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]]>Process capability measures how well the process performs to meet given specified outcome. It indicates the conformance of a process to meet given requirements or specifications. Capability analysis helps to better understand the performance of the process with respect to meeting customer’s specifications and identify the process improvement opportunities.

Process Capability Analysis Steps

- Determine the metric or parameter to measure and analyze.
- Collect the historical data for the parameter of interest.
- Prove the process is statistically stable (i.e. in control).
- Calculate the process capability indices.
- Monitor the process and ensure it remains in control over time. Update the process capability indices if needed.

Process capability can be presented using various indices depending on the nature of the process and the goal of the analysis. Popular process capability indices are:

- Cp
- Pp
- Cpk
- Ppk
- Cpm

The Cp index is process capability. It assumes the process mean is centered between the specification limits and essentially is the ratio of the distance between the specification limits to six process standard deviations. Obviously, the higher this value the better, because it means you can fit the process variation between the spec limits more easily. Cp measures the process’ potential capability to meet the two-sided specifications. It does not take the process average into consideration.

High Cp indicates the small spread of the process with respect to the spread of the customer specifications. Cp is recommended when the process is centered between the specification limits. Cp works when there are both upper and lower specification limits. The higher Cp the better, meaning the spread of the process is smaller relative to the spread of the specifications.

Note: Cpm can work only if there is a target value specified.

Data File: “Capability Analysis” tab in “Sample Data.xlsx”

Steps in Minitab to run a process capability analysis:

- Click Stat → Basic Statistics → Normality Test.
- A new window named “Normality Test” pops up.
- Select “HtBk” as the variable.

- Click “OK.”
- The histogram and the normality test results appear in the new window.

In this example, the p-value is 0.275, greater than the alpha level (0.05). We fail to reject the hypothesis and conclude that the data are normally distributed.

- Click Stat → Quality Tools → Capability Analysis→ Normal.
- A new window named “Capability Analysis(Normal Distribution)” pops up.
- Select “HtBk” as the single column and enter “1” as the subgroup size.
- Enter “6” as the “Lower spec” and “7” as the “Upper spec”

- Click “Options” button and another new window named “Capability Analysis(Normal Distribution) – Options” pops up.
- Enter “6.5” as the target and click “OK.”

- Click “OK” in the “Capability Analysis(Normal Distribution)” window.
- The capability analysis results appear in the new window.

If the p-value of the previous normality test is smaller than the alpha level (0.05), we would reject the null hypothesis and conclude that the data are not normally distributed. Thus, we would perform a Non-Normal Capability analysis as follows:

- Click Stat → Quality Tools → Capability Analysis→ Non-Normal
- A new window named “Capability Analysis(Non-Normal Distribution)” pops up.
- Select “HtBk” as the single column.
- Enter “6” as the “Lower spec” and “7” as the “Upper spec.”

- Click “Options” button and another new window named “Capability Analysis(Non-Normal Distribution) – Options” pops up.
- Enter “6.5” as the target and click “OK.”

- Click “OK” in the “Capability Analysis(Non-Normal Distribution)” window.
- The capability analysis results appear in the new window.

Model summary: With P_{pk} of less than 1.0 we can conclude that the capability of this process is not very good. Anything less than 1.0 should be considered not capable and we should strive for P_{pk} to reach levels of greater than 1 and preferably over 1.67.

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]]>The post Attribute MSA with Minitab appeared first on Deploy OpEx.

]]>Data File: “Attribute MSA” tab in “Sample Data.xlsx” (an example in the AIAG MSA Reference Manual, 3rd Edition).

Steps in Minitab to run an attribute MSA:

Step 1: Reorganize the original data into four new columns (i.e., Appraiser, Assessed Result, Part, and Reference).

- Click Data → Stack → Blocks of Columns.
- A new window named “Stack Blocks of Columns” pops up.
- Select “Appraiser A,” “Part,” and “Reference” as block one.
- Select “Appraiser B,” “Part,” and “Reference” as block two.
- Select “Appraiser C,” “Part,” and “Reference” as block three.
- Select the radio button of “New worksheet” and name the sheet “Data.”
- Check the box “Use variable names in subscript column.”
- Click “OK.”

- The stacked columns are created in the new worksheet named “Data.”

- Name the four columns from left to right in worksheet “Data”: Appraiser, Assessed Result, Part, and Reference.

Step 2: Run a MSA using Minitab

- Click Stat → Quality Tools → AttributeAgreement Analysis.
- A new window named “AttributeAgreement Analysis” pops up.
- Click in the blank box next to “Attributecolumn” and the variables appear in the list box on the left.
- Select “Assessed Result” as “Attribute”
- Select “Part” as “Sample.”
- Select “Appraiser” as “Appraisers.”
- Select “Reference” as “Known standard/attribute.”

- Click the “Options” button and another window named “AttributeAgreement Analysis – Options” pops up.
- Check the boxes of both “Calculate Cohen’s kappa if appropriate” and “Display disagreement table.”

- Click “OK” in the window “AttributeAgreement Analysis – Options.”
- Click “OK” in the window “AttributeAgreement Analysis.”
- The MSA results appear in the newly-generated window and the session window.

The rater scores represent how the raters agree with themselves. Appraiser A, for instance, agreed with himself on 84% of the measurements made.

The important numbers are called out here. Of the 50 total measurements performed, for 78% of those (#39) the appraisers agreed with both themselves and the other appraisers.

*Kappa statistic* is a coefficient indicating the agreement percentage above the expected agreement by chance. Kappa ranges from −1 (perfect disagreement) to 1 (perfect agreement). When the observed agreement is less than the chance agreement, Kappa is negative. When the observed agreement is greater than the chance agreement, Kappa is positive. Rule of thumb: If Kappa is greater than 0.7, the measurement system is acceptable. If Kappa is greater than 0.9, the measurement system is excellent.

Model summary: In all cases the Kappa indicates that the measurement system is acceptable.

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]]>The post Variable Gage R&R with Minitab appeared first on Deploy OpEx.

]]>Whenever something is measured repeatedly or by different people or processes, the results of the measurements will vary. Variation comes from two primary sources:

- Differences between the parts being measured
- The measurement system

We can use a variable Gage R&R to conduct a measurement system analysis to determine what portion of the variability comes from the parts and what portion comes from the measurement system. There are key study results that help us determine the components of variation within our measurement system.

Measurement System Analysis (MSA) is a systematic method to identify and analyze the variation components of a measurement system. It is a mandatory step in any Six Sigma project to ensure the data are reliable before making any data-based decisions. A MSA is the check point of data quality before we start any further analysis and draw any conclusions from the data. Some good examples of data-based analysis where MSA should be a prerequisite:

- Correlation analysis
- Regression analysis
- Hypothesis testing
- Analysis of variance
- Design of experiments
- Statistical process control

You will see where and how the analysis techniques listed above are used. It is critical to know that any variation, anomalies, or trends found in your analysis are actually due to the data and not due to the inaccuracies or inadequacies of a measurement system. Therefore the need for a MSA is vital.

A measurement system is a process used to obtain data and quantify a part, product or process. Data obtained with a measurement device or measurement system are the observed values. Observed values are comprised of two elements

- True Value = Actual value of the measured part
- Measurement Error = Error introduced by the measurement system.

The true value is what we are ultimately trying to determine through the measurement system. It reflects the true measurement of the part or performance of the process.

Measurement error is the variation introduced by the measurement system. It is the bias or inaccuracy of the measurement device or measurement process.

The observed value is what the measurement system is telling us. It is the measured value obtained by the measurement system. Observed values are represented in various types of measures which can categorized into two primary types discrete and continuous. Continuous measurements are represented by measures of weight, height, money and other types of measures such as ratio measures. Discrete measures on the other hand are categorical such as Red/Yellow/Green, Yes/No or Ratings of 1–10 for example.

The guidelines for acceptable or unacceptable measurement systems can vary depending on an organizations tolerance or appetite for risk. The common guidelines used for interpretation are published by the Automotive Industry Action Group (AIAG). These guidelines are considered standard for interpreting the results of a measurement system analysis using Variable Gage R&R. Table 1.0 summarizes the AIAG standards.

Data File: “Variable MSA” tab in “Sample Data.xlsx”

Let’s take a look at an example of a Variable MSA using the data in the Variable MSA tab in your “Sample Data.xlsx” file. In this exercise we will first walk through how to set up your study using Minitab and then we will perform a Variable Gage MSA using 3 operators who all measured 10 parts three times each. The part numbers and operators and measurement trials are all generic so that you can apply the concept to your given industry. First we need to set up the study:

- Click on Stat → Quality Tools → Gage R&R → Create Gage R&R Study Worksheet.
- A new window named “Create Gage R&R Study Worksheet” pops up.

- Select 10 as the “Number of Parts.”

Select 3 as the “Number of Operators.”

Select 3 as the “Number of Replicates.”

Enter the part name (e.g., Part 01, Part 02, and Part 03).

Enter the operator name (e.g., Operator A, Operator B, Operator C).

Click on the “Options” button,

another window named “Create Gage R&R Study Worksheet – Options” pops up.

- Select the radio button “Do not randomize.”
- Click “OK” in the window “Create Gage R&R Study Worksheet – Options.”
- Click “OK’ in the window “Create Gage R&R Study Worksheet.”
- A new data table is generated.

Step 2: Data collection

In the newly-generated data table, Minitab has provided the data layout for your data collection for your variable MSA study. We have added the header “Measurement” for this example. You would have to do something similar.

When you conduct your variable MSA in your work environment it would be necessary to set up your study just as we have in the previous steps and then you could collect your measurement data properly. However, for our purposes today, we have provided you with an MSA that is setup with data already collected. We will use our “Variable MSA” tab in “Sample Data.xlsx,” for the next steps.

Step 3: Activate the Minitab worksheet with our Variable MSA data prepopulated.

Step 4: Implement Gage R&R

- Click Stat → Quality Tools → Gage Study → Gage R&R Study (Crossed).
- A new window named “Gage R&R Study (Crossed)” appears.
- Select “Part” as “Part numbers.”
- Select “Operator” as “Operators.”
- Select “Measurement” as “Measurement data.”

- Click on the “Options” button and another new window named “Gage R&R Study (Crossed) – ANOVA Options” pops up.
- Enter 5.15 as the “Study variation (number of standard deviations)”.

The value 5.15 is the recommended standard deviation multiplier by the Automotive Industry Action Group (AIAG). It corresponds to 99% of data in the normal distribution. If we use 6 as the standard deviation multiplier, it corresponds to 99.73% of the data in the normal distribution.

- Click “OK” in the window “Gage R&R Study (Crossed) – ANOVA Options.”
- Click “OK” in the window “Gage R&R Study (Crossed).”
- The MSA analysis results appear in the new window and the session window.

Step 5: Interpret the MSA results

The result of this Gage R&R study leaves room for consideration on one key measure. As noted in previous pages, the targeted percent contribution R&R should be less than 9% and study variation less than 30%. With % contribution at 7.76% it is below our 9% unacceptable threshold and similarly, Study variation at 26.86% is also below the threshold of 30% but this result is at best marginal and should be heavily scrutinized by the business before concluding that the measurement system does not warrant further improvement.

Visual evaluation of this measurement system is another effective method of evaluation but can at times be misleading without the statistics to support it. Diagnosing the mean plots above should help in the consideration of measurement system acceptability, you may benefit from taking a closer look at operator C.

Model summary: Visual evaluation of this measurement system alone might mislead you to a conclusion of a passing gage study (most of the variation seems to be in part to part, which is what we hope to see). However, an experienced practitioner will note such things such as the range chart being out of control. This may help provide clues regarding what to look for when trying to further diagnose the validity of this measurement system. An out of control Range chart in a variable MSA suggests that an operator or all operators are too inconsistent in their repeated measures causing wide ranges and out of control conditions.

Whenever a range chart is out of control, the accuracy of the X chart is automatically called into question. You will learn in future lessons that the control limits of an X chart are calculated using the mean of the R chart. If the R chart shows out-of-control conditions, then the mean is likely misrepresented and any calculation using it should be questioned.

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]]>A run chart is a chart used to present data in time order. These charts capture process performance over time. The X axis indicates time and the Y axis shows the observed values. A run chart is similar to a scatter plot in that it shows the relationship between X and Y. Run charts differ however, because they show how the Y variable changes with an X variable of time.

Run charts look similar to control charts except that run charts do not have control limits and they are much easier to produce than a control chart. A run chart is often used to identify anomalies in the data and discover pattern over time. They help to identify trends, cycles, seasonality and other anomalies.

Steps to plot a run chart in Minitab:

Data File: “Run Chart” tab in “Sample Data.xlsx”

- Click Stat → Quality Tools → Run Chart.
- A new window named “Run Chart” pops up.
- Select “Measurement” as the “Single Column.”
- Enter “1” as the “Subgroup Size.”

- Click “OK.”

The figure above is a run chart created with Minitab. The time series displayed by this chart appears stable. There are no extreme outliers, no visible trending or seasonal patterns. The data points seem to vary randomly over time.

Now, let us look at another example which may give us a different perspective. We will create another run chart using the data listed in the column labeled “Cycle”. This column is in the same file used to generate the figure above. Follow the steps used for the first run chart and instead of using “Measurement” use “Cycle” in the Run Chart dialog box pictured above.

In this figure above, the data points are clearly exhibiting a pattern. It could be seasonal or it could be something cyclical. Imagine that the data points are taken monthly and this is a process performing over a period of 2.5 years. Perhaps the data points represent the number of customers buying new homes. The home buying market tends to peak in the summer months and dies down in the winter.

Using the same data tab lets create a final run chart. This time use the “Trend” data. Again, follow the steps outlined previously to generate a run chart.

In this example, the process starts out randomly, but after the seventh data point almost every data point has a lower value than the one before it. This clearly illustrates a downward trend. What might this represent? Perhaps a process winding down? Product sales at the end of a product’s life cycle? Defects decreasing after introducing a process improvement?

Model summary: It should be clearly evident through our review of Histograms, Scatterplots and Run Charts, that there is great value in “visualizing” the data. Graphical displays of data can be very telling and offer excellent information.

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]]>A scatter plot is a diagram to present the relationship between two variables of a data set. A scatter plot consists of a set of data points. On the scatter plot, a single observation is presented by a data point with its horizontal position equal to the value of one variable and its vertical position equal to the value of the other variable. A scatter plot helps us to understand:

- Whether the two variables are related to each other or not
- What the strength of their relationship
- The shape of their relationship
- The direction of their relationship
- Whether outliers are present

Data File: “Scatter Plot” tab in “Sample Data.xlsx”

Steps to render a Scatterplot in Minitab:

- Click Graph → Scatterplot.
- A new window named “Scatterplots” pops up.

- Leaving “Simple” selected, click “OK”

A new window “Scatterplot– Simple” pops up. - Select “MPG” as the “Y variables.”
- Select “weight” as the “X variables.”

- Click “OK.”
- The scatter plot appears in the new window.

Model summary: The figure above is Minitab’s output of the scatterplot data. You can immediately see the value of graphical displays of data. This information obtainable by viewing this output shows a relationship between weight and MPG. This scatterplot shows that the heavier the weight the lower the MPG value and vice versa.

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]]>A histogram is a graphical tool to present the distribution of the data. The X axis represents the possible values of the variable and the Y axis represents the frequency of the value occurring. This graphical summary consists of adjacent rectangles erected over intervals with heights equal to the frequency density of the interval. The total area of all the rectangles is the number of data values.

A histogram can also be normalized. In the case of normalization, the X axis still represents the possible values of the variable, but the Y axis represents the percentage of observations that fall into each interval on the X axis. The total area of all the rectangles in a normalized histogram is 1. When using these graphical representations, we have a better understanding of the shape, location, and spread of the data.

Data File: “Histogram” tab in “Sample Data.xlsx”

Steps to render in Minitab:

- Click Stat → Basic Statistics → Graphical Summary.
- A new window named “Graphical Summary” pops up.
- Select “HtBk” as the “Variables.”

- Click “OK.”
- The histogram appears in the new window.

Model summary: The output from the previous steps has generated a graphical summary report of the data set HtBk. Among the information provided is a histogram. The image shows the frequency of the data for the numerical categories ranging from 5.7 to approximately 6.9. You can see the shape of the data roughly follows the bell curve.

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]]>A box plot is a graphical method to summarize a data set by visualizing the minimum value, 25th percentile, median, 75th percentile, the maximum value, and potential outliers. A percentile is the value below which a certain percentage of data fall. For example, if 75% of the observations have values lower than 685 in a data set, then 685 is the 75th percentile of the data. At the 50th percentile, or median, 50% of the values are lower and 50% are higher than that value.

The figure above describes how to read a box plot. Here are a few explanations that may help. The middle part of the plot, or the “interquartile range,” represents the middle quartiles (or the 75th minus the 25th percentile). The line near the middle of the box represents the median (or middle value of the data set). The whiskers on either side of the IQR represent the lowest and highest quartiles of the data. The ends of the whiskers represent the maximum and minimum of the data, and the individual dots beyond the whiskers represent outliers in the data set.

Data File: “Box Plot” tab in “Sample Data.xlsx”

Steps to render a box plot in Minitab:

- Click Graph → Boxplot.
- A new window named “Boxplots” pops up.

- Click “OK” in the window “Boxplots.”
- Another new window named “Boxplot– One Y, Simple” pops up.

- Select “HtBk” as the “Graph Variables.”
- Click the “Data View” button and a new window named “Boxplot– Data View” pops up.
- Check the boxes “Mediansymbol” and “Mean ”
- Click “OK” in the window “Boxplot– Data View.”

- Click “OK” in the window “Boxplot– One Y, Simple.”
- The box plot appears automatically in the new window.

Model summary: The above figure demonstrates the result of the boxplot after navigating through the Minitab menus to yield this output. Notice the interquartile range between the 25^{th} and 75^{th} quartiles, the median line, the mean, and the whiskers.

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]]>In statistics, a t test is a hypothesis test in which the test statistic follows a Student’s t distribution if the null hypothesis is true. We apply a one sample t test when the population variance (σ) is unknown and we use the sample standard deviation (s) instead. A hypothesis test is a statistical method in which a specific hypothesis is formulated about a population, and the decision of whether to reject the hypothesis is made based on sample data. Hypothesis tests help to determine whether a hypothesis about a population or multiple populations is true with certain confidence level based on sample data. Hypothesis testing is a critical tool in the Six Sigma tool belt. It helps us separate fact from fiction, and special cause from noise, when we are looking to make decisions based on data.

One sample t test is a hypothesis test to study whether there is a statistically significant difference between a population mean and a specified value.

- Null Hypothesis (H0): μ = μ0
- Alternative Hypothesis (Ha): μ ≠ μ0

Where:

- μ is the mean of a population of our interest
- μ0 is the specific value we want to compare against

- The sample data of the population of interest are unbiased and representative.
- The data of the population are continuous.
- The data of the population are normally distributed.
- The variance of the population of our interest is unknown.
- One sample t-test is more robust than the z-test when the sample size is small (< 30).

To check whether the population of our interest is normally distributed, we need to run normality test. While there are many normality tests available, such as Anderson–Darling, Sharpiro–Wilk, and Jarque–Bera, our examples will default to using the Anderson-Darling test for normality.

- Null Hypothesis (H0): The data are normally distributed.
- Alternative Hypothesis (Ha): The data are not normally distributed.

**Test Statistic and Critical Value of One Sample t-Test**

To understand what is happening when you run a t-test with your software, the formulas here will walk you through the key calculations and how to determine if the null hypothesis should be accepted or rejected. To determine significance, you must calculate the t-statistic and compare it to the critical value, which is a reference value based on the alpha value and degrees of freedom (n – 1). The t-statistic is calculated based on the sample mean, the sample standard deviation, and the sample size.

Test statistic is calculated with the formula:

(Y ) ̅is the sample mean, n is the sample size, and s is the sample standard deviation

- tcrit is the t-value in a Student’s t distribution with the predetermined significance level α and degrees of freedom (n –1).
- tcrit values for a two-sided and a one-sided hypothesis test with the same significance level α and degrees of freedom (n – 1) are different.

**Decision Rules of One Sample T-Test**

Based on the sample data, we calculated the test statistic tcalc, which is compared against tcrit to make a decision of whether to reject the null.

- Null Hypothesis (H0): μ = μ0
- Alternative Hypothesis (Ha): μ ≠ μ0

If |tcalc| > tcrit, we reject the null and claim there is a statistically significant difference between the population mean μ and the specified value μ0.

If |tcalc| < tcrit, we fail to reject the null and claim there is not any statistically significant difference between the population mean μ and the specified value μ0.

Case study: We want to compare the average height of basketball players against 7 feet.

Data File: “One Sample T-Test” tab in “Sample Data.xlsx”

- Null Hypothesis (H0): μ = 7
- Alternative Hypothesis (Ha): μ ≠ 7

Step 1: Test whether the data are normally distributed

- Click Stat → Basic Statistics → Normality Test.
- A new window named “Normality Test” pops up.
- Select “HtBk” as the variable.

- Click “OK.”
- A new window named “Probability Plot of HtBk” appears, which covers the results of the normality test.

- Null Hypothesis(H
_{0}): The data are normally distributed. - Alternative Hypothesis(H
_{a}): The data are not normally distributed.

Since the p-value of the normality is 0.275, which is greater than alpha level (0.05), we fail to reject the null and claim that the data are normally distributed. If the data are not normally distributed, you need to use hypothesis tests other than the one sample t-test. Now we can run the one-sample t-test, knowing the data are normally distributed.

Step 2: Run the one-sample t-test

- Click Stat → Basic Statistics → 1 Sample
- A new window named “One Samplet for the Mean” pops up.
- Click the blank drop-down box and select “One or more samples, each in a column”.
- Select “HtBk” as the “Samples in columns.”
- Check the box of “Perform hypothesis test.”
- Enter the hypothesized value “7” into the box next to “Perform hypothesis test.”

- Click “OK.”
- The one-sample t-test result appears automatically in the session window.

Model summary: Since the p-value is smaller than alpha level (0.05), we reject the null hypothesis and claim that average height of our basketball players is statistically different from 7 feet.

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]]>The post Central Limit Theorem with Minitab appeared first on Deploy OpEx.

]]>The Central Limit Theorem is one of the fundamental theorems of probability theory. It states a condition under which the mean of a large number of independent and identically-distributed random variables, each of which has a finite mean and variance, would be approximately normally distributed. Let us assume Y1, Y2 . . . Yn is a sequence of n i.i.d. random variables, each of which has finite mean μ and variance σ2, where σ2 > 0. When n increases, the sample average of the n random variables is approximately normally distributed, with the mean equal to μ and variance equal to σ2/n, regardless of the common distribution Yi follows where i = 1, 2 . . . n.

A sequence of random variables is independent and identically distributed (i.i.d.) if each random variable is independent of others and has the same probability distribution as others. It is one of the basic assumptions in Central Limit Theorem. Consider the law of large numbers (LLN)—It is a theorem that describes the result of performing the same experiment a large number of times. According to the LLN, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. The following example will explain this further.

Let us assume we have 10 fair die at hand. Each time we roll all 10 die together we record the average of the 10 die. We repeat rolling the die 50 times until we will have 50 data points. Upon doing so, we will discover that the probability distribution of the sample average approximates the normal distribution even though a single roll of a fair die follows a discrete uniform distribution. Knowing that each die has six possible values (1, 2, 3, 4, 5, 6), when we record the average of the 10 dice over time, we would expect the number to start approximating 3.5 (the average of all possible values). The more rolls we perform, the closer the distribution would be to

The confidence interval is an interval where the true population parameter would fall within a certain confidence level. A 95% confidence interval, the most commonly used confidence level, indicates that the population parameter would fall in that region 95% of the time or we are 95% confident that the population parameter would fall in that region. The confidence interval is used to describe the reliability of a statistical estimate of a population parameter.

The width of a confidence interval depends on the:

- Confidence level—The higher the confidence level, the wider the confidence interval
- Sample size—The smaller the sample size, the wider the confidence interval
- Variability in the data—The more variability, the wider the confidence interval

Data File: “Central Limit Theorem” tab in “Sample Data.xlsx”

- Click Stat → Basic Statistics → Graphical Summary.
- A new window named “Graphical Summary” pops up.
- Select “Cycle Time(Minutes)” as the variable.
- The confidence level is 0.95 by default.

- Click “OK.”
- A new window named “Summary for Cycle Time(Minutes)” pops up.

The 95% confidence interval of the mean is shown in the newly-generated “Summary for Cycle Time (Minutes).” The confidence level is 95% by default. In order to see the confidence interval of “Cycle Time (Minutes)” at other confidence level, we need to enter the confidence level of our interest in the window “Graphical Summary” and click “OK.”

The following example shows how to generate 90% confidence interval of the mean.

Model summary: Here the example shows what selections to make in Minitab to get the confidence intervals around the mean and standard deviation. While Minitab has the default confidence interval at 95%, you can see that other confidence levels can be selected as well.

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]]>Multi Vari analysis is a graphic-driven method to analyze the effects of categorical inputs on a continuous output. It studies how the variation in the output changes across different inputs and helps us quantitatively determine the major source of variability in the output. Multi-Vari charts are used to visualize the source of variation. They work for both crossed and nested hierarchies.

- Y: continuous variable
- X’s: discrete categorical variables. One X may have multiple levels

Hierarchy is a structure of objects in which the relationship of objects can be expressed similar to an organization tree. Each object in the hierarchy is described as above, below, or at the same level as another one. If object A is above object B and they are directly connected to each other in the hierarchy tree, A is B’s parent and B is A’s child. In Multi-Vari analysis, we use the hierarchy to present the relationship between categorical factors (inputs). Each object in the hierarchy tree indicates a specific level of a factor (input). There are generally two types of hierarchies: crossed and nested.

**Crossed Hierarchy**– In the hierarchy tree, if one child item has more than one parent item at the higher level, it is a crossed relationship.**Nested Hierarchy –**In the hierarchy tree, if one child item only has one parent item at the higher level, it is a nested relationship.

Case study: ABC Company produces 10 kinds of units with different weights. Operators measure the weights of the units before sending them out to customers. Multiple factors could have an impact on the weight measurements. The ABC Company wants to have a better understanding of the main source of variability existing in the weight measurement. The ABC Company randomly selects three operators (Joe, John, and Jack) each of whom measures the weights of 10 different units. For each unit, there are three items sampled.

Data File: “Multi-Vari” tab in “Sample Data.xlsx.”

Steps in Minitab to perform a Multi-Vari Analysis:

- Click Stat → Quality Tools → Multi-Vari Chart.
- A window named “Multi-Vari Chart” pops up.
- Select “Measurement” as the “Response,” “Unit” as “Factor 1,” and “Operator” as “Factor 2.”

- Click “Options” button.
- A new window named “Multi-Vari Chart” appears.
- Check the box of “Display individual data points.”

- Click “OK.”
- Analyze the Multi-Vari chart to determine the major source of the variation in the output.

Model summary: Based on the Multi-Vari chart ouptu, the measurement of units ranges from 0.4 to 1.1. Joe’s and John’s mean measurements stay between 0.8 and 0.9. Jack’s mean is slightly lower than both Joe’s and John’s. John has the worst variation when measuring the same kind of unit because John has the highest difference between the maximum and minimum bars for any kind of unit. By observing the black lines of three operators, it seems like all three operators’ measurements follow the same pattern. The operator-to-operator variability is not large. The unit-to-unit variability is large and it could be the main source of variation in measurements.

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