.
The best fits are as follows:
‒ Weight: Lognormal
‒ Thickness: Johnson Sb (see the note immediately beneath the Thickness(Johnson) Capability report title)
‒ Purity: Weibull
Construct the Simulation Column
To use the Simulate utility to estimate Ppk confidence limits, you need to construct a simulation formula that reflects the fitted Weibull distribution. If you prefer not to follow the steps below, you can obtain the results in this section by running the Add Simulation Column table script.
1. Scroll to the Purity (Weibull) Capability report and find the Parameter Estimates report.
Figure 10.26 Weibull Parameter Estimates for Purity
These are the parameter estimates for the best fitting distribution, which is Weibull.
1. In the Tablet Measurements.jmp sample data table, select Cols > New Columns.
2. Next to Column Name, enter Simulated Purity.
3. From the Column Properties list, select Formula.
4. In the formula editor, select Random > Random Weibull.
5. The placeholder for beta is selected. Click the insertion element (^).
Figure 10.27
This adds a placeholder for the parameter alpha.
6. Right-click in the Parameter Estimates report table and select Make into Data Table.
7. Copy the entry in Row 2 in the Estimate column (1589.7167836).
8. In the formula editor window, select the placeholder for beta in the Random Weibull formula and paste 1589.7167836 into the placeholder for beta.
9. In the data table that you created from the Parameter Estimates report, copy the entry in Row 1 in the Estimate column (99.918708989).
10. In the formula editor window, select the placeholder for alpha in the Random Weibull formula and paste 99.918708989 into the placeholder for alpha.
Figure 10.28 Completed Formula Window
11. Click OK in the formula editor window.
The Simulated Purity column contains a formula that simulates values from the best-fitting distribution.
Simulate Confidence Intervals for Purity Ppk and Expected % Nonconforming
When you use Simulate, the entire analysis is run the number of times that you specify. To shorten the computing time, you can minimize the computational burden by running only the required analysis. In this example, because you are interested only in Purity with a fitted Weibull distribution, you perform only this analysis before running Simulate.
1. In the Process Capability report, select Relaunch Dialog from the Process Capability red triangle menu.
2. (Optional) Close the Process Capability report.
3. In the launch window, from the Cast Selected Columns into Roles list, select Weight&Dist(Lognormal) and Thickness&Dist(Johnson).
4. Click Remove.
5. Click OK.
6. Select Individual Detail Reports from the Process Capability red triangle menu,
Both Ppk and Ppl values are provided, but they are identical because Purity has only a lower specification limit.
7. In the Overall Sigma Capability report, right-click the Estimate column and select Simulate.
In the Column to Switch Out list, Purity is selected. In the Column to Switch In list, Simulated Purity is selected.
8. Next to Number of Samples, enter 500.
Note: The next step is not required. However, it ensures that you obtain exactly the simulated values shown in this example.
9. (Optional) Next to Random Seed, enter 12345.
10. Click OK.
The calculation might take several seconds. A data table entitled Process Capability Simulate Results (Estimate) appears. The Ppk and Ppl columns in this table each contain 500 values calculated based on the Simulated Purity formula. The first row, which is excluded, contains the values for Purity obtained in your original analysis. Because Purity has only a lower specification limit, the Ppk values are identical to the Ppl values.
11. In the Process Capability Simulate Results (Estimate) data table, click the green triangle next to the Distribution script.
Figure 10.29 Distribution of Simulated Ppk Values for Purity
Two Distribution reports are shown, one for Ppk and one for Ppl. But Purity has only a lower specification limit, so that the Ppk and Ppl values are identical. For this reason, the Distribution reports are identical.
A 95% confidence interval for Ppk based on the 2.5% and 97.5% quantiles is 0.9092 to 1.1446. The true Ppk value could be above 1.0, which would place Purity above the Ppk = 1 line in the Capability Index Plot you constructed in “Nonnormal Capability Analysis”.
12. In the Process Capability report, right-click the Expected Overall % column in the Nonconformance report and select Simulate.
13. Next to Number of Samples, enter 500.
14. (Optional) Next to Random Seed, enter 12345.
15. Click OK.
The calculation might take several seconds. A data table entitled Process Capability Simulate Results (Expected Overall %) appears. Because Purity has only a lower specification limit, the Below LSL values are identical to the Total Outside values.
16. In the Process Capability Simulate Results (Expected Overall %) data table, click the green triangle next to the Distribution script.
Figure 10.30 Distribution of Simulated Total Outside Values for Purity
Again, two identical Distribution reports appear. A 95% confidence interval for the Expected Overall % nonconforming, based on the 2.5% and 97.5% quantiles, is 0.055 to 0.238.
Statistical Details for the Process Capability Platform
The following sections provide statistical details for the Process Capability platform:
Variation Statistics
Denote the standard deviation of the process by σ. The Process Capability platform provides two types of capability indices. The Ppk indices are based on an estimate of σ that uses all of the data in a way that does not depend on subgroups. This overall estimate can reflect special cause as well as common cause variation. The Cpk indices are based on an estimate that attempts to capture only common cause variation. The Cpk indices are constructed using within-subgroup, or short-term, estimates of σ. In this way, they attempt to reflect the true process standard deviation. When a process is not stable, the overall and within estimates of σ can differ markedly.
Overall Sigma
The overall sigma does not depend on subgroups. JMP calculates the overall estimate of σ as follows:
The formula uses the following notation:
N = number of nonmissing values in the entire data set
yi = value of the ith observation
= mean of nonmissing values in the entire data setCaution: When the process is stable, the Overall Sigma estimates the process standard deviation. If the process is not stable, the overall estimate of σ is of questionable value, since the process standard deviation is unknown.
Estimates of Sigma Based on Within-Subgroup Variation
An estimate of σ that is based on within-subgroup variation can be constructed in one of three ways:
• Within sigma estimated by average of ranges
• Within sigma estimated by average of unbiased standard deviations
• Within sigma estimated by moving range
If you specify a subgroup ID column or a constant subgroup size on the launch window, you can specify your preferred within-subgroup variation statistic. See “Launch the Process Capability Platform”. If you do not specify a subgroup ID column, a constant subgroup size, or a historical sigma, JMP estimates the within sigma using the third method (moving range of subgroups of size two).
Within Sigma Based on Average of Ranges
Within sigma estimated by the average of ranges is the same as the estimate of the standard deviation of an X/R chart:
The formula uses the following notation:
Ri = range of ith subgroup
ni = sample size of ith subgroup
d2(ni) = expected value of the range of ni independent normally distributed variables with unit standard deviation
N = number of subgroups for which
Within Sigma Based on Average of Unbiased Standard Deviations
Within sigma estimated by the average of unbiased standard deviations is the same as the estimate for the standard deviation in an X/S chart:
The formula uses the following notation:
ni = sample size of ith subgroup
c4(ni) = expected value of the standard deviation of ni independent normally distributed variables with unit standard deviation
N = number of subgroups for which
si = sample standard deviation of the ith subgroup.
Within Sigma Based on Moving Range
Within sigma estimated by moving range is the same as the estimate for the standard deviation for Individual Measurement and Moving Range charts:
The formula uses the following notation:
MR = the mean of the nonmissing moving ranges computed as (MR2+MR3+...+MRN)/(N-1) where MRi = |yi - yi-1|.
d2(2) = expected value of the range of two independent normally distributed variables with unit standard deviation.
Notation for Goal Plots and Capability Box Plots
The formulas for the Goal Plot and Capability Box Plots use the following notation:
Yij = ith observation for process j
= mean of the observations on process j
= standard deviation of the observations on process jTj = target value for process j
LSLj = lower specification limit for process j
USLj = upper specification limit for process j
SD(Yj) = standard deviation for process j
Goal Plot
This section provides details about the calculation of the mean shift and standard deviation standardized to specification quantities plotted in the Goal Plot. This section uses the notation defined in “Notation for Goal Plots and Capability Box Plots”.
The mean shift and the standard deviation standardized to the specification limits for the jth column are defined as follows:
Mean Shift Standardized to Spec = 
Std Dev Standardized to Spec = 
Note: If either LSLj or USLj is missing, twice the distance from the target to the nonmissing specification limit is used in the denominators of the Goal Plot coordinates.
Goal Plot Points for Processes with Missing Targets
Suppose that the process has both a lower and an upper specification limit but no target. Then the formulas given in “Goal Plot” are used, replacing Tj with the midpoint of the two specification limits.
Suppose that the process has only one specification limit and no target. To obtain (x,y) coordinates for a point on the Goal Plot, the capability indices of the process are used. (See “Capability Indices for Normal Distributions” for definitions in terms of the theoretical mean and standard deviation.) For sample observations, the following relationships hold:
If a process has two specification limits and a target at the midpoint of the limits, then the (x,y) coordinates for the point on the Goal Plot satisfy these relationships:
To obtain coordinates when there is only one specification limit and no target, these relationships are used. To identify a unique point requires an assumption about the slope of the line from the origin on which the points fall. A slope of 0.5 is assumed for the case of an upper specification limit and of -0.5 for a lower specification limit. When capability values are equal to one and the Ppk slider for the goal plot triangle is set to 1, these slopes place the points on the goal plot triangle lines.
Consider the case of only an upper specification limit and no target. Using the assumption that the (x,y) coordinates fall on a line from the origin with slope 0.5, solving for x and y gives the following coordinates:
Consider the case of only a lower specification limit and no target. Using the assumption that the (x,y) coordinates fall on a line from the origin with slope -0.5, solving for x and y gives the following coordinates:
Note: If either Cpu or Cpl is less than -0.6, then it is set to -0.6 in the formulas above. At the value -2/3, the denominator for x assumes the value 0. Bounding the capability values at -0.6 prevents the denominator from assuming the value 0 or switching signs.
Capability Box Plots for Processes with Missing Targets
A column with no target can have both upper and lower specification limits, or only a single specification limit. This section uses the notation defined in “Notation for Goal Plots and Capability Box Plots”.
Two Specification Limits and No Target
When no target is specified for the jth column, the capability box plot is based on the following values for the transformed observations:
Single Specification Limit and No Target
Suppose that only the lower specification limit is specified. (The case where only the upper specification limit is specified in a similar way.)
When no target is specified for the jth column, the capability box plot is based on the following values for the transformed observations:
Note: When a column has only one specification limit and no target value, and the sample mean falls outside the specification interval, no capability box plot for that column is plotted.
Capability Indices for Normal Distributions
This section provides details about the calculation of capability indices for normal data.
For a process characteristic with mean μ and standard deviation σ, the population-based capability indices are defined as follows. For sample observations, the parameters are replaced by their estimates:
Cp =
Cpl =
Cpu =
Cpk =
Cpm =
The formulas use the following notation:
LSL = Lower specification limit
USL = Upper specification limit
T = Target value
For estimates of Within Sigma capability, σ is estimated using the subgrouping method that you specified. For estimates of Overall Sigma capability, σ is estimated using the sample standard deviation. With the default AIAG (Ppk) Labeling, the indices based on Overall Sigma are denoted by Pp, Ppl, Ppu, and Ppk. The labeling for the index Cpm does not change when Overall Sigma is used.
If either of the specification limits is missing, the capability indices containing the missing specification limit are reported as missing.
Capability Indices for Nonnormal Distributions: Percentile and Z-Score Methods
This section describes how capability indices are calculated for nonnormal distributions. Two methods are described: the Percentile (also known as ISO/Quantile) method and the Z-Score (also known as Bothe/Z-scores) method. When you select a distribution for a nonnormal process variable, you can fit a parametric distribution or a nonparametric distribution. You can use either the Percentile or the Z-Score methods to calculate capability indices for the process variable of interest. However, unless you have a very large amount of data, a nonparametric fit might not accurately reflect behavior in the tails of the distribution.
Note: For both the Percentile and the Z-Score methods, if the data are normally distributed, the capability formulas reduce to the formulas for normality-based capability indices.
The descriptions of the two methods use the following notation:
LSL = Lower specification limit
USL = Upper specification limit
T = Target value
Percentile (ISO/Quantile) Method
The percentile method replaces the mean in the standard capability formulas with the median of the fitted distribution and the 6σ range of values with the corresponding percentile range. The method is described in AIAG (2005).
Denote the α*100th percentile of the fitted distribution by Pα. Then Percentile method capability indices are defined as follows:
Z-Score (Bothe/Z-Scores) Method
The Z-Score method transforms the specification limits to values that have the same probabilities on a standard normal scale. It computes capability measures that correspond to a normal distribution with the same risk levels as the fitted nonnormal distribution.
Let F denote the fitted distribution for a process variable with lower and upper specification limits given by LSL and USL. Define equivalent standard normal specification limits as follows:
Then the Z-Score method capability indices are defined as follows:
Note: Because Cpm is a target-based measure, it cannot be calculated using the Z-Scores method.
Note: For very capable data, F(LSL) or F(USL) can be so close to zero or one, respectively, that LSLF or USLF cannot be computed. In these cases, the corresponding capability indices are defined as Infinity.
Parameterizations for Distributions
This section gives the density functions f for the distributions used in the Process Capability platform. It also gives expected values and variances for all but the Johnson distributions.
Normal
, 
, 
, σ > 0E(X) = μ
Var(X) = σ2
Gamma
, x > 0, α > 0, σ > 0E(X) = ασ
Var(X) = ασ2
Johnson
Johnson Su
, 
, θ > 0, δ > 0Johnson Sb
, θ < x < θ+σ, σ > 0Johnson Sl
, for x > θ if σ = 1, x < θ if σ = -1where 
is the standard normal probability density function.
is the standard normal probability density function.Lognormal
, x > 0, 
, σ > 0E(X) =
Var(X) =
Weibull
, α > 0, β > 0E(X) =
Var(X) =
, where 
is the gamma function.
, where is the gamma function...................Content has been hidden....................
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