Not long ago, I couldn’t abide statistics. I did respect it, but in much the same way a gazelle respects a lion. Most of my early experiences with statistics indicated that close encounters resulted in pain, so I avoided further contact whenever possible.

So how is it that today I write about statistics? That’s simple: it merely required completely reinventing the way I thought about and approached the discipline. When I decided to approach it as a language rather than a purely mathematical set of skills, the doors opened.

Why does my experience matter to you? If you're a statistician yourself, you know all too well the typical reactions people have when they learn we work with statistics and data analysis: blank stares, uncomfortable silence, horrible jokes, or some variant of, “Oh, how nice." Followed quickly by, "Excuse me, I'm going somewhere I don’t have to talk about statistics.”

People react this way because they’re intimidated by statistics. Maybe you're intimidated by statistics. I certainly used to be...I thought it was too hard to understand. I thought that I'd forgotten what I needed to know. Sometimes I suspected that maybe I just wasn't smart enough to get it. Then I realized I was in a Tower of Babel situation: I just didn't speak the language of statistics. 

Maybe my experience in actually coming to love statistics will resonate with you. Approaching statistics as a kind of conceptual language—rather than a peculiarly ambivalent branch of mathematics—may offer a path to make data analysis more accessible to more people, or at least help us do a better job of communicating with our fellow humans who don’t love statistics.

Straight out of college, I was hired as a feature writer for a science magazine. A few years later I was editing the magazine myself. But in some respects I felt like a gazelle glimpsing a lion’s tail in the grass: my environment delivered constant reminders that statistics existed. The science journals were full of them, scientists cited statistics constantly, and I needed to write about them in every article I did.

I realized I needed to confront my dysfunctional relationship with statistics. So as a seasoned, professional editor, filled with trepidation, I enrolled in a basic statistics course. Now I felt like a gazelle trying to tiptoe quietly through the lion’s den. I was terrified but determined to pass, at least. When I received an A, I couldn’t believe it. What had changed? 

I realized I no longer saw statistics through a mathematical lens. I had come to recognize statistics as a way to describe, understand, and communicate about the world, just like other languages. 

Once I began thinking of statistics as a language that enriches how we know and experience life, it immediately became less threatening. I enrolled in subsequent statistics courses, and completed a master’s degree in applied statistics almost before I realized it.

Mathematics was a core element of these studies, of course, but I loved that simply solving equations wasn’t the ultimate goal: the meaning of the solution was what counted, and the numbers were just a tool to get there. I had never enjoyed math, but I loved statistics. The difference was that in statistics, doing the math correctly is only the beginning.

The real effort comes next: understanding, interpreting, and communicating the implications of our results, including any conditions, caveats, and shortcomings. Given that statistics deals with probability, every analysis has elements of ambiguity and uncertainty. Our models are never complete. There is always another factor to consider, another way to evaluate and dissect the data, another sample to take, or another method that could be applied. That's not unlike the study of literature, where there is always another lens through which to refract the text, another frame of reference through which it can be interpreted. 

Statisticians know the challenges involved in communicating what it is we do. Many people see statistics as inaccessible, esoteric, and intimidating—and in fairness, many statistical concepts are difficult to grasp. 

Maybe it’s incumbent on us to be better translators for this strange language we’ve adopted. One of the ways we've tried make data analysis more accessible for more people is by adding the Assistant to Minitab Statistical Software, so people can get their statistical results in plain language. 

What else could we, as companies and as individuals, be doing to make more people more comfortable with our data-driven world? 

When you work in data analysis, you quickly discover an irrefutable fact: a lot of people just can't stand statistics. Some people fear the math, some fear what the data might reveal, some people find it deadly dull, and others think it's bunk. Many don't even really know why they hate statistics—they just do. Always have, probably always will. 

Problem is, that means we who analyze data need to communicate our results to people who aren't merely inexperienced in statistics, but may have an actively antagonistic relationship with it. 

Given that situation, which side do you think needs to make concessions? 

When I was a kid, I used to see this T-shirt in the Spencer Gifts store in the Monroeville Mall: "If you can't dazzle them with brilliance, baffle them with bull---." I thought it was an amusing little saying then, but as I've grown older I've realized an underlying truth about it:

In terms of substance, brilliance and bull often are identical.

Whether what you're saying is viewed as one or the other depends on how well you're getting your ideas across. Given the ubiquity of the "Lies, damned lies, and statistics" quip, it would seem that most statistically-minded people aren't doing that well. We so often forget that most of the people we need to reach just don't get statistics. But when we do that, we're putting another layer on the Tower of Babel. 

In sharing the results of an analysis with people who aren’t statistics-savvy, we have two obligations. First, to make a concerted effort to convey clearly the results of our analysis, and its strengths and limitations. Most of us do this to some degree. But second, we should take every opportunity we can to demystify and humanize statistics, to help people appreciate not just the complexity but also the art that goes into analysis. To promote statistical literacy. I think most of us can do better in this regard.

There is an impression among those not well versed in statistical methods that the discipline is something of a black box: statisticians know the magic buttons that will transform a spreadsheet full of data into something meaningful. 

A good statistician knows the formulas and methods inside out, and very smart ones expand the discipline with new techniques and applications. But an effective statistician is sensitive to the relationship between the language of statistics and the language the audience speaks, and able to bridge that gap.    

Statisticians who are trying to communicate about their work with the uninitiated are like ambassadors: they need to be completely cognizant of local knowledge, customs, and beliefs, and present their message in a way that will be understood by the recipients. 

In other words, unless we're speaking to a room full of other statisticians, we should stop talking like statisticians. 

The language of statistics can seem particularly impenetrable and obtuse. That's hard to deny, given that  the method we use to compare means is called “Analysis of Variance.” And when it comes to distributions, right-skewed data are clustered on the left side of a bar graph and left-skewed data clustered on the right. Not exactly intuitive. That's why the Assistant in Minitab 17 uses plain language in its output and dialog boxes, and avoids confusing statistical jargon.

Indeed, some statistical language can seem like outright obfuscation, like the notion that a statistical test "failed to reject the null hypothesis." From an editorial viewpoint, "failing to reject the null hypothesis" would seem a needlessly circular equivalent to the word accept.

Of course from a statistical perspective, replacing "failure to reject" with "accept" would be very wrong. So we’re left with a phrase that’s precise, correct, and also confusing. It takes only seconds to compare "failing to reject the null" to a jury saying "not guilty." When evidence against the accused isn’t convincing, that doesn’t prove innocence. But how often is "failure to reject the null" presented to lay audiences with no explanation?

Another difficulty with statistical language, ironically, is that it includes so many common words. Unfortunately, their meanings in statistics are not the same as their common connotations, so when we use them in a statistical context, we often connote unintended ideas. Consider just a few of the terms that mean one thing to statisticians, and quite another to everyone else.

• Significant. For most, this word equates to "important." Statisticians know that significant things may have no importance at all.
• Normal. People take this to mean it something is ordinary or commonplace, not that it follows a Gaussian distribution.
• Regression. To "regress" is to shrink or move backwards. Most people won’t relate that idea to estimating an output variable based on its inputs.
• Average. People hear this not as a mathematical value but as a qualitative judgment, meaning common or fair. 
• Error. Statisticians mean the measure of an estimate's precision, but people hear "mistake." 
• Bias. For statisticians, it doesn't mean attitudinal prejudice, but rather the accuracy of a gauge compared to a reference value.
• Residual. People think residuals are leftovers, not the differences between observed and fitted values.
• Power. A statistical test can be powerful without being influential. Seems like a contradiction, unless you know it refers to the probability of finding a significant (there we go again…) effect when it truly exists.   
• Interaction. An act of communication for most, rather than effects of one factor being dependent on another.
• Confidence. This word carries an emotional charge, and can leave a non-statistical audience thinking statistical confidence means the researchers really believe in their results. 

And the list goes on...statistical terms like sample, assumptions, stability, capability, success, failure, risk, representative, and uncertainty can all mean different things to the world outside the statistical circle.

Statisticians frequently lament the common misunderstandings and lack of statistical awareness among the population at large, but we are responsible for making our points clear and complete when we reach out to non-statistical audiences—and almost any audience we reach will have a non-statistical contingent. 

Making an effort to help the people we communicate with appreciate the technical meanings of these terms as we use them is an easy way to begin promoting higher levels of statistical literacy.

Many of us have data stored in a database or file that we need to analyze on a regular basis. If you're in that situation and you're using Minitab Statistical Software, here's how you can save some time and effort by automating the process.

When you're finished, instead of using File > Query Database (ODBC) each time you want to perform analysis on the most up-to-date set of data, you can add a button to a menu or toolbar that will update the data. To do this you will need to:

A. Create an Exec (.MTB) file that retrieves the data and replaces the current data.
B. Add a shortcut to that file to either a menu or toolbar.

First, I'll create a Minitab script or "exec" that pulls in new data to my worksheet. This is easier than it might sound. 

1. Use File > Query Database (ODBC) to import the desired data. I have several fields that need to be updated, so I can just use File > Query Database (ODBC) repeatedly to pull required fields from multiple tables.

2. Open the History window by clicking the yellow notepad icon and select the ODBC commands/subcommands.

3. Right-click the selected commands and choose Save As...

4. In the Save As... dialog box, choose Exec Files (*MTB) from the Save as Type: drop-down. Choose a filename and location—for example, I'm going to save this as GetData.MTB on my desktop.

5. In Minitab, choose Tools > Notepad.

6. In Notepad, choose File > Open. Change Files of Type: to All Files, and open the .MTB file you just created.

7. Do the following for each ODBC command and corresponding subcommands: 

• Replace the period (.) at the end of the last subcommand with a semi-colon (;).
   
• Add the following below the last subcommand, including the period (In this example, 'Date' and 'Measurement' are the columns I want to store the imported data in. Typically, these share the same name as the fields they are imported from):

Columns 'Date' 'Measurement'.

For example:

Make sure the column names you specify in the Columns subcommand already exist in the Minitab worksheet. You also can use column numbers such as C1 C2, without single-quotes. If you're importing many columns, instead of naming each one individually, you can specify a range like this: Columns C1-C10. 

8. Choose File > Save and then close Notepad. This exec will run the commands and update my data sheet each time it is run.

But I want to make it even easier. Instead of opening the script when I want to use it, I want to be able to just select it from a menu.

To add the .MTB file to a menu in Minitab, I do the following:

1. Choose Tools > Customize.

2. Click the Tools tab.

3. Click  for New (Insert) as shown. If you hover the cursor over the button, the ToolTip displays New (Insert).

4. Enter a name for the button, and then press [Enter]. (For example, enter Get My Data.)

5. Click  to view the Open files dialog box. From Files of type, choose All Files (*.*) then navigate to the .MTB file and double-click it. The dialog box will look like this:

6. Click Close. Now I can run the macro by choosing Tools > Get My Data.

I can also add the macro to a menu other than Tools. 

But now that I think about it, I really don't even want to bother with a menu. I'd prefer to just click on a button and have my data updated automatically. It's easy to do. 

7. Choose Tools > Customize.

8. On the Commands tab, under Categories, choose Tools. Note: If you did not complete steps 5 and 6, the macro will not yet appear in the list.

9. Click and drag Get My Data to the desired place on a menu or toolbar.

Basically, that's it. However, you can change what is displayed on the toolbar by right-clicking the button or text while the Tools > Customize dialog box is open. You can select Image, Text, or Image and Text. 

To change the image that is displayed, choose Edit Button Image. To change the text that is displayed, choose Name Button. As shown below, I have inserted a red button with a circular arrow in the main toolbar, and named it "Get My Data."  

Now I can update my data at any time by clicking on the new button. And if you've been following along, so can you!  If you don't already have Minitab Statistical Software and you'd like to give it a try, download the free 30-day trial. 

Each year Santa’s Elves have to take all the information provided by family, friends and teachers to determine if all the children of the world have been “Naughty” or “Nice.” This is no small task, as according to the website www.santafaqs.com Santa delivers over 5 billion presents per year.

Not only is it a large task in terms of size, but it is critical that the Elves have a consistent approach to this assessment. Santa does not want to give presents to naughty children, but he is adamant that he would rather mistakenly give a present to a naughty child than run the risk of not giving a present to a nice child. 

For this reason, every summer Santa trains all his staff on separating people into the “Naughty” and “Nice” categories, and then he gives them a final test on a set of characters where their behaviour category is already known. For each of these 50 characters, Santa gives the Elves details of their behaviour as reported by their family, friends and work colleagues, and they give them a Naughty or Nice grade. To set up and analyse his new Elf recruits performance, Santa uses an Attribute Agreement Analysis.

The full list of characters and their grades can be seen in this Minitab project file: elf-test.mpj. If you don't already have Minitab and you'd like to give Attribute Agreement Analysis a try with this data set, you can download the free 30-day trial. 

The first thing Santa has to do is create an Attribute Agreement Worksheet, which ensures that each Elf evaluates all the characters in a random order and creates a Minitab worksheet that includes expected category (Naughty or Nice) for each person so that Santa or one of his helpers can quickly enter the Elves assessments. 

To avoid any pre-judgement the Elves do not see the name of the person they are assessing—only their Sample No and the information from family and friends.

The steps he follows are:

1. File > Open Project > Elf-Test.mpj
2. Assistant > Measurement System Analysis (MSA) > Attribute Agreement Worksheet

Santa completes the dialog box as follows and clicks OK. He then prints of the collection datasheets and gets the new Elves to assess the information for each of the people of the list and categorise them as Naughty or Nice.  Once he has this information it is input into the Minitab Worksheet.

Once Santa has collected all this data, he runs the Attribute Agreement Analysis in the Assistant and gets the following results:

Santa is happy with the overall error rate. However, he is very concerned that the percentage of Nice people being rated as Naughty is higher than the overall error rate. This means that there are some good people that may not get presents. This is not acceptable, so he uses another report produced by Minitab to investigate which people are being mis-classified.

This chart shows which samples were misclassified as Naughty.

Santa is worried because every Elf said person 26 was Naughty when the standard was Nice. When Santa looks at the Elf-Test Worksheet, he can see that person 26 was Sherlock Holmes. Santa checks the information on him and can see why the Elves think he is naughty: he smokes and the neighbours have complained that he plays his violin (badly) at all hours of the day and night. Santa provides extra training to the Elves to help them realise that musicians only improve if they practise regularly, so the neighbours will have to suffer.

Characters, 24, 40 and 49 (Little Red Riding Hood, Stuart Little and Shrek, respectively) were only misclassified once apiece, so Santa wants to investigate which Elves made the wrong decision in these cases and again he uses one of the reports the Assistant produces as a standard.

From this report Santa, can see that Berry is the strictest elf—and the one who has made the most mistakes classifying Nice people as Naughty. For this reason, Santa decides to reassign Berry to the reindeer welfare department.

Jingle and Sparkle are now full time Niceness monitors, and Santa is sure—thanks to his training program and the Attribute Agreement Assessment Analysis completed in Minitab—that everyone will get the presents they deserve this year.

If, like Santa, you have to make qualitative assessments on your products or services, an Attribute Agreement Analysis is a good way to verify and improve the performance of you assessors.

This is an era of massive data. A huge amount of data is being generated from the web, from customer relations records but also from sensors used in the manufacturing industry (semiconductor, pharmaceutical, petrochemical companies and many other industries).

Univariate Control charts

In the manufacturing industry, critical product characteristics get routinely collected to ensure that all products at every step of the process remain well within specifications. Dedicated univariate control charts are deployed to ensure that any drift gets detected as early as possible to avoid negative effects on the final product performance. Ideally, when a special cause gets identified, the equipment should be immediately stopped until the issue gets resolved.

Monitoring tool process parameters

In modern plants, many manufacturing tools are connected to IT networks so that tool process parameters can be collected and stored in real time (pressures, temperatures etc…). Unfortunately, this type of data is, very often, not continuously monitored, although we might expect process parameters to play an important role in terms of final product quality. When a quality incident occurs, data from these numerous upstream process parameters are sometimes retrieved from databases, to investigate ( A Posteriori ) why this incident took place in the first place.

A more efficient approach would be to monitor these process parameters in real time and try to understand how they affect complex manufacturing processes: which process parameters are really important which ones are not ? what are their best settings etc… ?

Multivariate control charts

Monitoring upstream tool parameters might lead to a huge increase in the number of control charts though. In this context, process engineers might benefit from using multivariate charts which would enable them to monitor up to 7 or 8 parameters together in a single chart. Rather than using equipment process parameter data in a fire fighting mode to investigate the causes of previous quality incidents, this approach would focus on long term improvements.

Multivariate control charts are based on squared standardized (generalized) multivariate distances from the general mean. In Minitab, the T² Hotelling method is used to generate multivariate charts.

An obvious advantage of using multivariate charts is that they enable one to minimize the total number of control charts, but there are some additional related benefits involved as well:

• Analyzing process parameters jointly: Many process parameters are related to one another, for example, for a particular process step we might expect the pressure value to be large when temperature is high. Considering every process parameter separately is not necessarily a good option and might be misleading. Detecting any mismatch between parameter settings may be very useful.

  In the graph below the Y1 and Y2 parameter values are correlated (high values for Y1 associated to high values for Y2) so that the red point in the lower right corner appears to be out-of-control (beyond the control ellipse) from a multivariate point of view. From a univariate perspective, this red point remains within the usual fluctuation bounds for both Y1 and Y2, though. This point clearly represents a mismatch between Y1 and Y2. The squared generalized multivariate distance from the red point to the scatterplot mean is unusually large.

• Overall rate of false alarms : The probability of a false alarm with three sigma standard limits in a control chart is 0,27%. If 100 charts are monitored at the same time, the probability of a false alarm automatically increases to 27% (0.27% * 100).

  However, when numerous variables are monitored simultaneously using a single multivariate chart, the overall / family rate of false alarms remains close to 0.27%.

  3-D measurements : When three dimensional measurements of a product are performed, the amount of data can get pretty large to ensure that all dimensions (the X, Y and Z dimensions) of a 3-D object remain within specifications. If the product gets damaged in a particular area, this will usually impact more than one dimension so the three dimensions should not be considered separately from one another. If a multivariate charts simultaneously monitors deviations from the ideal planned X, Y, Z values, their combined effects will be taken into account.

A simple example :

Eight process parameters have been monitored using eight univariate Xbar control charts. No out of control has been detected (see below):

The eight control charts above, may be replaced by a single multivariate chart. The associated multivariate chart displayed below, monitors the eight variables simultaneously. Although no out of control point had been detected in the univariate charts, subgroup number 12 turns out to be out of control in the multivariate chart.

Interpretation : to investigate why an out-of-control point (subgroup 12) occurred in the multivariate chart, I used simple graphs (scatterplots) to analyze time trends. Note that as far as the X3, X4 and X5 parameters are involved, subgroup 12 is positioned far away from the other points.

Conclusion :

When Process parameters have no direct critical effect, a univariate dedicated chart is not necessarily required. Multivariate charts would enable one to routinely monitor many tool process parameters with fewer charts.  The objective would be to better understand whether out of control points in a multivariate chart may be used to anticipate quality issues as far as the product characteristics are concerned.

To better control a process, we need to assess how upstream tool parameters affect the final product. Multivariate charts are also very useful to monitor 3-D measurements. Interpreting the reason for an out of control point in a multivariate chart, is a key aspect.

While Minitab 17 is currently a Windows-only application, there are people who only have a Mac available for the installation who also find they need to use Minitab 17. 

It is possible to run Minitab 17 on a Macintosh, though the steps involved in the installation can seem a little daunting at first. In the Technical Support department, we sometimes hear reluctance in people’s voices when we throw out terms like ‘Apple Boot Camp’ or ‘desktop virtualization software.'

NOTE: Minitab products have not been rigorously tested in either Apple Boot Camp or desktop virtualization software, and assistance from Minitab Technical Support is limited. Before purchasing a Minitab product, it is strongly recommended that you download the trial version of the software to test in your environment.  Because the trial version will not run in a virtual environment without a product key, please contact Technical Support for help running the trial on a virtual machine.

So for those of you who would like to install Minitab 17 on your Mac, my goal is to show you exactly how to do it. In this first post in a 2-part series, I’ll walk through the installation using Apple's Boot Camp software.

In addition to your Mac, you’ll need:

• A flash drive
• A copy of Windows 7 or later version ISO
• And of course, Minitab 17 Statistical Software

Boot Camp is a product from Apple that allows you to install and dual-boot your Intel-based Mac in OS X or Windows. With this software, you must boot solely in OS X or Windows—you cannot use OS X and Windows at the same time. The advantage of using Boot Camp is that Windows runs natively, which reduces the likelihood of technical issues.

Boot Camp is included for free with Mac OS X Leopard (v10.5) or higher. If you plan to install Windows 7, it is recommended that you use the Boot Camp software included with Snow Leopard (v10.6) or above.  For more information about Boot Camp, see this Apple support page: https://support.apple.com/en-us/HT201468

To get started, save your Windows ISO to the desktop on your Mac, and plug in your flash drive.

To install Minitab 17 on a Mac using Apple Boot Camp, locate the Apple Boot Camp Assistant in Finder on your Mac, and double-click on the icon to get started.  Click Continue in the Introduction window:

Select the options you’d like to use in the window below (I’m using all three), make sure your USB drive is plugged in, then click Continue again:

The Boot Camp Assistant found the Windows ISO on my desktop in the ISO image path shown below, and will create the bootable USB in with the flash drive listed under Destination disk.  Click Continue to confirm:

Once the bootable USB step is complete, the Boot Camp Assistant will download and save the Windows support software to the flash drive as shown below.  Click Continue again:

The Task Status window will show during the download:

After the download above completes, choose how to partition the hard drive for the third step in the window shown below, then click Install:

Once the hard drive is partitioned, the machine will require a restart and will boot up in Windows. 

Once the Windows installation process begins, follow the on-screen instructions to complete the installation (you’ll have to accept several prompts to continue installing Windows, including language, license agreement, etc.).

Once the Windows installation is complete and your Mac is booted up in Windows, save the Minitab 17 installer to the desktop, then double-click on mtben1721su to launch the setup:

On the first setup screen, click Next:

Check the box to accept the license agreement and click Next again:

Enter your 18-digit product key and click Next:

On the Ready to Install Minitab 17 screen, click Install:

Minitab will show a progress bar during the installation:

When the setup is complete, click Finish:

Next, double-click on the Minitab 17 icon on the desktop to launch Minitab 17.  You’ll be prompted to enter an email address to register, then click OK:

If a product key wasn’t entered during the installation (or, if the license wasn’t activated), you’ll have the option to use the free trial or to activate with a Product key.  Complete the activation, and Minitab 17 is ready to go on your Mac!

Each time you want to switch between the Mac and Windows partition on your hard drive, you’ll need to restart your Mac while holding down the alt (option) key on your keyboard. 

That will give you the option of which partition to use in a screen similar to the one above- when you boot up the machine in Windows you can use Minitab 17.  When you want to go back to your Mac OS make that selection when you reboot the machine.

When using Windows on your Mac, it is best to shut down Windows when you are finished using your software. Closing the lid of your notebook computer may cause Windows to hibernate or suspend, which can cause an error about a tampered license. If that happens, try shutting down Windows and rebooting the virtual computer.

Stay tuned for part II of this post, where I’ll explain how to install Minitab 17 on a Mac using desktop virtualization software.

In my last post, I walked through the steps to install Minitab 17 on a Mac using Apple Boot Camp.  Minitab 17 can also be installed on a Mac using desktop virtualization software.

In addition to your Mac, you’ll need:

• A copy of Windows 7 or later version ISO
• Minitab 17 Statistical Software

Desktop virtualization software allows you to install and use Windows on your Intel-based Mac without requiring a reboot. This allows you to be in OS X and Windows at the same time (this isn’t possible if Minitab is installed using BootCamp, as explained in the previous post). Mac virtualization software is available from industry leaders such as VMware (VMware Fusion), Parallels (Parallels Desktop for Mac), and Oracle (VirtualBox).

NOTE: Minitab products have not been rigorously tested in either Apple Boot Camp or desktop virtualization software, and assistance from Minitab Technical Support is limited. Before purchasing a Minitab product, it is strongly recommended that you download the trial version of the software to test in your environment.  Because the trial version will not run in a virtual environment without a product key, please contact Technical Support for help running the trial on a virtual machine.

In this post, I’ll walk through the steps to install Minitab 17 on a Mac using Oracle's VirtualBox software.

To get started, click the VirtualBox link above to download VirtualBox from the Oracle website.

Choose the amd64 link that corresponds to VirtualBox for OS X to download.  Once the download completes, locate the VirtualBox download (shown below) and double-click to open:

In the VirtualBox window shown below, follow the instructions by double-clicking on the icon in the upper-left corner to begin the installation:

 Click Continue if the message below is presented:

Next click Continue in the installation window below:

Click Install to accept the defaults and being the installation:

You’ll see a progress window during the installation:

The screen below will show when the VirtualBox installation is complete.  Click Close:

With Virtualbox now installed on your Mac, run the program by double-clicking on the icon show below in the Applications folder:

When you launch Virtualbox, the Welcome screen below will be displayed:

Since we haven’t created any virtual machines yet, the area to the left side of the screen above will be blank.  Click on the New button near the upper-left side of the above screen to create a virtual machine.

In the new window (shown below), name the virtual machine and select your version of Windows, then click Continue:

Now select how much space you want to allocate to the virtual machine (I’m using 1024 MB), then click Continue again:

Leave the default option of creating a new hard disk and click Create to enter that wizard:

Choose VDI as the hard drive file type and click Next:

Make the disk Dynamically allocated, then click Continue:

Name the virtual machine in the field below and adjust the size to about 30GB (more or less if desired), then click Create:

Now that the virtual machine has been created, the VirtualBox Manager window will show the new machine on the left side (marked in red below).  With the virtual machine selected on the left, click on the Settings button the top (marked in pink below). 

In the next window (shown below), you can use the buttons at the top to make changes to the default settings- I’m keeping the defaults. 

Now you’ll need your Windows ISO to install- I’ve saved my Windows 7 ISO to the desktop.  To install from an ISO saved to the desktop, click on the Storage button at the top, and choose Empty as shown below, then click on the CD icon to the right and choose Virtual Optical Disk File:

In the window below, I’m browsing to the Windows ISO on my desktop, then clicking Open:

Now double-click on the virtual machine to boot up:

When you boot up, follow the instructions to complete the Windows installation:

When the Windows installation is complete, the virtual machine will reboot.  Once rebooted, we’re almost there- now we can download and install Minitab 17 (for more detailed instructions on installing Minitab 17, please see Part I in this series). 

If you're just getting started in the world of quality improvement, or if you find yourself in a position where you suddenly need to evaluate the quality of incoming or outgoing products from your company, you may have encountered the term "acceptance sampling." It's a statistical method for evaluating the quality of a large batch of materials from a small sample of items, which statistical software like Minitab can make much easier.

Basic statistics courses usually teach sampling in the context of surveys: you administer the survey to a representative sample of individuals, then extrapolate from that sample to make inferences about the entire population the samples comes from. We hear the results of such sampling every day in the news when the results of polls are discussed.

The idea behind acceptance sampling is similar: we inspect or test a sample of a product lot, then extrapolate from that sample to make an inference about whether the entire batch is acceptable, or whether it needs to be rejected.

You can see why this is useful for safeguarding quality. If you work for an electronics manufacturer that is receiving a shipment of 500 capacitors, inspecting and testing every one will take too much time and cost too much money. It's much more efficient to examine a few to determine whether the full shipment is ready to use, or if you should send the lot back to your supplier.

But how many do you need to look at? Acceptance sampling will help you determine how many capacitors to examine, and how many defectives you can tolerate and still accept the shipment.

But it's important to remember that acceptance sampling won't give estimates of quality levels, and because you're inspecting items that are already complete, it doesn't give you any direct process control.

If you want to use acceptance sampling to evaluate a batch of products, you first need to decide which method is best for your situation: acceptance sampling by attributes, or by variables. 

Acceptance sampling by attributes assesses either the number of defects or the number of defective items in a sample. You might tally the total number of defects, in which case each defect in a single item with multiple defects is counted. Alternatively, you can count defective items, in which case the first problem makes an item defective, and you move on to evaluate the next item in your sample. 

In Minitab, you can choose Stat > Quality Tools > Acceptance Sampling by Attributes to either create a new sampling plan or to compare various plans.

Earlier, I shared an overview of acceptance sampling. Now we'll look at how to do acceptance sampling by variables, facilitated by the tools in Minitab Statistical Software. If you're not already using it and you'd like to follow along, you can get the free 30-day trial version. 

In contrast to acceptance sampling by attributes, where inspectors make judgment calls about defective items, acceptance sampling by variables involves the evaluating sampled items on properties that can be measured—for example, the diameter of a hole in a circuit board, or the length of a camshaft.

When you go to Quality > Acceptance Sampling by Variables you will find two options to select from. 

Create / Compare lets you either create a new sampling plan or compare several different ones. Accept / Reject lets you evaluate and make an acceptance decision about a batch of items based on data you've collected according to a sampling plan. 

In this post, we'll look at what you can do with the Create / Compare tools. 

Suppose your electronics company receives monthly shipments of 1,500 LEDs, which are used to indicate whether a device is switched on or off. The soldering leads that go through the devices' circuit boards need to be a certain length. You want to use acceptance sampling to verify the length of the soldering leads. 

Select Stat > Quality Tools > Acceptance Sampling by Variables >Create / Compare, then choose Create a sampling plan. Since we're creating and comparing variable sampling plans, we don't need any real data yet. Instead, we'll just enter information about our process into the dialog box, which you can fill out as shown below. 

For Units for quality levels, choose the appropriate units for your measurement type. You and your supplier have agreed to use defectives per million to represent the number of defectives in your sample.

You and your supplier also have agreed on the poorest process average that would be an Acceptable quality level (AQL), as well as the poorest average you will tolerate before a lot reaches the Rejectable quality level (RQL or LTDP). You and the supplier agree that for LEDs, the AQL is 100 defectives per million, and the RQL is 400 defectives per million

You set the probability of accepting a poor lot (Consumer's risk) at 10 percent, and the chances of rejecting a good lot (Producer's risk) at 5 percent.

You also can enter upper and/or lower specification for your measured property, as well as optionally the historical standard deviation of your process. The lower specification for your LEDs leads is 2 cm.

The lot size refers to the entire population of units that the sample will be taken from. In this case, the size of your monthly LED shipment is 1500.

After you complete the dialog box as shown above and click OK, Minitab produces the following output in the Session Window. 

You need to randomly select and inspect 64 items from each batch of 1500 LEDs. You'll use the mean and standard deviation of your random sample to calculate the Z value, where Z = (mean - lower spec)/ standard deviation. You also can use historical data about the standard deviation, if available.

If Z.LSL is greater than the critical distance, in this case k = 3.51320, you can accept the entire batch of LEDs. If the Z value is less than the critical distance, reject the shipment. 

The probability of accepting a shipment at the AQL is 95%, and when the sampling plan was set up, you and your supplier agreed that lots of 100 defectives per million would be accepted approximately 95% of the time. You also have a 90% probability of rejecting a batch of LEDs at the RQL. This fits your agreement with the supplier that lots with 400 defectives per million would be rejected most of the time for your protection.

If, after a lot is rejected, the supplier's corrective action is to perform 100% inspection and rework any defective items, the Average Outgoing Quality (AOQ) represents the average quality of the lot and the Average Total Inspection (ATI) represents the average number of inspected items after additional screening.

The Average Outgoing Quality (AOQ) level is 91 defectives per million at the AQL and 38.2 defectives per million at the RQL. As we discussed in the overview of acceptance sampling, this is because outgoing quality will be good for lots that are either very good to begin with, or that undergo rework and reinspection due to a poor initial inspection. The Average outgoing quality limit (AOQL) represents the worst-case outgoing quality level, which usually occurs when a batch is neither very good nor very bad.

The Average Total Inspection (ATI) per lot represents the average number of LEDs inspected at a particular quality level and probability of acceptance. For the quality level of 100 defectives per million, the average total of inspected LEDs per lot is 135.6. For the quality level of 400 defectives per million, the average total number of items inspected is 1356.8.

You believe this is a reasonable and acceptable plan to follow, but your supervisor isn't sure, and asks you to see how this plan stacks up against some other possible options. I'll show you how to do that in my next post. 

In my last post, I showed how to use Minitab Statistical Software to create an acceptance sampling plan by variables, using the scenario of a an electronics company that receives monthly shipments of LEDs that must have soldering leads that are at least 2 cm long. This time, we'll compare that plan with some other possible options. 

The variables sampling plan we came up with to verify the lead length called for the random selection and inspection of 64 items from each batch of 1,500 LEDs. You and the supplier agree that the AQL is 100 defectives per million and the RQL is 400 defectives per million. If the calculated Z value is greater than the critical distance (3.51320), you'll accept the entire lot. Sounds great, right? 

But that's not what your boss thinks. He believes that inspecting 64 LEDs is a waste of time, and says he's got a gut feeling that you could probably get by with inspecting half of that number, maybe even fewer. Your gut tells you that's too low.

Fortunately, Minitab makes it easy to consider a few possible plans very easily, so neither of you need to place a bet on whose gut feeling is correct. If you'd like to follow along and you're not already using Minitab, please download the free 30-day trial. 

Start by selecting Stat > Quality Tools > Acceptance Sampling by Variables > Create/Compare..., and when the dialog box appears, choose the option for Compare User Defined Sampling Plans. 

In Units for quality levels, choose Defectives per million. Since you and your supplier have already established the acceptable and rejectable quality levels, for Acceptable quality level (AQL) enter 100, and for Rejectable quality level (RQL or LTDP), enter 400. However, you don't need to enter AQL and RQL when you are simply comparing sampling plans.

In Sample sizes, enter 32 50 64 75 100, and enter 3.55750 in Critical distances (k values). All that's left is to enter the lower spec for each item, the historical standard deviation of .145, and the lot size of 1500. 

When you press OK, Minitab's Session Window displays the following output: 

The table shown above shows the probabilities of accepting and rejecting lots of LEDs at quality levels of 100 and 400 defects per million opportunities for different sample sizes. 

Your boss suggested cutting the number you sampled in half, to 32 items. Under that scenario, however, the producer's risk of having a good shipment rejected has more than doubled, from 5% to 12.2%. You know your supplier won't accept that. Moreover, your odds of properly rejecting a poor-quality shipment have fallen from 90% to just 81%, a level you aren't comfortable with. It's clear that a sample size of 32 does not give you or supplier sufficient protection. 

Minitab also produces graphs that make it easy to see and understand the differences between sampling plans visually. In the graph below, the solid blue line represents the 32-item sampling plan, the dotted red line represents a plan that samples 50 items, and the green line represents the original sampling plan you devised, which called for evaluating 64 items. 

Comparing these curves, it's easy to see how far the blue line representing a 32-item sample diverges from the others. But the lines for the 50- and 64-item sampling plans are quite close; the chance of rejecting a good lot only rises 2%, from 5 to 7%, while the odds of correctly rejecting a poor lot only fall 3%, from 90 to 87%.

Evaluating 14 fewer LEDs would save a fair amount of time without adding much additional risk for either your or your supplier, so the 50-item sampling plan may be the best option for keeping yourself, your supplier, and your boss amenable to the inspection process. 

In my next post, we'll use that sampling plan to evaluate the next lot of 1,500 LEDs, and make a decision about whether to accept the shipment, or reject it and return it to the supplier for corrective action.  

Now that we've seen how easy it is to create plans for acceptance sampling by variables, and to compare different sampling plans, it's time to see how to actually analyze the data you collect when you follow the sampling plan. 

If you'd like to follow along and you're not already using Minitab, please download the free 30-day trial. 

If you'll recall from the previous post, after comparing several different sampling plans, you decided that sampling 50 items from your next incoming lot of 1,500 LEDs would be the best option to satisfy your supervisor's desire to sample as few items as possible while at the same time providing sufficient protection to you and your supplier. That protection stems from an acceptable probability that lots will not be accepted or rejected in error. Under this plan, you have just a 7% chance of rejecting a good lot, and an 87% chance to rejecting a poor lot. 

So, on the day your next shipment of LEDs arrives, you select 50 of them and carefully measure the soldering leads. To make sure the sampling process will be effective, you're diligent about taking samples from throughout the entire lot, at random. You record your measurements and place the data into a Minitab worksheet. 

This time, when you go to Stat > Quality Tools > Acceptance Sampling by Variables, choose the Accept/Reject Lot... option. 

The goal of this analysis is to determine whether you should accept or reject this latest batch of LEDs, based on your sample data. If the calculated Z value is greater than the critical distance (3.5132), you will accept the entire lot. Otherwise, the lot goes back to your supplier for rework and correction.

In Measurement data, enter 'Lead Length'.  In Critical distance (k value), enter 3.5132. In Lower spec, enter 2. Finally, for Historical standard deviation, enter 0.145.  Your dialog box will look like this: 

When you click OK, the Session Window provides the following output: 

From the measurements of the 50 LEDs, that you sampled, the mean length of the solder leads is 2.52254 centimeters, and the historical standard deviation is 0.145 inches. The lower specification of the pipe thickness is 2 inches. 

When you created the sampling plan, the critical distance was determined to be 3.5132. Because this is smaller than the calculated Z.LSL (3.60375), you will accept the lot of 1,500 LEDs.

In an earlier post, I shared an overview of acceptance sampling, a method that lets you evaluate a sample of items from a larger batch of products (for instance, electronics components you've sourced from a new supplier) and use that sample to decide whether or not you should accept or reject the entire shipment. 

There are two approaches to acceptance sampling. If you do it by attributes, you count the number of defects or defective items in the sample, and base your decision about the entire lot on that. The alternative approach is acceptance sampling by variables, in which you use a measurable characteristic to evaluate the sampled items. Doing it by attributes is easier, but sampling by variables requires smaller sample sizes. 

In this post, we'll do acceptance sampling by attributes using Minitab Statistical Software. If you're not already using it and you'd like to follow along, you can get our free 30-day trial version. 

You manage components for a large consumer electronics firm. In that role, you're responsible for sourcing the transistors, resistors, integrated circuits, and other components your company uses in its finished products. You're also responsible for making sure your vendors are supplying high-quality products, and rejecting any batches that don't meet your standards.

Recently, you've been hearing from the assembly managers about problems with one of your suppliers of capacitors. You order these components in batches of 1,000, and it's just not feasible to inspect every individual item coming in. When the next batch of capacitors arrives from this supplier, you decide to use sampling so you can make a data-driven decision to either accept or reject the entire lot.

Before you can devise your sampling plan, you need to know what constitutes an acceptable quality level (AQL) for a batch of capacitors, and what is a rejectable quality level (RQL). As you might surmise, these are figures that need to be discussed with and agreed to by your supplier. You'll also need to settle on levels of the "producer's risk," which is the probability of incorrectly rejecting a lot that should have been accepted, and the "consumer's risk," which the probability that a batch which should have been rejected is accepted. In many cases, the Consumer's Risk is set at a higher level than the Producer's Risk.

Your agreement with the supplier is that the AQL is 1%, and the RQL is 8%. The producer's risk has been set at 5%, which means that about 95% of the time, you'll correctly accept a lot with a defect level of 1% or lower. You've agreed to accept a consumer's risk level of 10%, which means that about 90% of the time you would correctly reject a lot that has a defect level of 8% or higher. 

Now we can use Minitab to determine an appropriate sampling plan. 

1. Choose Stat > Quality Tools > Acceptance Sampling by Attributes.
2. Choose Create a sampling plan.
3. In Measurement type, choose Go / no go (defective).
4. In Units for quality levels, choose Percent defective.
5. In Acceptable quality level (AQL), enter 1. In Rejectable quality level (RQL or LTPD), enter 8.
6. In Producer's risk (Alpha), enter 0.05. In Consumer's risk (Beta), enter 0.1.
7. In Lot size, enter 1000.
8. Click OK.

Minitab produces the following output in the Session Window: 

For each lot of 1,000 capacitors, you need to randomly select and inspect 65. If you find more than 2 defectives among these 65 capacitors, you should reject the entire lot. If you find 2 or fewer defective items, accept the entire lot.

Minitab plots an Operating Characteristic Curve to show you the probability of accepting lots at various incoming quality levels. In this case, the probability of acceptance at the AQL (1%) is 0.972, and the probability of rejecting is 0.028. When the sampling plan was set up, you and your supplier agreed that lots of 1% defective would be accepted approximately 95% of the time to protect the producer. 

The probability of accepting a batch of capacitors at the RQL (8%) is 0.099 and the probability of rejecting is 0.901. The consumer and supplier agreed that lots of 8% defective would be rejected most of the time to protect the consumer.

When the next batch of capacitors arrives at the dock, you pick out 65 at random and test them. Five of the 65 samples are defective. 

Based on your plan, you reject the lot. Now what? Typically, the supplier will need to take some corrective action, such as inspecting all units and reworking or replacing any that are defective.

Minitab produces two graphs that can tell you more. If we assume that rejected lots will be 100% inspected and all defects rectified, the Average Outgoing Quality (AOQ) plot represents the relationship between the quality of incoming and outgoing materials. The Average Total Inspection (ATI) shows the correlation between the quality of incoming materials and the number of items that need to be inspected.

When incoming lots are very good or very bad, the outgoing quality will be good because poor lots get reinspected and fixed, and good lots are already good. In the graph below, the AOQ level is 1.4% at the AQL and 1.0% at the RQL. But when incoming quality is neither very good or very bad, the number of bad parts that gets through rises, so outgoing quality gets worse. The maximum % defective level for outgoing quality is called the Average Outgoing Quality Limit (AOQL). This figure is included in the session window output above, and you can see it in the graph below: At about 3.45% defective, Average Outgoing Quality Limit(AOQL) = 1.968, the worst-case outgoing quality level.

The ATI per lot represents the average number of capacitors you will need to inspect at a particular quality level. 

In the graph above, you can see that if the lot's actual % defective is 2%, the average total number of capacitors inspected per lot will approach 200 (including re-inspections after the supplier has rectified a rejected lot). If the quality level of 10% defective, the average total number of capacitors inspected per lot is 907.3.

Check out my earlier posts for a walk through of performing acceptance sampling by variables. 

At the start of a new year, I like to look for data that’s labeled 2016. While it’s not necessarily new for 2016, one of the first data sets I found was healthcare.gov’s data about qualified health and stand-alone dental plans offered through their site.

Now, there’s lots of fun stuff to poke around in a data set this size—there are over 90,000 records on more than 140 variables. But to start out I used Minitab to do some exploratory graphical analysis.

One statistic you might be interested in is the mean cost of the plans available. Minitab makes this easy because Minitab’s bar chart automatically computes the means, and other statistics, to plot them. This is a chart of the means by state for premiums paid by 21-year old adults. I colored Utah in red because it’s going to do something none of the other states do.

Here’s a bar chart of the means for a couple with 2 children, aged 40:

See how Utah moved? For 21-year olds, Utah was the second-cheapest. For the category of couple+2 children, age 40, Utah’s not radically different in price from many other states, but its rank changed. In fact, of all the states, Utah is the only one that changed position relative to any others.

We’re not talking about large differences in the means, but what makes the change seem really odd is this: Utah is the only state where the mean price for a couple+2 children, age 40, is not completely determined by the price for adults at the age of 21.

Here’s a scatterplot of the means of all plan premiums in each state for the two example groups from the dataset. Utah is the red dot:

If you remove Utah from the data set (Minitab makes excluding points easy) the R2 value is 100%

Does the difference have to do with the plans? Do the providers in Utah do something different? Is this simply a quirk of how the data are recorded? Does it have to do with Utah’s history of providing a healthcare exchange before the Affordable Care Act? It’s hard to say without looking a little deeper. But Minitab’s easy exploratory graphs make it simple to find the points in a data that show the need for further investigation.

I’ll do my own follow-up, because my natural curiosity can’t be satisfied otherwise. If you have your own hypothesis, feel free to share it in the comments section.

If you perform linear regression analysis, you might need to compare different regression lines to see if their constants and slope coefficients are different. Imagine there is an established relationship between X and Y. Now, suppose you want to determine whether that relationship has changed. Perhaps there is a new context, process, or some other qualitative change, and you want to determine whether that affects the relationship between X and Y.

For example, you might want to assess whether the relationship between the height and weight of football players is significantly different than the same relationship in the general population.

You can graph the regression lines to visually compare the slope coefficients and constants. However, you should also statistically test the differences. Hypothesis testing helps separate the true differences from the random differences caused by sampling error so you can have more confidence in your findings.

In this blog post, I’ll show you how to compare a relationship between different regression models and determine whether the differences are statistically significant. Fortunately, these tests are easy to do using Minitab statistical software.

In the example I’ll use throughout this post, there is an input variable and an output variable for a hypothetical process. We want to compare the relationship between these two variables under two different conditions. Here is the Minitab project file with the data.

When the constants (or y intercepts) in two different regression equations are different, this indicates that the two regression lines are shifted up or down on the Y axis. In the scatterplot below, you can see that the Output from Condition B is consistently higher than Condition A for any given Input value. We want to determine whether this vertical shift is statistically significant.

To test the difference between the constants, we just need to include a categorical variable that identifies the qualitative attribute of interest in the model. For our example, I have created a variable for the condition (A or B) associated with each observation.

To fit the model in Minitab, I’ll use: Stat > Regression > Regression > Fit Regression Model. I’ll include Output as the response variable, Input as the continuous predictor, and Condition as the categorical predictor.

In the regression analysis output, we’ll first check the coefficients table.

This table shows us that the relationship between Input and Output is statistically significant because the p-value for Input is 0.000.

The coefficient for Condition is 10 and its p-value is significant (0.000). The coefficient tells us that the vertical distance between the two regression lines in the scatterplot is 10 units of Output. The p-value tells us that this difference is statistically significant—you can reject the null hypothesis that the distance between the two constants is zero. You can also see the difference between the two constants in the regression equation table below.

When two slope coefficients are different, a one-unit change in a predictor is associated with different mean changes in the response. In the scatterplot below, it appears that a one-unit increase in Input is associated with a greater increase in Output in Condition B than in Condition A. We can see that the slopes look different, but we want to be sure this difference is statistically significant.

How do you statistically test the difference between regression coefficients? It sounds like it might be complicated, but it is actually very simple. We can even use the same Condition variable that we did for testing the constants.

We need to determine whether the coefficient for Input depends on the Condition. In statistics, when we say that the effect of one variable depends on another variable, that’s an interaction effect. All we need to do is include the interaction term for Input*Condition!

In Minitab, you can specify interaction terms by clicking the Model button in the main regression dialog box. After I fit the regression model with the interaction term, we obtain the following coefficients table:

The table shows us that the interaction term (Input*Condition) is statistically significant (p = 0.000). Consequently, we reject the null hypothesis and conclude that the difference between the two coefficients for Input (below, 1.5359 and 2.0050) does not equal zero. We also see that the main effect of Condition is not significant (p = 0.093), which indicates that difference between the two constants is not statistically significant.

It is easy to compare and test the differences between the constants and coefficients in regression models by including a categorical variable. These tests are useful when you can see differences between regression models and you want to defend your conclusions with p-values.

If you're learning about regression, read my regression tutorial!

How deeply has statistical content from Minitab blog posts (or other sources) seeped into your brain tissue? Rather than submit a biopsy specimen from your temporal lobe for analysis, take this short quiz to find out. Each question may have more than one correct answer. Good luck!

1. Which of the following are famous figure skating pairs, and which are methods for testing whether your data follow a normal distribution?
   
   a. Belousova-Protopopov
   b. Anderson-Darling
   c. Kolmogorov-Smirnov
   d. Shen-Zhao
   e. Shapiro-Wilk
   f. Salé-Pelletier
   g. Ryan-Joiner
   
     Figure skaters are a, d, and f. Methods for testing normality are b, c, e, and g. To learn about the different methods for testing normality in Minitab, click here.
   
    
2. A t-value is so-named because...
   
   a. Its value lies midway between the standard deviation(s) and the u-value coefficient (u).
   b. It was first calculated in Fisher’s famous “Lady Tasting Tea” experiment.
   c. It comes from a t-distribution.
   d. It’s the first letter of the last name of the statistician who first defined it.
   e. It was originally estimated by reading tea leaves.
   
     The correct answer is c. To find out what the t-value means, read this post.

   
    

3. How do you pronounce µ, the mean of the population, in English?
   
   a. The way a cow sounds
   b. The way a kitten sounds
   c. The way a chicken sounds
   d. The way a sheep sounds
   e. The way a bullfrog sounds
   
     The correct answer is b. For the English pronunciation of µ and, more importantly, to understand how the population mean differs from the sample mean, read this post.
   
   
4. What does it mean when we say a statistical test is “robust” to the assumption of normality?
   
   a. The test strongly depends on having data that follow normal distribution.
   b. The test can perform well even when the data do not strictly follow a normal distribution.
   c. The test cannot be used with data that follow a normal distribution.
   d. The test will never produce normal results.
   
     The correct answer is b. To find out which commonly used statistical tests are robust to the assumption of normality, see this post.
   
    
5. A Multi-Vari chart is used to...
   
   a. Study patterns of variation from many possible causes.
   b. Display positional or cyclical variations in processes.
   c. Study variations within a subgroup, and between subgroups.
   d. Obtain an overall view of the factor effects.
   e. All of the above.
   f. Ha! There’s no such thing as a “Multi-Vari chart!”
   
     The correct answer is e (or, equivalently, a, b, c, and d). To learn how you can use a Multi-Vari chart, see this post.
   
    
6. How can you identify a discrete distribution?
   
   a. Determine whether the probabilities of all outcomes sum to 1.
   b. Perform the Kelly-Banga Discreteness Test.
   c. Assess the kurtosis value for the distribution.
   d. You can’t—that’s why it’s discrete.
   
   
     The correct answer is a. To learn how to identify and use discrete distributions, see this post. For a general description of different data types, click here. If you incorrectly answered c, see this post.

   
    

7. Which of these events can be modeled by a Poisson process?
   
   a. Getting pooped on by a bird
   b. Dying from a horse kick while serving in the Prussian army
   c. Tracking the location of an escaped zombie
   d. Blinks of a human eye over 24-hour period
   e. None of the above.
   
     The correct answer is a, b, and c. To understand how the Poisson process is used to model rare events, see the the following posts on Poisson and bird pooping, Poisson and escaped zombies, and Poisson and horse kicks.
   
    
8. Why should you examine a Residuals vs. Order Plot when you perform a regression analysis?

   a. To identify non-random error, such as a time effect.
   b. To verify that the order of the residuals matches the order of data in the worksheet.
   c. Because a grumpy, finicky statistician said you have to.
   d. To verify that the residuals have constant variance.
   
    

   The correct answer is a. For examples of how to interpret the Residuals vs Order plot in regression, see the following posts on snakes and alcohol, independence of the residuals, and residuals in DOE.

   
    

9. The Central Limit Theorem says that...

   a. If you take a large number of independent, random samples from a population, the distribution of the samples approaches a normal distribution.
   b. If you take a large number of independent, random samples from a population, the sample means will fall between well-defined confidence limits.
   c. If you take a large number of independent, random samples from a population, the distribution of the sample meansapproaches a normal distribution.
   d. If you take a large number of independent, random samples from a population, you must put them back immediately.
    

   The correct answer is c, although it is frequently misinterpreted as a. To better understand the central limit theorem, see this brief, introductory post on how it works, or this post that explains it with bunnies and dragons.

    

   
   

10. You notice an extreme outlier in your data. What do you do?
    
    a. Scream. Then try to hit it with a broom.
    b. Highlight the row in the worksheet and press [Delete]
    c. Multiply the outlier by e-1
    d. Try to figure out what’s going on
    e. Change the value to the sample mean
    f.  Nothing. You’ve got bigger problems in life.
     

    The correct answer is d. Unfortunately, a, b, and f are common responses in practice. To see how to use brushing in Minitab graphs to investigate outliers, see this post. To see how to handle extreme outliers in a capability analysis, click here. To read about when it is and isn't appropriate to delete data values, see this post. To see what it feels like, statistically and personally, to be an outlier, click here.

    
     

11. Which of the following are true statements about the Box-Cox transformation?
    
    a. The Box-Cox transformation can be used with regression analysis.
    b. You can only use the Box-Cox transformation with positive data.
    c. The Box-Cox transformation is not as powerful as the Johnson transformation.
    d. The Box-Cox transformation transforms data into 3-dimensional cube space.
      a, b, and c are true statements. To see how the Box-Cox uses a logarithmic function to transform non-normal data, see this post. For an example of how to use the Box-Cox transformation when performing a regression analysis, see this post. For a comparison of the Box-Cox and Johnson transformations, see this post.

    
     

12. When would you use a paired t-test instead of a 2-sample t-test?
    
    a. When you don’t get significant results using a 2-sample t test.
    b. When you have dependent pairs of observations.
    c. When you want to compare data in adjacent columns of the worksheet.
    d. When you want to analyze the courtship behavior of exotic animals.
     

    The correct answer is b. For an explanation of the difference between a paired t test and a 2-sample t-test, click here.

    
     

13. Which of these are common pitfalls to avoid when interpreting regression results?
    
    a. Extrapolating predictions beyond the range of values in the sample data.
    b. Confusing correlation with causation.
    c. Using uncooked spaghetti to model linear trends.
    d. Adding too much jitter to points on the scatterplot.
    e. Assuming the R-squared value must always be high.
    f. Treating the residuals as model errors.
    g. Holding the graph upside-down.
     

    The correct answers are a, b, and e. To see an amusing example of extrapolating beyond the range of sample data values, click here. To understand why correlation doesn't imply causation, see this post. For another example, using NFL data, click here, and for yet another, using NBA data, click here. To understand what R-squared is, see this post. To learn why a high R-squared is not always good, and a low R-squared is not always bad, see this post.

    
     

14. Which of the following are terms associated with DOE (design of experiment), and which are terms associated with a BUCK? 
    
    a. Center point
    b. Crown tine
    c. Main effect
    d. Corner point
    e. Pedicle
    f.  Split plot
    g. Block
    h. Burr
    i. Main beam
    j. Run
     

    The design of experiment (DOE) terms are a, c, d, f, g, and j. The parts of a buck's antlers are b, e, and h. The Minitab blog contains many great posts on DOE, including several step-by-step examples that provide a clear, easy-to-understand synopsis of the process to follow when you create and analyze a designed experiment in Minitab. Click hereto see a complete compilation of these DOE posts.

    
     

15. Which of these are frequently cited as common statistical errors?

    a. Assuming that a small amount of random error is OK.
    b. Assuming that you've proven the null hypothesis when the p-value is greater than 0.05.
    c. Assuming that correlation implies causation.
    d. Assuming that statistical significance implies practical significance.
    e. Assuming that inferential statistics is a method of estimation.
    f. Assuming that statisticians are always right.
     

    The correct answers are b, c, and d. To see common statistical mistakes you should avoid click here. And here.

For more information on the concepts covered in this quiz—as well as many other statistical concepts—check out the Minitab Topic Library.

On the Topic Library Overview page, click Menu to access topic of your choice.
For example, for more information on interpreting residual plots in regression analysis, click Modeling Statistics > Regression and correlation > Residuals and residual plots.

Ahh, bottled water. Refreshing, convenient...and sometimes pricey. Or in my case, I should say usually pricey. Confession: I’m a sucker for water that comes in the “pretty” plastic bottles, and my experience is that the pretty-bottle brands are usually the pricier ones. Does bottled water cost increase with the fanciness of the bottle? Well, that could be an analysis for a different day …

My colleague recently shared some interesting stats about the buying and disposing of plastic bottled water containers (Maybe she’s noticed my excessive use of “pretty” bottled waters …?).

According to the organization Ban the Bottle, making bottles to meet America’s demand for bottled water requires more than 17 million barrels of oil annually, which is enough to fuel 1.3 million cars for 1 year. They also cite that Americans consume more than 48 billion bottles of water annually, which is enough bottles to circle the earth 230 times!

While these bottled water facts are certainly enough to convince me to scale back on my “pretty” bottled water habit, a visual representation of data in the form of a Minitab graph is also compelling.

Check out the overlaid time series plot below that shows data published by the Container Recycling Institute on the number of plastic bottled water containers sold, recycled, and wasted for 1991-2013:

You can see that bottled water sales have largely risen over the past 20+ years, although it’s interesting to note that they were potentially impacted by the economic downturn in 2008. And while recycling rates have seen a gentle increase, they have not seen enough of an increase to come even close to the volume of bottles wasted (and not recycled) over the years.

This is certainly food for thought when considering whether or not you should fork over a buck or two for a bottle of water—or in my case, $4 or $5 for the “pretty” bottled water—and whether or not you should throw those bottles in the garbage can!

For more on creating time series plots in Minitab, visit this article from Minitab Support.

Unless you live under a black country rock, you’ve no doubt heard that the world recently lost one of the greatest artists of our time, David Bowie. My memories of the Thin White Duke go all the way back to my formative years. I recall his music echoing through the halls of our house as I crooned along whilst doing the chores. Then as now, Bowie’s creativity and energy inspired me and helped me do what I do.

Since his death, I’ve been reflecting on the many prophetic works that this prolific and visionary artist contributed to the world. In the old days, songs were released in collections called “albums.” This was an artifact of an inefficient and technologically unsophisticated delivery system that relied on large, unwieldy disks that were prone to scratches, warping, and other defect modalities. But I digress. Like a true artist, Bowie often used the media at hand as a vehicle for his art.

In addition, his albums often told stories, which many different audiences have interpreted in many different ways. When I listen to Bowie, I hear stories about life, love...and process quality control.

You might be surprised to discover that David Bowie was a proponent of quality process improvement. For example, you may be familiar with one of David’s earlier classics, “The Man Who Sold the World.” But did you know that David’s original title for the album was The Man Who Sold the World on the Benefits of Continuous Quality Improvement? Of course, that's never been publicly acknowledged. Unfortunately, cigar-chomping executives at the record company forced him to shorten the title because, in their words, “Kids don’t dig quality improvement.” Fools.

Bowie’s subsequent album, Hunky Dory, was an ode to the happy state of affairs that can be achieved if one practices continuous quality improvement. Don’t believe me? Then I challenge you to explain why I hear these lines from the song “Changes”:

I watch the ripples change their size
but never leave the value stream of warm impermanence

For decades I’ve struggled to understand these inscrutable lyrics, but now I realize that they are about control charts. Of course! You see, by ripples, David refers to the random fluctuations of varying sizes that occur naturally in any process. And he asserts that if the process is in control, then the ripples don’t wander outside of the control limits (a.k.a. the stream). Whilst acknowledging that such control makes us feel warm and fuzzy, David also reminds us that process stability is impermanent unless one is dedicated to continuous process improvement and control.

If Hunky Dory is an homage to quality utopia, then Diamond Dogs surely represents the dysphoric chronicles of a harrowing dystopia in which the pursuit of quality has been abandoned. (Fun fact: some claim the original album title was Your Business Is a Diamond in the Rough; Don’t Let Quality Go to the Dogs.) Perhaps jarred by the panic in Detroit, David warned us to pay careful attention to issues of quality in our economic and social institutions. And he warned of an Orwellian future in which individuals are unable to pursue and maintain quality in their organizations because they are stifled by an authoritative ‘big brother’ who gives them neither the attention nor the resources to do so effectively.

By the time his album Young Americans was released, David appeared to be feeling cautiously optimistic about improvements in the quality of quality improvements, as I am reminded every time I hear these lyrics from the song “Golden Years”:

Some of these days, and it won't be long
Gonna’ drive back down where you once belonged
In the back of a dream car twenty foot long
Don't cry my sweet, don't break my heart
Doing all right, but you gotta work smart
Shift upon, shift upon, day upon day, I believe oh Lord
I believe Six Sigma is the way

Some might question Bowie’s insistence on Six Sigma methodology, but I believe none would question his assertion that we must “work smart,” and that dedication to quality is absolutely essential.

As one final piece of evidence, I present the following quote from Bowie's song, "Starman." I personally believe this song is about a quality analyst from an advanced civilization in another galaxy. Gifted songwriter that he was, David realized that "quality analyst from an advanced civilization in another galaxy" was too many syllables to belt out on stage, so he used the "starman" as a metaphor. I've taken the liberty of making the substitution below; I think you'll agree, the veracity of my interpretation is inescapable. 

There's a [quality analyst from an advanced civilization in another galaxy] waiting in the sky
He'd like to come and meet us
But he thinks he'd blow our minds
There's a [quality analyst from an advanced civilization in another galaxy​] waiting in the sky
He's told us not to blow it
'Cause he knows it's all worthwhile

So, so obvious when you know what you're looking for. Kind of gives you goosebumps.

I took a few moments with fellow Minitab blogger and Bowie fan, Eston Martz, to brainstorm about what made Bowie such a monumental and influential artist. I collected our notes and created this fishbone diagram in Minitab Statistical Software. This is only a partial listing of Bowie's albums, musical collaborators, personas, and topics that he covered in his music. It would take many more fish with many more bones to cover all of his artistic collaborations, movie roles, and other artistic endeavors. Thanks for the music, David, and thanks for the inspiration, past, present, and future.

Any time you see a process changing, it's important to determine why. Is it indicative of a long term trend, or is it a fad that you can ignore since it will be gone shortly? 

For example, in the 2014 NBA Finals, the San Antonio Spurs beat the two-time defending champion Miami Heat by attempting more 3-pointers (23.6 per game) than any championship team in league history. In the 2015 regular season, the Golden State Warriors made more 3-pointers than any NBA team event attempted from 1980-1988. And this season Steph Curry, by himself, has attempted more 3-pointers than the average NBA team attempted from 1980-1994. Oh yeah, and there is this shot, too...

As I said, when you see a process changing, it's important to determine why. Are you seeing a long-term trend, or is it a soon-to-fade fad? If it's the former, you don't want to be left behind as everybody else moves on without you. But if it's the latter, you don't want to waste time and money changing your entire process for something that won't help you in the long run.

Of course, this applies outside the world of sports, too. Whether you're trying to remove defects from your process, determine how the market for your product is changing, or develop the best strategy for your basketball team, it's always good to know all the details on the changes going on around you. So let's see if the increased use of the 3-pointer in the NBA is here to stay, or if it is a fad that might fade away once Steph Curry leaves the league.

The NBA introduced the 3-pointer in the 1979-80 season. At first it was considered a "gimmick" and wasn't heavily used. But as time went on, teams become more and more reliant on the 3-point shot. In fact, the number of 3-point attempts per game has increased from 2.8 in 1980 to 23.7 in 2016!

This increase in the 3-point shot isn't some new fad. It's actually been going on since the 3-pointer was introduced to the league! (The bump you see from 1995-97 resulted from the NBA shortening the 3-point line before reverting the line to its original distance in 1998.) Now, with the success Golden State has had in implementing a strategy that emphases 3-point shots, it's likely that other teams will follow suit and increase the number of 3-pointers per game even further in the coming years.

So what is the driving force behind this increase? Well, it's just simple math! Since 1980, teams have pretty consistently made about 48% of their 2-point shots. That means when you shoot a 2-point shot, your expected points are 0.48 * 2 = 0.96. Now, this number is actually a little higher, since it doesn't include times you're fouled shooting a 2-point shot (which happens much more often than being fouled shooting a 3-pointer), and you get to shoot resulting free throws. So let's just call the number of expected points "1" to make the math easy.

If you can expect to score 1 point every time you shoot a 2-pointer, you would need to make at least 33% of your 3-pointers to have the same expected value. So do NBA shooters consistently shoot above 33% on their 3-pointers? I used Minitab Statistical Software to create the following time series plot of the data:

We see that it took awhile for NBA players to consistently make more than 33% of their 3-point shots. Coaches were actually correct in not using the 3-pointer too frequently in the 80s and early 90s. But since 1995, the NBA has averaged a percentage that warrants an increased use of the 3-point line. And if you want to explain the reason behind the amazing start that Golden State has been off to this season, look no further than the amount of 3-pointers they attempt and the percentage they make. They average almost 30 attempts per game, and they make a ridiculous 42.4% of their 3-point attempts! You would have to make 63.6% of your 2-pointers to have the same expected number of points as the Warriors' 3-point shots! For some perspective on how hard that is, in his best season, (2013-14) LeBron James made only 62.2% of his 2-point shots.

So 3-point shooting has been steadily increasing from the start, NBA players have consistently made over 33% of their 3-point shots since 1995, and now 3-point shooting has Golden State on track to have the best record in the history of the NBA. Add it all up, and there is only one conclusion:

3-point shooting isn't going away anytime soon.

In my time at Minitab, I’ve gotten a good understanding of what types of graphs users create. Everyone knows about histograms, bar charts, and time series plots. Even relatively less familiar plots like the interval plot and individual value plot are still used quite often. However, one of the most underutilized graphs we have available is the area graph. If you’re not familiar with an Area Graph, here’s the example from the Minitab help menu of what it looks like:

As you can see, an area graph is a great way to be able to view multiple time series trends in one plot, especially if those plots form a part of one whole. There are numerous ways this can be used to visualize things. Anytime you are interested in multiple series that make up a whole, an area graph can do the job. You could use it to show enrollment rates by gender, precipitation rates by county, population totals by city, etc.

I’m going to show you how to go about creating one in Minitab. First, we need to put our data in our worksheet. For this graph, we need each of the series, or sections, in a separate column. An additional constraint on this graph is that we need all of the columns to be of equal length, so be sure that’s the case. In our example we will use sales data different branches, and show that an area graph can be an improvement over a simple time series plot.

Once it’s in your worksheet, we can go to Graph > Time Series Plot, and look at the data in a basic time series plot.  As you can see, there are a few challenges with interpreting this plot. 

First, the plot looks extremely messy. While it gives a good look at the sales from the individual branches, it is very hard to track an individual branch through time. And it’s not much better to look at 4 (or more) separate individual plots, because it then makes it harder to compare. Additionally, when you make separate plots, an important piece of information is lost: total sales. For example, in August, Philadelphia, London, and Seattle had a total sales increase, while New York had its worst month of the year. Was this an overall gain or overall loss? We can’t really tell from individual plots. 

Instead, let’s look at an Area Graph. You can find this by going to Graph > Area Graph, and entering the series the same way as we did the time series plot. Take a look at our output below:

For starters, it looks much cleaner. We are able to see clear trends in the overall pattern. We can see that overall sales spiked in August, answering our question from above. We can use this to evaluate trends in multiple series, as well as the contribution of each series to the total quantity. We get all the information about total sales month-to-month, as well as the individual series for each location, in one plot, instead of in the messy, hard-to-read Time Series plot we created first.

Next time you need to evaluate multiple series together, considering taking a look at the Area Graph to get a cleaner picture of your data!

In the world of linear models, a hierarchical model contains all lower-order terms that comprise the higher-order terms that also appear in the model. For example, a model that includes the interaction term A*B*C is hierarchical if it includes these terms: A, B, C, A*B, A*C, and B*C.

Fitting the correct regression model can be as much of an art as it is a science. Consequently, there's not always a best model that everyone agrees on. This uncertainty carries over to hierarchical models because statisticians disagree on their importance. Some think that you should always fit a hierarchical model whereas others will say it's okay to leave out insignificant lower-order terms in specific cases.

Beginning with Minitab 17, you have the flexibility to specify either a hierarchical or a non-hierarchical linear model for a variety analyses in regression, ANOVA, and designed experiments (DOE). In the example above, if A*B is not statistically significant, why would you include it in the model? Or, perhaps you’ve specified a non-hierarchical model, have seen this dialog box, and you aren’t sure what to do?

In this blog post, I’ll help you decide between fitting a hierarchical or a non-hierarchical regression model.

Reason 1: The terms are all statistically significant or theoretically important

This one is a no-brainer—if all the terms necessary to produce a hierarchical model are statistically significant, you should probably include all of them in the regression model. However, even when a lower-order term is not statistically significant, theoretical considerations and subject area knowledge can suggest that it is a relevant variable. In this case, you should probably still include that term and fit a hierarchical model.

If the interaction term A*B is statistically significant, it can be hard to imagine that the main effect of A is not theoretically relevant at all even if it is not statistically significant. Use your subject area knowledge to decide!

Reason 2: You standardized your continuous predictors or have a DOE model

If you standardize your continuous predictors, you should fit a hierarchical model so that Minitab can produce a regression equation in uncoded (or natural) units. When the equation is in natural units, it’s much easier to interpret the regression coefficients.

If you standardize the predictors and fit a non-hierarchical model, Minitab can only display the regression equation in coded units. For an equation in coded units, the coefficients reflect the coded values of the data rather than the natural values, which makes the interpretation more difficult.

You should always consider a hierarchical model for DOE models because they always use standardized predictors. Starting with Minitab 17, standardizing the continuous predictors is an option for other linear models.

Even if you aren’t using a DOE model, this reason probably applies to you more often than you realize in the context of hierarchical models. When your model contains interaction terms or polynomial terms, you have a great reason to standardize your predictors. These higher-order terms often cause high levels of multicollinearity, which can produce poorly estimated coefficients, cause the coefficients to switch signs, and sap the statistical power of the analysis. Standardizing the continuous predictors can reduce the multicollinearity and related problems that are caused by higher-order terms.

Read my blog post about multicollinearity, VIFs, and standardizing the continuous predictors.

Models that contain too many terms can be relatively imprecise and can have a lessened ability to predict the values of new observations.

Consequently, if the reasons to fit a hierarchical model do not apply to your scenario, you can consider removing lower-order terms if they are not statistically significant.

In my view, the best time to fit a non-hierarchical regression model is when a hierarchical model forces you to include many terms that are not statistically significant. Your model might be more precise without these extra terms.

However, keep an eye on the VIFs to assess multicollinearity. VIFs greater than 5 indicate that multicollinearity might be causing problems. If the VIFs are high, you may want to standardize the predictors, which can tip the balance towards fitting a hierarchical model. On the other hand, removing the interaction terms that are not significant can also reduce the multicollinearity.

You can fit the hierarchical model with standardization first to determine which terms are significant. Then, fit a non-hierarchical model without standardization and check the VIFs to see if you can trust the coefficients and p-values. You should also check the residual plots to be sure that you aren't introducing a bias by removing the terms.

Keep in mind that some statisticians believe you should always fit a hierarchical model. Their rationale, as I understand it, is that a lower-order term provides more basic information about the shape of the response function and a higher-order term simply refines it. This approach has more of a theoretical basis than a mathematical basis. It is not problematic as long as you don’t include too many terms that are not statistically significant.

Unfortunately, there is not always a clear-cut answer to the question of whether you should fit a hierarchical model. I hope this post at least helps you sort through the relevant issues.