# XLSTAT-Statistical Process Control

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XLSTAT-Statistical Process Control (formerlly called XLSTAT-SPC) is an Excel add-in which has been developed to provide XLSTAT users with a powerful solution for Statistical Process Control (SPC). XLSTAT-Statistical Process Control is the ideal module for companies who apply Six Sigma methods to control and improve the quality of their production or sales processes.

XLSTAT-Statistical Process Control analytical control charts and Measurement System Analysis (MSA) tools, such as Gage Repeatability and Reproducibility for quantitatitave and qualitative data or Pareto plots, enable you to detect any process deviation.

All XLSTAT-Statistical Process Control functions have been intensively tested against other software to guarantee the users fully reliable results, and to allow you to integrate this software in your Six Sigma business improvement process.

## Demo version

A trial version of XLSTAT-Statistical Process Control is included in the main XLSTAT download.

## Prices and ordering

These analyses are included in the XLStat-Quality and XLStat-Premium packages.

# Subgroup Charts

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### What is a subgroup control chart

A subgroup chart is a type of control chart that focus on the quality characteristic measurement within one subgroup.

It is used to supervise production quality, in the case where you have a group of measurements for each point in time. The measurements need to be quantitative data. This tool is useful to recap the mean and the variability of the measured production quality.

### Subgroup chart in XLSTAT

Integrated in this tool, you will find Box-Cox transformations, calculation of process capability and the application of rules for special causes and Westgard rules (an alternative set of rules to identify special causes) to complete your analysis.
The subgroup charts tool offers you the following chart types alone or in combination:

• An X bar chart is useful to follow the mean of a production process. Mean shifts are easily visible in the diagrams.
• An R chart (Range chart) is useful to analyze the variability of the production. A large difference in production, caused for example by the use of different production lines, will be easily visible.
• S and S’² charts are also used to analyze the variability of production. The S chart draws the standard deviation of the process and the S’² chart draws the variance (which is the square of the standard deviation).

Note: During the interpretation of the different indicators for the process capability please pay attention to the fact that some indicators suppose normality or at least symmetry of the distribution of the measured values. By the use of a normality test, you can verify these premises (see the Normality Tests in XLSTAT-Pro).

### Results for subgroup charts in XLSTAT

#### Box-Cox transformation

Estimates of the parameters of the model: This table is available only if the Lambda parameter has been optimized. It displays the estimator for Lambda.
Series before and after transformation: This table displays the series before and after transformation. If Lambda has been optimized, the transformed series corresponds to the residuals of the model. If it hasn’t then the transformed series is the direct application of the Box-Cox transformation

#### X bar/ R/ S/ S² table

This table contains information about the center line and the upper and lower control limits of the selected chart. There will be one column for each phase.

#### Process capabilities:

Process capabilities: These tables are displayed, if the "process capability" option has been selected. There is one table for each phase. A table contains the following indicators for the process capability and if possible the corresponding confidence intervals: Cp, Cpl, Cpu, Cpk, Pp, Ppl, Ppu, Ppk, Cpm, Cpm (Boyle), Cp 5.5, Cpk 5.5, Cpmk, and Cs (Wright).
For Cp, Cpl, and Cpu, information about the process performance is supplied and for Cp a status information is given to facilitate the interpretation.
Interpretation of the Cp

• Cp values have the following status based on Ekvall and Juran (1974):
• "not adequate" if Cp < 1
• "adequate" if 1 ≤ Cp ≤ 1.33
• "more than adequate" if Cp > 1.33
• Based on Montgomery (2001), Cp needs to have the following minimal values for the process performance to be as expected:
• 1.33 for existing processes
• 1.50 for new processes or for existing processes when the variable is critical
• 1.67 for new processes when the variable is critical
• Based on Montgomery (2001), Cpu and Cpl need to have the following minimal values for process performance to be as expected:
• 1.25 for existing processes
• 1.45 for new processes or for existing processes when the variable is critical
• 1.60 for new processes when the variable is critical

Capabilities chart
This chart contains information about the specification and control limits. A line between the lower and upper limits represents the interval with an additional vertical mark for the center line. The different control limits of each phase are drawn separately.

#### Normality tests:

For each of the four tests, the statistics relating to the test are displayed including, in particular, the p-value which is afterwards used in interpreting the test by comparing with the chosen significance threshold.
If requested, a Q-Q plot is then displayed.

### Subgroup charts in XLSTAT

The following results are displayed separately for each requested chart. Charts can be selected alone or in combination with the X bar chart.

• Observation details: This table displays detailed information for each subgroup. For each subgroup the corresponding phase, the size, the mean, the minimum and the maximum values, the center line, and the lower and upper control limits are displayed. If the information about the zones A, B and C are activated, then the lower and upper control limits of the zones A and B are displayed as well.
• Rule details: If the rules options are activated, a detailed table about the rules will be displayed. For each subgroup, there is one row for each rule that applies. "Yes" indicates that the corresponding rule was fired for the corresponding subgroup and "No" indicates that the rule does not apply.
• X bar/ R/ S/ S² chart: If the charts are activated, then a chart containing the information of the two tables above is displayed. Each subgroup is displayed. The center line and the lower and upper control limits are displayed as well. If the corresponding options have been activated, the lower and upper control limits for the zones A and B are included and there are labels for the subgroups for which rules were fired. A legend with the activated rules and the corresponding rule number is displayed below the chart.
• Histograms: The histograms are displayed. If desired, you can change the color of the lines, scales, titles as with any Excel chart.
• Run chart: The chart of the last data points is displayed.

# Individual Charts

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### What is an individual chart

Individual charts are a type of control chart used to monitor the processes.
Control charts were first mentioned in a document by Walter Shewhart that he wrote during his time working at Bell Labs in 1924. He described his methods completely in his book (1931).

For a long time, there was no significant innovation in the area of control charts. With the development of CUSUM, UWMA and EWMA charts in 1936, Deming expanded the set of available control charts.

Control charts were originally used in area of goods production. Therefore the wording is still from that domain. Today this approach is being applied to a large number of different fields, for instance services, human resources, and sales. In the following lines, we use the wording from the production and shop floors.

### Individual charts in XLSTAT

The individual charts tool offers you the following chart types alone or in combination:

• X Individual: An X individual chart is useful to follow the moving average of a production process. Mean shifts are easily visible in the diagrams.
• MR moving range: An MR chart (moving range diagram) is useful to analyze the variability of the production. Large difference in production, caused by the use of different production lines, will be easily visible.

Note 1: If you want to investigate smaller mean shifts, then you can also use CUSUM individual charts which are often preferred in comparison with the individual control charts, because they can detect smaller mean shifts.

Note 2: If you have more than one measurement for each point in time, then you should use the control charts for subgroups.

Note 3: If you have measurements in qualitative values (for instance ok, not ok, conform not conform), then use the control charts for attributes.

XLSTAT-SPC Individual charts option offers you the following options for the estimation of the standard deviation (sigma) of the data set, given n measurements:

• Average moving range: The estimator for sigma is calculated based on the average moving range using a window length of m measurements.
Ŝ = m / d2,
where d2 is the control chart constant according to Burr (1969).
• Median moving range: The estimator for sigma is calculated based on the median of the moving range using a window length of m measurements.
Ŝ = median / d4,
where d4 is the control chart constant according to Burr (1969).
• standard deviation: The estimator for sigma is calculated based on the standard deviation of the n measurements.
Ŝ = s / c4,
where c4 is the control chart constant according to Burr (1969).

#### Process capability

Process capability describes a process and informs if the process is under control and the distribution of the measured variables are inside the specification limits of the process. If the distributions of the measured variables are in the technical specification limits, then the process is called "capable".

Note: During the interpretation of the different indicators for the process capability please pay attention to the fact that some indicators suppose normality or at least symmetry of the distribution of the measured values. By the use of a normality test, you can verify these premises (see the Normality Tests in XLSTAT-Pro).

If the data are not normally distributed, you have the following possibilities to obtain results for the process capabilities.
Use the Box-Cox transformation to improve the normality of the data set. Then verify again the normality using a normality test.
Use the process capability indicator Cp 5.5.

Once the capability is computed, a table contains the following indicators for the process capability and if possible the corresponding confidence intervals: Cp, Cpl, Cpu, Cpk, Pp, Ppl, Ppu, Ppk, Cpm, Cpm (Boyle), Cp 5.5, Cpk 5.5, Cpmk, and Cs (Wright).

For Cp, Cpl, and Cpu, information about the process performance is supplied and for Cp a status information is given to facilitate the interpretation.

Interpretation of the Cp

• Cp values have the following status based on Ekvall and Juran (1974):
• "not adequate" if Cp < 1
• "adequate" if 1 ≤ Cp ≤ 1.33
• "more than adequate" if Cp > 1.33
• Based on Montgomery (2001), Cp needs to have the following minimal values for the process performance to be as expected:
• 1.33 for existing processes
• 1.50 for new processes or for existing processes when the variable is critical
• 1.67 for new processes when the variable is critical
• Based on Montgomery (2001), Cpu and Cpl need to have the following minimal values for process performance to be as expected:
• 1.25 for existing processes
• 1.45 for new processes or for existing processes when the variable is critical
• 1.60 for new processes when the variable is critical

#### Chart rules

XLSTAT offers you the possibility to apply rules for special causes and Westgard rules on the data set. Two sets of rules are available in order to interpret control charts. You can activate and deactivate separately the rules in each set.

Test special causes:

• 1 point more than 3s from center line
• 9 points in a row on same side of center line
• 6 points in a row, all increasing or all decreasing
• 14 points in a row, alternating up and down
• 2 out of 3 points > 2s from center line (same side)
• 4 out of 5 points > 1s from center line (same side)
• 15 points in a row within 1s of center line (either side)
• 8 points in a row > 1s from center line (either side)

Apply Westgard rules:

• Rule 1 2s
• Rule 1 3
• Rule 2 2s
• Rule 4s
• Rule 4 1s
• Rule 10 X

### Plots for individual charts in XLSTAT

The following results are displayed separately for each requested chart. Charts can be selected alone or in combination with the X individual chart.

• X Individual / MR moving range table: This table contains information about the center line and the upper and lower control limits of the selected chart. There will be one column for each phase.
• Observation details: This table displays detailed information for each observation. For each observation, the corresponding phase, the mean or median, the center line, the lower and upper control limits are displayed. If the information about the zones A, B and C are activated, then the lower and upper control limits of the zones A and B are displayed as well.
• Rule details: If the rules options are activated, a detailed table about the rules will be displayed. For each observation, there is one row for each rule that applies. "Yes" indicates that the corresponding rule was fired, and "No" indicates that the rule does not apply.
• X Individual / MR moving range Chart: If the charts are activated, then a chart containing the information of the two tables above is displayed. Each observation is displayed. The center line and the lower and upper control limits are displayed as well. If the corresponding options have been activated, the lower and upper control limits for the zones A and B are included and there are labels for the observations for which rules were fired. A legend with the activated rules and the corresponding rule number is displayed below the chart.

# Attribute Charts

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Attribute charts are a type of quality control charts that study the non-conformity in a process.

Use this tool to supervise the production quality, in the case where you have a single measurement for each point in time. The measurements are based on attribute or attribute counts of the process.

This tool is useful to recap the categorical variables of the measured production quality.

Integrated in this tool, you will find Box-Cox transformations, calculation of process capability and the application of rules for special causes and Westgard rules (an alternative rule set to identify special causes) available to complete your analysis.

The attribute charts tool includes the following chart types:

P chart: A P chart is useful to follow the fraction of non conforming units of a production process.

NP chart: An NP chart is useful to follow the absolute number of non conforming units of a production process.

C chart: A C chart is useful to follow the number of non conforming units per inspection unit of a production process having a constant size of a inspection unit.

U chart: A U chart is useful to follow the number of non conforming units per inspection unit of a production process having a non constant size of a inspection unit.

# Time Weighted Charts

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### What are time weighted charts

Time weighted charts are a type quality control charts used in the area of good production.
They are used to supervise production quality, in the case where you have a group of measurements or a single measurement for each point in time. The measurements need to be made on quantitative variables.

The weighted time charts are useful to recap the mean and the variability of the measured production quality.

### Time weighted charts in XLSTAT

XLSTAT-SPC time weighted charts tool offers you the following chart types:

• CUSUM (Cumulative sum of quality characteristic measurement) or CUSUM individual
• UWMA (Uniformly weighted moving average) or UWMA individual
• EWMA (Exponentially weighted moving average) or EWMA individual

A CUSUM, UWMA or EWMA chart is useful to follow the mean of a production process. Mean shifts are easily visible in the diagrams.

#### UWMA and EWMA charts

These charts are not directly based on the raw data. They are based on the smoothed data.
In the case of UWMA charts, the data is smoothed using a uniform weighting in a moving window. Then the chart is analyzed like Shewhart charts.
In the case of EWMA charts, the data is smoothed using a exponentially weighting. Then the chart is analyzed like Shewhart charts.

#### CUSUM charts

These charts are not directly based on the raw data. They are based on the normalized data.
These charts help to detect mean shifts of a user defined granularity. The granularity is defined by the design parameter k. K is the half of the mean shift to be detected. Normally to detect 1 sigma shifts, k is set to 0.5.
Two kinds of CUSUM charts can be drawn: one and two sided charts.

# Pareto charts

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### What is a Pareto chart

A Pareto plot displays both the frequency of occurrence, and the cumulative percentage of the total number of occurrences of level of a category variable. It draws its name from an Italian economist, but J. M. Juran is credited with being the first to apply it to industrial problems.

### Purpose of a Pareto plot

The purpose of the Pareto chart is to highlight the most important among a (typically large) set of factors. The causes that should be investigated (e. g., nonconforming items) are listed and percentages assigned to each one so that the total is 100 %. The percentages are then used to construct the diagram that is essentially a bar or pie chart. Pareto analysis uses the ranking of causes to determine which of them should be pursued first.

### Pareto chart in XLSTAT

XLSTAT offers you a large number of descriptive statistics and charts which give you a useful and relevant insight of your data, for a sample made up of N qualitative values:

• Number of observations
• Number of missing values
• Sum of weights*
• Mode*
• Frequency of mode*
• Category
• Frequency by category*
• Relative frequency by category*
• Cumulated relative frequency by category*

(*) Statistics followed by an asterisk take the weight of observations into account.

Although you can select several variables (or samples) at the same time, XLSTAT calculates all the descriptive statistics for each of the samples independently.

Several types of chart are available for qualitative data:

• Bar charts and Pie charts: Check this option to represent the frequencies or relative frequencies of the various categories of qualitative variables as bars or pieces of pie.
• Double pie charts: These charts are used to compare the frequencies or relative frequencies of sub-samples with those of the complete sample.
• Doughnuts and Stacked bars: these options are only checked if a column of sub-samples has been selected. These charts are used to compare the frequencies or relative frequencies of sub-samples with those of the complete sample.

# Gage Repeatability and Reproducibility

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### What is Gage R&R for quantitative variables (Measurement System Analysis)

Measurement System Analysis (MSA) or Gage R&R (Gage Repeatability and Reproducibility) is a method to control and assess a measurement process. It is useful to determine which sources are responsible for the variation of the measurement data. Variability can be caused by the measurement system, the operator or the parts. Gage R&R applied to quantitative measurements is based on two common methods: ANOVA and R control charts.

The word "gage" (or gauge) refers to the fact that the methodology is aimed at validating instruments or measurement methods. For quantitative data, two characteristics are of interest:

• Repeatability: A measurement is "repeatable" if the measures taken by a given operator for the same object (product, unit, part, or sample, depending of the field of application) repeatedly, do not vary above a given threshold. If the repeatability of a measurement system is not satisfactory, one should question the quality of the measurement system, or train the operators that do not obtain repeatable results if the measurement system does not appear to be responsible for the high variability.
• Reproducibility: A measurement is "reproducible" if the measures obtained for a given object (product, unit, part, or sample, depending of the field of application) by several operators do not vary above a given threshold. If the reproducibility of a measurement system is not satisfactory, one should train the operators so that their results are more homogeneous.

The goal of a Gage R&R analysis is to identify the sources of variability and to take the necessary actions to reduce them if necessary.

### Gage R&R for quantitative variables in XLSTAT

The Gage R&R for quantitative variables XLSTAT tools should be used to control and validate your measurement method and measurement systems, in the case where you have several quantitative measures taken by one or more operators on several parts.

When the measures are quantitative data, two alternative methods are available for Gage R&R analysis. This first is based on analysis of variance (ANOVA) and the second on R control charts (Range and average).

#### ANOVA to compute Gage R&R for quantitative variables in XLSTAT

When the ANOVA model is used in R&R analysis, one can statistically test whether the variability of the measures is related to the operators, and/or to the parts being measured themselves, and/or to an interaction between both (some operators might give for some parts significantly higher or lower measures), or not. Two designs are available when doing gage R&R analysis: the crossed design (balanced) and the nested design. XLSTAT includes as of today the crossed and designs.

#### R charts to compute Gage R&R for quantitative variables

While less powerful than the ANOVA method, the Gage R&R analysis based on Range and Average analysis, is easy to compute and produces control charts (R charts). As the ANOVA method, it allows to compute the repeatability and the reproducibility of the measurement process. To use this method you need to have several parts, operators and repetitions (typically 10 parts, 3 operators, and 2 repetitions).

#### Indicators for Gage R&R for quantitative variables

XLSTAT offers several indicators derived from the variances to describe the measurement system.

• Study variation: The study variation for the different sources is calculated as product of the corresponding standard deviation of the source and the used defined factor k Sigma:
Study variation = k*Ŝ
• % tolerance: The tolerance in percent is defined as the ratio of the variance in the study and the user defined tolerance:
% tolerance = Study variation / tolerance
• The process sigma in percent is defined as ratio of the standard deviation of the source and the user defined historic process sigma:
% process = standard deviation of the source / process sigma
• Precision to tolerance ratio (P/T):
P / T = k* ŜR&R² / tolerance
• Rho P (Rho Part):
ρpart = Ŝpart² / Ŝ²
• Rho M:
ρM = ŜR&R² / Ŝ²
• Signal to noise ratio (SNR):
SNR = [2 ρpart / (1- ρpart)]1/2
• Discrimination ratio (DR):
DR = (1 + ρpart) / (1 - ρpart)
• Bias:
Bias = µmeasurements- target
• Bias in percent:
Bias % = µmeasurements- target / tolerance
• Resolution:
Resolution = Bias + 3* ŜR&R²

# Gage Repeatability and Reproducibility for Attributes

### What is Gage Gage Repeatability and Reproducibility for Attributes

Measurement System Analysis (MSA) or Gage R & R (Gage Repeatability and Reproducibility) is a method to control and judge a measurement process. It is useful to determine which sources are responsible for the variation of the measurement data. Variability can be caused by the measurement system, the operator or the parts.

The word "gage" (or gauge) refers to the fact that the methodology is aimed at validating instruments or measurement methods.

In contrast to the Gage R & R for quantitative measurements, the analysis based on attributes gives information on the "agreement" and on the "correctness". The concepts of variance, repeatability and reproducibility are not relevant in this case.

A high "agreement" of the measures taken by a given operator for the same object (product, unit, part, or sample, depending of the field of application) repeatedly, shows that the operator is consistent. If the agreement of a measurement system is low, one should question the quality of the measurement system or protocol, or train the operators that do not obtain a high agreement, if the measurement system does not appear to be responsible for the lack of agreement.

A high "correctness" of the measures taken by an operator for the same object (product, unit, part, or sample, depending of the field of application) in comparison to the given reference or standard value shows that the operator comes to correct results. If the correctness of a measurement system is low, one should train the operators so that their results are more correct.

Correctness can be computed using the Kappa or the Kendall statistics. Kappa coefficients can be used in the case of qualitative and ordinal quantitative measurements. Kendall coefficients can be used in the case of ordinal measurements with at least 3 categories.

The two concepts "agreement" and "correctness" can be computed for a given operator, for a given operator compared to the standard, between two operators and for all operators compared to the standard.

The goal of a Gage R & R analysis for attributes is to identify the sources of low agreement and low correctness, and to take the necessary actions if necessary.

### Gage Gage Repeatability and Reproducibility for Attributes in XLSTAT

The XLSTAT-SPC Gage R & R for Attributes tool is used to control and validate your measurement method and measurement systems, in the case where you have several qualitative measures taken by one or more operators on several parts.

#### Results for Gage Gage Repeatability and Reproducibility for Attributes in XLSTAT

When the measures are qualitative or ordinal quantitative data, the Gage R&R analysis for attributes is based on the following statistics to evaluate the agreement and correctness:

• agreement statistics
• disagreement statistics
• Kappa statistics
• Kendall statistics

If possible, the following comparisons are performed:

• Intra rater
• Operator vs. standard
• Inter rater and
• All Operators vs. standard

Agreement statistics

It is possible to calculate these statistics in all of the sections.
In the intra rater section, XLSTAT computes for each operator the number of cases where he agrees with himself for a given part across repetitions.

Additionally the ratio of the number of cases and the total number of inspections of the operator is computed.

In the Operator vs. standard section, XLSTAT gives the number of cases where an operator agrees with the standard across repetitions. Additionally the ratio of the number of cases and the total number of inspections of the operator is computed.

In the inter rater section, XLSTAT computes the number of cases where all operators agree for a given part and across repetitions. Additionally the ratio of the number of cases and the total number of inspections of all the operators is computed.

In the all operators vs. standard section, XLSTAT computes the number of cases where all operators agree with the standard, across all repetitions. Additionally the ratio of the number of cases and the total number of inspections of all the operators is computed.

In addition, confidence intervals are calculated. For proportions, XLSTAT allows you to use the simple (Wald, 1939) or adjusted (Agresti and Coull, 1998) Wald intervals, a calculation based on the Wilson score (Wilson, 1927), possibly with a correction of continuity, or the Clopper-Pearson (1934) intervals. Agresti and Caffo recommend using the adjusted Wald interval or the Wilson score intervals.

Disagreement statistics

This statistic is only calculated in the Operator vs. standard section in the case where the measurement variable is binary (for example, success or failure). Three different kinds of disagreements statistics are calculated for each operator:

• False Negatives: This statistic counts the number of cases where a given operator systematically evaluates a part as category 0 while the standard evaluates it as category 1. Additionally the proportion of false negatives across all parts of category 0 is displayed.
• False Positives: This statistic counts the number of cases where a give operator systematically evaluates a part as category 1 while the standard evaluates it as category 0. Additionally the proportion of false positive across all parts of category 1 is displayed.
• Mixed: This statistic counts the number of cases where an operator will be inconsistent in the rating of a given part across repetitions. The proportion of such cases computed as the ratio between Mixed and the total number of parts is displayed.

Kappa coefficients

Cohen’s and Fleiss Kappa are well suited for qualitative variables. These coefficients are calculated on contingency tables obtained from paired samples. The Fleiss’ kappa is a generalization of the Cohen’s kappa. The kappa coefficient varies between -1 and 1. The closer the kappa is to 1, the higher the association.

In the case of an intra rater analysis, it is necessary that 2 or more measures have been taken by an operator for a given part.
In the case of operator vs. standard, the number of the measures for each operator must be the same as the number of measures for the standard.

In the case of inter rater, the number of the investigations for the two operators being compared must be the same.
In the case of all operators vs. standard the number of investigations for each operator for a given part has to be the same.

Kendall coefficients

These indicators are available for ordinal quantitative variables with at least 3 categories.

Kendall’s tau: This coefficient, also referred to as tau-b, allows to measure on a -1 to 1 scale the degree of concordance between two ordinal variables.
The Kendall’s coefficient of concordance: This coefficient measures on a 0 (no agreement) to 1 (perfect agreement) scale the degree of concordance between two ordinal variables.

The coefficients are computed to evaluate the measurement system by comparing each operator to the standard, operators between each other, and all operators vs. standard

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5 May, 2016