Depending on how often and when you use statistical software like Minitab, there may be specific tools or a group of tools you find yourself using over and over again. You may have to do a monthly report, for instance, for which you use one tool in our Basic Statistics menu, another in Quality Tools, and a third in Regression. 

But there are a lot of functions and capabilities in our software, and if you don't use Minitab every day, it might be hard to remember where specific tools are located. While the menus are easy to navigate, you might benefit from grouping all of those commands you use most often in one place. In Minitab, you can do this by creating a custom toolbar to fit your exact needs. 

To add a toolbar, go to Tools> Customize and choose the Toolbars tab. Now click New… and enter a name. At this point, a blank box will appear, like this:

You can leave it hovering, or you can dock it by dragging it next to an existing toolbar. From here, we can add menus and commands for different tasks. This is done by switching to the Commands tab and picking out the commands you want included.

For example, if you wanted a Capability Analysis to be included in your toolbar, you can choose the Stat category on the left and then find the Capability command from the right. You can then drag this command into your toolbar for easy use. 

You can add any command you wish that appears in Minitab's menu.

In addition to custom toolbars, there is a New Menu command, which can give you even more control over organization. The picture above illustrates how I accomplished this while building a DMAIC toolbar. 

You can drag this into the toolbar, and then right click and choose to rename it to anything you wish. This is helpful if you want to organize your toolbar into steps. Step 1 may be preliminary graphs, while Step 2 may be analysis and results. 

Once you have your toolbar built, it will become a part of your active profile. Anytime you make this profile active in your Minitab session, you will have access to this toolbar, which you can use to quickly navigate the commands you use most often. 

T-tests are handy hypothesis tests in statistics when you want to compare means. You can compare a sample mean to a hypothesized or target value using a one-sample t-test. You can compare the means of two groups with a two-sample t-test. If you have two groups with paired observations (e.g., before and after measurements), use the paired t-test.

How do t-tests work? How do t-values fit in? In this series of posts, I’ll answer these questions by focusing on concepts and graphs rather than equations and numbers. After all, a key reason to use statistical software like Minitab is so you don’t get bogged down in the calculations and can instead focus on understanding your results.

In this post, I will explain t-values, t-distributions, and how t-tests use them to calculate probabilities and assess hypotheses.

T-tests are called t-tests because the test results are all based on t-values. T-values are an example of what statisticians call test statistics. A test statistic is a standardized value that is calculated from sample data during a hypothesis test. The procedure that calculates the test statistic compares your data to what is expected under the null hypothesis.

Each type of t-test uses a specific procedure to boil all of your sample data down to one value, the t-value. The calculations behind t-values compare your sample mean(s) to the null hypothesis and incorporates both the sample size and the variability in the data. A t-value of 0 indicates that the sample results exactly equal the null hypothesis. As the difference between the sample data and the null hypothesis increases, the absolute value of the t-value increases.

Assume that we perform a t-test and it calculates a t-value of 2 for our sample data. What does that even mean? I might as well have told you that our data equal 2 fizbins! We don’t know if that’s common or rare when the null hypothesis is true.

By itself, a t-value of 2 doesn’t really tell us anything. T-values are not in the units of the original data, or anything else we’d be familiar with. We need a larger context in which we can place individual t-values before we can interpret them. This is where t-distributions come in.

When you perform a t-test for a single study, you obtain a single t-value. However, if we drew multiple random samples of the same size from the same population and performed the same t-test, we would obtain many t-values and we could plot a distribution of all of them. This type of distribution is known as a sampling distribution.

Fortunately, the properties of t-distributions are well understood in statistics, so we can plot them without having to collect many samples! A specific t-distribution is defined by its degrees of freedom (DF), a value closely related to sample size. Therefore, different t-distributions exist for every sample size. You can graph t-distributions using Minitab’s probability distribution plots.

T-distributions assume that you draw repeated random samples from a population where the null hypothesis is true. You place the t-value from your study in the t-distribution to determine how consistent your results are with the null hypothesis.

The graph above shows a t-distribution that has 20 degrees of freedom, which corresponds to a sample size of 21 in a one-sample t-test. It is a symmetric, bell-shaped distribution that is similar to the normal distribution, but with thicker tails. This graph plots the probability density function (PDF), which describes the likelihood of each t-value.

The peak of the graph is right at zero, which indicates that obtaining a sample value close to the null hypothesis is the most likely. That makes sense because t-distributions assume that the null hypothesis is true. T-values become less likely as you get further away from zero in either direction. In other words, when the null hypothesis is true, you are less likely to obtain a sample that is very different from the null hypothesis.

Our t-value of 2 indicates a positive difference between our sample data and the null hypothesis. The graph shows that there is a reasonable probability of obtaining a t-value from -2 to +2 when the null hypothesis is true. Our t-value of 2 is an unusual value, but we don’t know exactly how unusual. Our ultimate goal is to determine whether our t-value is unusual enough to warrant rejecting the null hypothesis. To do that, we'll need to calculate the probability.

The foundation behind any hypothesis test is being able to take the test statistic from a specific sample and place it within the context of a known probability distribution. For t-tests, if you take a t-value and place it in the context of the correct t-distribution, you can calculate the probabilities associated with that t-value.

A probability allows us to determine how common or rare our t-value is under the assumption that the null hypothesis is true. If the probability is low enough, we can conclude that the effect observed in our sample is inconsistent with the null hypothesis. The evidence in the sample data is strong enough to reject the null hypothesis for the entire population.

Before we calculate the probability associated with our t-value of 2, there are two important details to address.

First, we’ll actually use the t-values of +2 and -2 because we’ll perform a two-tailed test. A two-tailed test is one that can test for differences in both directions. For example, a two-tailed 2-sample t-test can determine whether the difference between group 1 and group 2 is statistically significant in either the positive or negative direction. A one-tailed test can only assess one of those directions.

Second, we can only calculate a non-zero probability for a range of t-values. As you’ll see in the graph below, a range of t-values corresponds to a proportion of the total area under the distribution curve, which is the probability. The probability for any specific point value is zero because it does not produce an area under the curve.

With these points in mind, we’ll shade the area of the curve that has t-values greater than 2 and t-values less than -2.

The graph displays the probability for observing a difference from the null hypothesis that is at least as extreme as the difference present in our sample data while assuming that the null hypothesis is actually true. Each of the shaded regions has a probability of 0.02963, which sums to a total probability of 0.05926. When the null hypothesis is true, the t-value falls within these regions nearly 6% of the time.

This probability has a name that you might have heard of—it’s called the p-value!  While the probability of our t-value falling within these regions is fairly low, it’s not low enough to reject the null hypothesis using the common significance level of 0.05.

Learn how to correctly interpret the p-value.

As mentioned above, t-distributions are defined by the DF, which are closely associated with sample size. As the DF increases, the probability density in the tails decreases and the distribution becomes more tightly clustered around the central value. The graph below depicts t-distributions with 5 and 30 degrees of freedom.

The t-distribution with fewer degrees of freedom has thicker tails. This occurs because the t-distribution is designed to reflect the added uncertainty associated with analyzing small samples. In other words, if you have a small sample, the probability that the sample statistic will be further away from the null hypothesis is greater even when the null hypothesis is true.

Small samples are more likely to be unusual. This affects the probability associated with any given t-value. For 5 and 30 degrees of freedom, a t-value of 2 in a two-tailed test has p-values of 10.2% and 5.4%, respectively. Large samples are better!

I’ve explained how t-values and t-distributions work together to produce probabilities. In my next post, I’ll show how each type of t-test works.

Getting your data from Excel into Minitab Statistical Software for analysis is easy, especially if you keep the following tips in mind.

To paste into Minitab, you can either right-click in the worksheet and choose Paste Cells or you can use Control-V. Minitab allows for 1 row of column headers, so if you have a single row of column info (or no column header info), then you can quickly copy and paste an entire sheet at once. However, if you have multiple rows of descriptive text at the top of your Excel file, then use the following steps:

    Step 1 - Choose a single row for your column headers and paste it into Minitab. 

    Step 2 - Go back to your Excel file to copy all of the actual data over.

And if you have any summary info at the end of your Excel file, you'll want to exclude that too, just like any extraneous column header info.

Copy/paste is ideal when you have only a few Excel sheets. But what if you have lots of sheets? In this case, try using File > Open. Another advantage of File > Open is the additional import options, should you need them. For example, you can specify which sheets and rows to include. And there are even options to handle messy data issues, such as case mismatches and leading and trailing spaces.

Did you know about the Minitab Network group on LinkedIn? It’s the one managed by Eston Martz, who also edits the Minitab blog. I like to see what the members are talking about, which recently got me into some discussions about Raman spectroscopy data.

Not having much experience with Raman spectroscopy data, I thought I’d learn more about it and found the RRUFFTM Project.

The idea is that if you have a Raman device, you can analyze a mineral sample and compare your results to information in the database so that you can identify your mineral. Not having a Raman device, the site is still exciting to me because all of the RRUFFTM data are available in ZIP files that you can download and use to illustrate some neat things in Minitab.

So let’s say that you download one of the ZIP files from the RRUFFTM Project. The ZIP file contains a few thousand text files with intensity data for different minerals. Some minerals have a small number of files. Some minerals, like beryl, have many files.

Turns out beryl’s pretty cool. In its pure form, it’s colorless, but it comes in a variety of colors. In the presence of different ions, beryl can be aquamarine, maxixe, goshenite, heliodor, and emerald.

I extracted just the beryl files into a folder on my computer. Now, I want to analyze the files in Minitab. If I open the worksheet in Minitab without any adjustments, I get something like this:

While I could certainly rearrange this with formulas, I need only a few steps to open the file ready to analyze.

1. Choose File > Open Worksheet.
2. Select the text file.
3. Click Open. Minitab automatically recognizes that you have a text file, opens common options, and lets you see a simple preview of your data.
4. Scroll down so that you can see the first row of numbers, in this case, row 13.
5. Uncheck Data has column names.
6. In First Row to import, enter the row that has the data. In this case, 13.

Now you’ve solved the problem of including identifying information about the mineral in the worksheet. The other problem is that Minitab places all of the data in a single column unless you tell it how to divide the data. You can see the problem, even in the simple preview. Finish thewith these steps:

7. In Field Delimiter, select Comma.
8. Click OK.

Now your data is in a nice, analyzable format. But remember that there are more than 30 files with data on beryl. To analyze them together in Minitab, the data need to be in same worksheet.

 First, open the remaining worksheets with the correct import settings. Then, try these steps:

1. Choose Data > Merge Worksheets > Side-by-Side.
2. Click  to move all of the data from Available worksheets to Worksheets to merge.
3. Name the new worksheet.
4. Click OK.

All of your data is ready to go in a single worksheet.

The options that Minitab provides for opening and merging data sources make it easy to get a wide variety of data ready for analysis. The data features are a good complement to the easy graphs and analyses that you can do in Minitab.

As a recent graduate from Arizona State University with a degree in Business Statistics, I had the opportunity to work with students from different areas of study and help analyze data from various projects for them.

One particular group asked for help analyzing online survey data they had gathered from other students, and they wanted to see if their new student program was beneficial. I would describe this request as them giving us a "pile of data" and saying, "Tell us what you can find out."  

There were numerous problems with this "pile of data" because it wasn't organized, in part because of the way the survey itself was set up. (Our statistics professor later told us that she asked this group to come in because she'd looked at their data before they presented it to us and she wanted to see how we would perform with a "real-world" situation.)

Unfortunately, the statistics department didn't have a time machine that would enable us to go back and set up the survey to have better data that was more organized (I guess if we did have a time machine there would be no need for predictive analytics), but we did have Minitab and its tools to help with the importing of data, reviewing the data, and putting it in a format that is best for analyzing. 

So let’s assume you have a pile of survey data that is:

• Unbiased
• Taken from a random sample
• Taken from the appropriate audience
• Contained enough respondents

Many online survey tools allow you to download your data to a .csv or Excel file, which would be perfect to import into Minitab.

In fact, Minitab 17.3 has recently included a new dialog box that shows you the data before it is opened so you can modify the data type, include/exclude certain columns, and see how many rows are within the data. Within options of that same dialog box you are able to choose what is done with missing data points, and missing data rows. All of these new functions give you the ability to bring a "pile of data" into Minitab a little cleaner with less headache.

Once the data is in Minitab reviewing the data is essential to uncover any irregularities that may be hiding in the data before analysis. Within the Project Manager Bar there is the information icon that allows you to be able to see each column name, column ID, row count, how many missing data points and the type of data of each column. This provides the ability to quickly scan the different columns to make sure that the online data you received correctly by checking the row count, any missing data irregularities, and data type. 

Minitab also has numerous tools to format the data before analysis, including coding, sorting and splitting worksheets. 

For example, occasionally survey data will use “0” in the place of a non-response. This can be a problem because any data analysis will make this a data point when it probably shouldn't be. Minitab can find those “0”s and replace them with missing data to remove them from your worksheet so they won't throw off your analysis (Editor > Find and Replace > Replace).

Before analysis you can also sort your data (Data > Sort) and choose the column you would like to sort the data to, and you can also create a new worksheet from the sorted data.  I also really like the Split and Subset Worksheet options in the event you have a lot of data and it would be easier to look at smaller sections of it for analysis (Data > Split Worksheet and Data > Subset Worksheet).

These are just a few tools that allow you to import data and then prepare the data without having to go back and forth between your spreadsheet software and statistical software. So when you have someone drop off a "pile of data," see how you can use your Minitab tools to shovel through and find the gems that are lying beneath the surface.

Along with the explosion of interest in visualizing data over the past few years has been an excessive focus on how attractive the graph is at the expense of how useful it is. Don't get me wrong...I believe that a colorful, modern graph comes across better than a black-and-white, pixelated one. Unfortunately, however, all the talk seems to be about the attractiveness and not the value of the information presented.

Although perhaps not the most egregious example, one that sticks out to me is the radar chart (also known as the spider chart). The web site Mock Draftable provides radar charts for every prospect in the NFL draft. For example, here is their radar chart for defensive end Dadi Nicolas of Virginia Tech:

This chart uses Dadi's percentiles among other defensive-end prospects on some body measurements and physical tests completed at the combine. It attempts to convey:

1. How well Dadi measures against the other prospects in each measurement, by providing a point on the axis pertaining to that measurement.
2. How good Dadi is overall, by connecting the dots and enclosing a polygon that has an area that increases as individual measurements increase.
3. How "well rounded" Dadi is by looking at how rounded the polygon is...more round indicates a more balanced player, and one with more peaks indicates a less balanced player.

There is no question that what the eye is immediately drawn to is the area covered by the shaded polygon. This is a very misleading graph because of that and I'll explain why. For starters, the order of the categories as you read each axis on the chart is arbitrary. In this example it begins with physical attributes and continues through physical tests in no meaningful order. Allow me to provide four examples of radar charts for Dadi Nicolas that plot the exact same information but change the order of the categories:

If I didn't tell you these were all the same player, you would have to carefully inspect the axes and specific numbers to figure it out. But more broadly, you could draw contradictory conclusions as you look through them:

1. These certainly give different impressions of how well-rounded Dadi is. Charts 1 and 4 appear to show a player that is exceptional in some categories and not very good at all in others. Charts 2 and 3 appear to show a much more balanced player.
2. The area of the polygon on the charts varied wildly and gives completely different impression of the overall skill of the player. Chart 4 covers 20% of the available area while chart 3 covers 40%...using the same information.

I could go into the mathematical details on why the area differs so much but I think the pictures above are worth 1000 words.

If I were asked to chart Dadi's statistics, I could quite easily use Minitab to provide one that conveys the information in a better format. To start, I would use an Individual Value Plot so that I can asses where the player lies on the distribution of prospects, rather than looking at the percentile. I would then create a grouping variable to highlight Nicolas' data on the graph. Then I would place the categories in order of importance—I'm obviously not an NFL scout, but I did a quick correlation on these stats for the 2015 prospects and their draft position to come up with a rough order.

With more work I might come up with some even better ideas, but the point here is to illustrate how quickly a more informative graph could be produced. My graph looks like this (after some editing for looks...that still matters, after all!):

Now I can quickly make the following assessments without being mislead:

1. Dadi is roughly average when all characteristics are combined, but not impressive on two of the three most important categories—40-yard sprint and weight.
2. By plotting the raw values and not just percentile, we see that Dadi not only had the highest vertical jump, but was well above all others. In fact, the gap from Dadi to the 2nd-highest equals the gap from 2nd to 10th.
3. Nicolas is in fact unique and not balanced.

Of course, instead of using an Individual Value Plot, you could also just watch a freshman Dadi Nicolas chase down future NFL wide receiver Brandon Coleman:

Just don't use a radar chart!

We often receive questions about moving ranges because they're used in various tools in our statistical software, including control charts and capability analysis when data is not collected in subgroups. In this post, I'll explain what a moving range is, and how a moving range and average moving range are calculated.

A moving range measures how variation changes over time when data are collected as individual measurements rather than in subgroups.

If we collect individual measurements and need to plot the data on a control chart, or assess the capability of a process, we need a way to estimate the variation over time. But when we have individual observations, we cannot calculate the standard deviation for each subgroup. In such cases, the average moving range across all subgroups is an alternative way to estimate process variation.

Consider the 10 random data points plotted in the graph below:

A moving range is the distance or difference between consecutive points. For example, MR1 in the graph below represents the first moving range, MR2 represents the second moving range, and so forth:

The difference between the first and second points (MR1) is 0.704, and that’s a positive number since the first point has a lower value than the second. The second moving range, MR2, is the difference between the second point (21.0494) and the third (19.6375), and that’s a negative number (-1.4119), since the third point has a lower value than the second.  If we continue that way, we’ll have 9 moving ranges for our 10 data points.

In Minitab, a moving range is easy to compute by "lagging" the data. Continuing the example with the 10 data points above, I can use Stat> Time Series> Lag, and then complete the dialog box as shown below:

Clicking OK in the dialog above will shift the data in C1 down by one row and store the results in C4.  Now we can use Calc> Calculator to subtract C4 from C1 and calculate all the moving ranges:

To calculate the average moving range, we need to use the absolute value of the moving ranges we calculated above.  We’ll take a look at how to do that later. 

When Minitab calculates the average of a moving range, the calculation also includes and unbiasing constant. The formula used to calculate the moving range is:

The table of unbiasing constants is available within Minitab and on this page.

We’ve already done most of the work. To finish, we’ll find the right value of d2 in the table linked above, and use Minitab’s calculator to get the answer.  We need the value of d2 that corresponds to a moving range of length 2 (that’s the number of points in each moving range calculation, but don’t worry, I’ll explain more about the length of the moving range later):

Now back to Minitab, and we can use Calc> Calculator to get our answer:

Using the formula above, we’re telling Minitab to use the absolute values (ABS calculator command) in C5 to calculate the mean, and then divide that by our unbiasing constant value of 1.128.

Now to check our results against Minitab, we can use Stat > Control Charts> Variables Charts for Individuals> I-MR and enter our original data column:

Next, choose I-MR Options> Storage, and check the box next to Standard deviations, then click OK in each dialog box:

The results show the same average moving range value we calculated, 0.602627. 

In this case, because we used a moving range of length 2, the average moving range gives us an estimate of the average distance between our consecutive individual data points.  A moving range of length 2 is Minitab’s default, but that can be changed by clicking the I-MR Options button in the I-MR chart dialog, and then choosing the Estimate tab:

Here we can type in a different value (let’s use 3 as an example), and Minitab will use that number of points to estimate the moving ranges. If we did that for the calculations above, we’d have to make two adjustments:

1. We’d need to choose the correct value for the unbiasing constant, d2, that corresponds with a moving range length of 3:

2. We’d have to adjust the number of points used for our moving ranges from 2 to 3. Using the same random data as before:

With three data points, we’ll use just the highest and the lowest values from the first 3 rows, so MR1 will be 21.0494 – 19.6375 = 1.4119.

If you’ve enjoyed this post, check out some of our other blog posts about control charts.

Working with healthcare-related data often feels different than working with manufacturing data. After all, the common thread among healthcare quality improvement professionals is the motivation to preserve and improve the lives of patients. Whether collecting data on the number of patient falls, patient length-of-stay, bed unavailability, wait times, hospital acquired-infections, or readmissions, human lives are stake. And so collecting and analyzing data—and trusting your results—in a healthcare setting feels even more critical.

Because delivering quality care efficiently is of utmost importance in the healthcare industry, understanding your process, collecting data around that process, and knowing what analysis to perform is key. Awareness about your process and opportunities to improve patient care and cut costs will benefit from using data to drive decisions in your organization that will result in better business and better care.

So, in the interest of using data to draw insights and make decisions that have positive impacts, I’d like to offer several tips for exploring and visualizing your healthcare data in a way that will prepare you for a formal analysis. For instance, graphing your data and examining descriptive statistics such as means and medians can tell you a lot about how your data are distributed and can help you visualize relationships between variables. These preliminary explorations can also reveal unusual observations in your data that should be investigated before you perform a more sophisticated statistical analysis, allowing you to take action quickly when a process, outcome, or adverse event needs attention.

In the first part of this series, I’ll offer two tips on exploring and visualizing data with graphs, brushing, and conditional formatting. In part 2, I’ll offer three more tips focusing on data manipulation and obtaining descriptive statistics.

If you’d like to follow along, you can download and explore the data yourself! If you don’t yet have Minitab 17, you can download the free, 30-day trial.

Let’s look at a case study where a hospital was seeking to examine—and ultimately improve—their room cleaning procedures.

The presence of adenosine triphosphate (ATP) on a surface indicates that bacteria exists. Hospitals can use ATP detection systems to ensure the effectiveness of their sanitization efforts and identify improvement opportunities.

Staff at your hospital used ATP swab tests to test 8 surfaces in 10 different hospital rooms across 5 departments, and recorded the results in a data sheet. ATP measurements below 400 units ‘pass’ the swab test, while measurements greater than or equal to 400 units ‘fail’ the swab test and require further investigation.

Here is a screenshot of part of the worksheet:

You can use a histogram to graph all eight surfaces that were tested in separate panels of the same graph. This helps you observe and compare the distribution of data across each touch point.

If you’ve downloaded the data, you can use the ATP Unstacked.MTW worksheet to create this same histogram by navigating to Graph > Histogram > Simple. In the Graph Variables window, select Door Knob, Light Switch, Bed Rails, Call Button, Phone, Bedside Table, Chair, and IV Pole. Click on the Multiple Graphs subdialog and select In separate panels of the same graph under Show Graph Variables. Click OK through all dialogs.

These histograms reveal that:

• For all test areas, the distribution is asymmetrical with some extreme outliers.
• Data are all right-skewed.
• Data do not appear to be normally distributed.

An individual value plot can be used to graph the ATP measurements collected across all eight surfaces. Identifying the outliers is quite easy with this plot.

And again, you can use the ATP Unstacked.MTW worksheet to create an individual value plot that looks just like mine. Navigate to Graph > Individual Value Plots > Multiple Y’s > Simple, and choose Door Knob, Light Switch, Bed Rails, Call Button, Phone, Bedside Table, Chair, and IV Pole as Graph variables. Click OK.

This individual value plot reveals that:

• Extreme outliers are present for ATP measurements on Bed Rails, Call Button, Phone, and Bedside Table.
• These extreme values are influencing the mean ATP measured for each surface.
• It may be more helpful to analyze differences in medians since the means are skewed by these outliers (judging by the histogram and individual value plot).

Once the outliers are identified, you can investigate them with Minitab’s brushing tool to uncover more insights by right-clicking anywhere in the individual value plot and selecting Brush. Setting ID variables also helps to reveal information about other variables associated with these outliers. To do this, right-click in the graph again and select Set ID Variables. Enter Room as the Variable and click OK. Click and drag the cursor to form a rectangle around the outliers as shown below.

Brushing can provide actionable insights:

• Brushing the extreme outliers on the individual value plot and setting ID variables reveals the room numbers associated with high ATP measurements.
• Quickly identifying rooms where surfaces have high levels of ATP enables faster follow-up and investigation on specific surfaces in specific rooms.

Finally, you can use conditional formatting and other cell properties to investigate and make notes about the outliers. To look at outliers across all surfaces tested, highlight columns C2 through C9, right-click in the worksheet, and select Conditional Formatting > Statistical > Outlier. Alternatively, you can highlight only the extreme outliers by right-clicking in the worksheet, selecting Conditional Formatting > Highlight Cell > Greater Than and entering 2000 (a value we know extreme outliers are above based on the individual value plot).

To make notes about individual outliers, right-click on the cell containing the extreme value, select Cell Properties > Comment, and enter your cell comment.

Conditional formats and cell properties offer:

• Quick insight into surfaces and rooms with high ATP measurements.
• More efficient investigation of problem areas in order to make process improvements.

By exploring and visualizing your data in these preliminary ways, you can see how easy it is to draw conclusions before even doing an analysis. The data is not normally distributed but is highly skewed by several extreme outliers, which greatly influence the mean ATP measurement recorded for each surface. The first graph created to visualize the data is helpful evidence that comparing medians instead of means may be a more effective way to determine if statistically significant differences exist across surfaces. Investigating these outliers both graphically and in the worksheet offers further evidence that analyzing differences in median measurements will be most effective. It is also obvious that bed rails, call buttons, phones, and bedside tables are highly contaminated surfaces—one might surmise this is because of the touch points’ close proximity to sick patients, and the frequency with which patients come into contact with these surfaces.

You can use these insights to focus our initial process improvement efforts on the most problematic touch points and hospital rooms. In part 2 of this blog post, I’ll share some tips for manipulating data, extracting even more information from the data, and displaying descriptive statistics about contamination levels.

In the first partof this series, we looked at a case study where staff at a hospital used ATP swab tests to test 8 surfaces for bacteria in 10 different hospital rooms across 5 departments. ATP measurements below 400 units pass the swab test, while measurements greater than or equal to 400 units fail the swab test and require further investigation.

I offered two tips on exploring and visualizing data using graphs, brushing, and conditional formatting.

1. Evaluate the shape of your data.
2. Identify and investigate outliers.

By performing these preliminary explorations on the swab test data, we discovered that the mean ATP measurement would not be effective for testing whether surfaces showed statistically significant differences in contamination levels. This was due to the data being highly skewed by extreme outliers.

We then identified where these unusually high-ATP measurements were discovered in the hospital. These findings provide valuable information for appropriately focusing process improvement efforts on particular hospital rooms, departments, and surfaces within those rooms.

Now that we've seen how much some simple exploration and visualization tools can reveal, let's run through three more tools that will help you explore your own healthcare data in order to draw actionable insights.

If you’d like to follow along and didn't already download the data from the first post, you can download and explore the data yourself! If you don’t yet have Minitab 17, you can download the free, 30-day trial.

The swab test data the hospital staff collected and recorded is unstacked—this simply means that all response measurements are contained in multiple columns rather than stacked together in one column. To do additional data visualization and a more formal analysis, you need to reconfigure or manipulate how the data is arranged. We can accomplish this by stacking rows.

The ATP Stacked.MTW worksheet in the downloadable Minitab project file above already has the data reshaped for you. But you can manipulate the data on your own using the ATP Unstacked.MTW worksheet. Just navigate to Data > Stack > Rows, and complete the dialog as shown:

Stacking all rows of your data and storing the associated column subscripts (or column names) in a separate column will result in all ATP measurements stacked into one column, a separate column containing categories for Surfaces, and another column containing the Room Number.

With stacked data, you are properly set up to perform formal analyses in Minitab—this is an important step as you work with your data, as most Minitab analyses require columns of stacked data. We won’t tackle a formal analysis here, but rest assured that you are set up to do so!

Once your data are stacked, you can use functions available in Calc > Calculator and Data > Recode to leverage information intrinsic to your original data to create new variables to explore and analyze.

For instance, we know the first character of each room number denotes the department. You can use the ‘left’ function in Calc > Calculator to extract the left-most character from the Room column, and store the result in a new column labeled Department. You can do this by filling out the Calculator dialog as shown:

You also know that ATP measurements below 400 ‘pass’ the ATP swab test. Recoding ranges of ATP values to text to indicate which values ‘Pass’ and which values ‘Fail’ can be useful when visualizing the data. You can do this by filling out the Data >Recode > To Text dialog as shown:

Finally, you can use this newly extracted data to create a stacked bar chart showing the counts of measurements that failed, passed, or were missing from the ATP swab test across Department and the recoded ATP. Using the ATP Stacked.MTW worksheet, navigate to Graph > Bar Chart > Stack. Verify that the Bars represent drop-down shows the default selection, Counts of unique values. Click OK. Select Department and Recoded ATP as Categorical variables, and click OK.

Minitab produces the following graph:
 

The bar chart reveals that:

• Department 4 has the highest count of ATP measurements that failed the swab test.
• The sanitation team should consider focusing initial efforts in department 4 as the investigation of problems with room-cleaning procedures continues.

Now that we’ve manipulated the data in a way that prepares us for more formal analyses, identified which department contains the most contaminated surfaces, and compared the portion of measurements in each department that passed or failed the ATP swab test, we can display descriptive statistics to get an idea of how mean or median bacteria levels differed or varied across surfaces and across departments.

Using the ATP Stacked.MTW worksheet, navigate to Stat > Basic Statistics > Display Descriptive Statistics. Enter ATP as the Variable, Department as the By variable, and click OK. Press Ctrl + E to re-enter the Display Descriptive Statistics dialog, and replace Department with Surface as the By variable. Click OK.  The following output displays in Minitab’s Session Window.

The descriptive statistics reveal helpful information:

• These statistics allow for easy comparison of mean and median ATP measurements as well as the variation of ATP measurements, either by department or by surface.
   
• Notice that mean ATP measurements are much higher than median ATP measurements for both sets of descriptive statistics. This is because the data are right-skewed. Certain analyses that assume you have normally distributed data—such as t-tests to compare means—might not be the best tool to formally analyze this data. Comparing medians might offer more insight.
   
• Both sets of descriptive statistics highlight which departments and surfaces to focus on for investigation and process improvement efforts. For instance, department 4 has the highest median ATP presence, while Bed Rails, Phone, and Call Button—the touch points closest to a sick patient in a hospital bed—appear to be the most problematic surfaces to sanitize. Process improvement efforts can begin with this information.

What you’ve seen in this two-part blog post is just the beginning. But consider how much of this initial exploration is actionable! By having this foundation for visualizing and manipulating your data, you’ll be well on your way to investigating and testing root causes, and more efficiently performing analyses that yield trustworthy results.

If you’re interested in how other healthcare organizations use Minitab for quality improvement, check out our case studies.

In statistics, t-tests are a type of hypothesis test that allows you to compare means. They are called t-tests because each t-test boils your sample data down to one number, the t-value. If you understand how t-tests calculate t-values, you’re well on your way to understanding how these tests work.

In this series of posts, I'm focusing on concepts rather than equations to show how t-tests work. However, this post includes two simple equations that I’ll work through using the analogy of a signal-to-noise ratio.

Minitab statistical software offers the 1-sample t-test, paired t-test, and the 2-sample t-test. Let's look at how each of these t-tests reduce your sample data down to the t-value.

Understanding this process is crucial to understanding how t-tests work. I'll show you the formula first, and then I’ll explain how it works.

Please notice that the formula is a ratio. A common analogy is that the t-value is the signal-to-noise ratio.

The numerator is the signal. You simply take the sample mean and subtract the null hypothesis value. If your sample mean is 10 and the null hypothesis is 6, the difference, or signal, is 4.

If there is no difference between the sample mean and null value, the signal in the numerator, as well as the value of the entire ratio, equals zero. For instance, if your sample mean is 6 and the null value is 6, the difference is zero.

As the difference between the sample mean and the null hypothesis mean increases in either the positive or negative direction, the strength of the signal increases.

Lots of noise can overwhelm the signal.

The denominator is the noise. The equation in the denominator is a measure of variability known as the standard error of the mean. This statistic indicates how accurately your sample estimates the mean of the population. A larger number indicates that your sample estimate is less precise because it has more random error.

This random error is the “noise.” When there is more noise, you expect to see larger differences between the sample mean and the null hypothesis value even when the null hypothesis is true. We include the noise factor in the denominator because we must determine whether the signal is large enough to stand out from it.

Both the signal and noise values are in the units of your data. If your signal is 6 and the noise is 2, your t-value is 3. This t-value indicates that the difference is 3 times the size of the standard error. However, if there is a difference of the same size but your data have more variability (6), your t-value is only 1. The signal is at the same scale as the noise.

In this manner, t-values allow you to see how distinguishable your signal is from the noise. Relatively large signals and low levels of noise produce larger t-values. If the signal does not stand out from the noise, it’s likely that the observed difference between the sample estimate and the null hypothesis value is due to random error in the sample rather than a true difference at the population level.

Many people are confused about when to use a paired t-test and how it works. I’ll let you in on a little secret. The paired t-test and the 1-sample t-test are actually the same test in disguise! As we saw above, a 1-sample t-test compares one sample mean to a null hypothesis value. A paired t-test simply calculates the difference between paired observations (e.g., before and after) and then performs a 1-sample t-test on the differences.

You can test this with this data set to see how all of the results are identical, including the mean difference, t-value, p-value, and confidence interval of the difference.

Understanding that the paired t-test simply performs a 1-sample t-test on the paired differences can really help you understand how the paired t-test works and when to use it. You just need to figure out whether it makes sense to calculate the difference between each pair of observations.

For example, let’s assume that “before” and “after” represent test scores, and there was an intervention in between them. If the before and after scores in each row of the example worksheet represent the same subject, it makes sense to calculate the difference between the scores in this fashion—the paired t-test is appropriate. However, if the scores in each row are for different subjects, it doesn’t make sense to calculate the difference. In this case, you’d need to use another test, such as the 2-sample t-test, which I discuss below.

Using the paired t-test simply saves you the step of having to calculate the differences before performing the t-test. You just need to be sure that the paired differences make sense!

When it is appropriate to use a paired t-test, it can be more powerful than a 2-sample t-test. For more information, go to Why should I use a paired t-test?

The 2-sample t-test takes your sample data from two groups and boils it down to the t-value. The process is very similar to the 1-sample t-test, and you can still use the analogy of the signal-to-noise ratio. Unlike the paired t-test, the 2-sample t-test requires independent groups for each sample.

The formula is below, and then some discussion.

For the 2-sample t-test, the numerator is again the signal, which is the difference between the means of the two samples. For example, if the mean of group 1 is 10, and the mean of group 2 is 4, the difference is 6.

The default null hypothesis for a 2-sample t-test is that the two groups are equal. You can see in the equation that when the two groups are equal, the difference (and the entire ratio) also equals zero. As the difference between the two groups grows in either a positive or negative direction, the signal becomes stronger.

In a 2-sample t-test, the denominator is still the noise, but Minitab can use two different values. You can either assume that the variability in both groups is equal or not equal, and Minitab uses the corresponding estimate of the variability. Either way, the principle remains the same: you are comparing your signal to the noise to see how much the signal stands out.

Just like with the 1-sample t-test, for any given difference in the numerator, as you increase the noise value in the denominator, the t-value becomes smaller. To determine that the groups are different, you need a t-value that is large.

Each type of t-test uses a procedure to boil all of your sample data down to one value, the t-value. The calculations compare your sample mean(s) to the null hypothesis and incorporates both the sample size and the variability in the data. A t-value of 0 indicates that the sample results exactly equal the null hypothesis. In statistics, we call the difference between the sample estimate and the null hypothesis the effect size. As this difference increases, the absolute value of the t-value increases.

That’s all nice, but what does a t-value of, say, 2 really mean? From the discussion above, we know that a t-value of 2 indicates that the observed difference is twice the size of the variability in your data. However, we use t-tests to evaluate hypotheses rather than just figuring out the signal-to-noise ratio. We want to determine whether the effect size is statistically significant.

To see how we get from t-values to assessing hypotheses and determining statistical significance, read the other post in this series, Understanding t-Tests: t-values and t-distributions.

The Pareto chart is a graphic representation of the 80/20 rule, also known as the Pareto principle. If you're a quality improvement specialist, you know that the chart is named after the early 20th century economist Vilfredo Pareto, who discovered that roughly 20% of the population in Italy owned about 80% of the property at that time.

You probably also know that the Pareto principle was later adopted and repurposed as a powerful business metric by Dr. Joseph Juran in the 1940s, to identify the "vital few" issues versus the "trivial many".

But most people don't realize that human use of the Pareto chart goes back much earlier than this. Archeological evidence suggests the chart could date back to the Middle Paleolithic era: using broken-off mastodon bones for bars, and hyena sinews for connect lines, it appears Stone Age humans constructed rudimentary Pareto charts to depict problems as they first began to cook with fire.

Based on the fossilized records, I used our statistical software to recreate a Stone Age Pareto chart:

Unfortunately, although Paleolithic humans were able to create a rough-hewn version of a Pareto chart, their brains were still too small to interpret it. Moreover, they didn't have follow-up tools to identify the root causes of the "vital few" problems identified by the chart. Early attempts at fishbone diagrams similarly failed because the bones were eaten before the diagram was completed. Thus, it would take another 400,000 years of evolution before humans could fry an egg, over-easy.

Fast forward to about 4500 BP (Before Pareto). Hieroglyphic documents unearthed in the tombs of the great pyramids reveal that Egyptian quality engineers in the sphinx manufacturing industry used Pareto charts to reveal the vital few defects in their product.

Unlike their Stone age predecessors, Egyptian quality engineers were able to identify root causes of the vital few issues shown on the chart. For example, they found that the poor grade of limestone used to make the sphinx was responsible for most nose and beard breakage. (The engineers recommended using a more durable, high-quality stone for construction. Unfortunately, the chief treasurer deemed this too costly, arguing that it would undercut the Pharaohs' short-term profit margin over the next few centuries.)

Based on the second-highest bar in the chart, product designers also recommended that the design be modified to make the sphinx either more like a lion, or more like a human. However, upper level priests and viziers noted that the current design was based on Nile delta marketing research that showed the 50% of customers wanted a gigantic lion, while 50% wanted a really, really big person. The current design was deemed a compromise.

None of the recommendations based on the chart were adopted, and the sphinx manufacturing industry went bankrupt soon thereafter.

By the 20th century, the Pareto chart had become a quintessential tool for quality improvement in the manufacturing and service industries. However, new applications were still being made in other diverse fields, including social work and psychotherapy.

On February 7, 1959, marriage therapist Dr. Sigma Freud was the first to apply a Pareto chart in the venue of couples counseling. Alfred and Gloria VanderCamp had sought help to save their crumbling marriage of 23 years. But the counseling sessions soon became bogged down in endless recriminations, as each spouted the innumerable, trivial flaws of the other.

To gain insight, the doctor suggested that the VanderCamps track and record each other's defects over a period of one month. Then the Pareto chart could be used to identify the vital few flaws from the trivial many, allowing the couple to focus on important issues in the marriage. The results are shown below.

Although Dr. Freud was hopeful that the VanderCamps could improve their relationship by focusing on vital flaws, her initial application of the Pareto chart overlooked two critical assumptions:

1. Pareto charts that track frequency assume that the more frequently something happens, the greater the impact it has on the outcome. If this is not the case, flaws should be scored by severity.

2. A Pareto analysis usually illuminates only a snapshot in time, and may not take into account changing conditions.

And so it was. Two months into therapy, Mrs. VanderCamp decided that nothing in life was as important as dancing, and ran off with a ballroom dance instructor.

Several months into the new relationship, upon creating a Pareto chart of her twinkle-toed partner, the former Mrs. VanderCamp was aghast to discover that his flaws—both the vital few and the trivial many—were essentially the same as Alfred's.

She did, however, dance more.

You can use contour plots, 3D scatterplots, and 3D surface plots in Minitab to view three variables in a single plot. These graphs are ideal if you want to see how temperature and humidity affect the drying time of paint, or how horsepower and tire pressure affect a vehicle's fuel efficiency, for example. Ultimately, these three graphs are good choices for helping you to visualize your data and examine relationships among your three variables. 

Contour plots display a 3-dimensional relationship in two dimensions, with x- and y-factors (predictors) plotted on the x- and y-scales and response values represented by contours. You can think of a contour plot like a topographical map, in which x-, y-, and z-values are plotted instead of longitude, latitude, and elevation.

For example, this contour plot shows how reheat time (y) and temperature (x) affect the quality (contours) of a frozen entrée (mac-n-cheese, anyone?). The darker regions indicate higher quality. The contour levels reveal a peak centered in the vicinity of 35 minutes (Time) and 425 degrees (Temp). Quality scores in this peak region are greater than 8.

To create a contour plot in Minitab, choose Graph> Contour Plot. Note that you can easily change the number and colors of contour levels by right-clicking in the graph area and choosing Edit Area.

A 3D scatterplot graphs the actual data values of three continuous variables against each other on the x-, y-, and z-axes. Usually, you would plot predictor variables on the x-axis and y-axis and the response variable on the z-axis.

You can create 3D scatterplots in Minitab by choosing Graph> 3D Scatterplot. Take the frozen entrée example from above—you can plot a simple 3D scatterplot to show how reheat time and temperature affect the quality of the entrée:

It’s also easy to rotate a 3D scatterplot to view it from different angles. Just click on your plot to activate it, then choose Tools> Toolbars> 3D Graph Tools.

Use a 3D surface plot to create a three-dimensional surface based on the x-, y-, and z-variables. The predictor variables are displayed on the x- and y-scales, and the response (z) variable is represented by a smooth surface (in a 3D surface plot) or a grid (in a 3D wireframe plot).

You may be thinking that the 3D surface plot looks very similar to the 3D scatterplot. The only difference between the two is that for the surface plot, Minitab displays a continuous surface or a grid (wireframe plot) of z-values instead of individual data points.

Here’s the frozen entrée data shown on a 3D surface plot:

To build a 3D surface plot in In Minitab, choose Graph> 3D Surface Plot. The same instructions above for rotating a 3D scatterplot apply here as well, making it just as easy to view your 3D surface plot from different angles.

It’s your lucky day! Here’s a bonus fourth way to graph 3 variables in Minitab: You can also use a bubble plot to explore the relationships among three variables on a single plot. Like a scatterplot, a bubble plot plots a y-variable versus an x-variable. However, the symbols ("bubbles") on this plot vary in size. The area of each bubble represents the value of a third variable. Visit this blog post to learn more!

If you want to try your hand at creating these graphs in Minitab and you don't already have it, we offer a full trial version—it's free for 30 days!

Once upon a time, when people wanted to compare the standard deviations of two samples, they had two handy tests available, the F-test and Levene's test.

Statistical lore has it that the F-test is so named because it so frequently fails you.1 Although the F-test is suitable for data that are normally distributed, its sensitivity to departures from normality limits when and where it can be used.

Levene’s test was developed as an antidote to the F-test's extreme sensitivity to nonnormality. However, Levene's test is sometimes accompanied by a troubling side effect: paradoxical dissociations. To see what I mean, take a look at these results from an actual test of 2 standard deviations that I actually ran in Minitab 16 using actual data that I actually made up:

Nothing surprising so far. The ratio of the standard deviations from samples 1 and 2 (s1/s2) is 1.414 / 1.575 = 0.898. This ratio is our best "point estimate" for the ratio of the standard deviations from populations 1 and 2 (Ps1/Ps2).

Note that the ratio is less than 1, which suggests that Ps2 is greater than Ps1. 

Now, let's have a look at the confidence interval (CI) for the population ratio. The CI gives us a range of likely values for the ratio of Ps1/Ps2. The CI below labeled "Continuous" is the one calculated using Levene's method:

What in Gauss' name is going on here?!? The range of likely values for Ps1/Ps2—1.046 to 1.566—doesn't include the point estimate of 0.898?!? In fact, the CI suggests that Ps1/Ps2 is greater than 1. Which suggests that Ps1 is actually greater than Ps2.

But the point estimate suggests the exact opposite! Which suggests that something odd is going on here. Or that I might be losing my mind (which wouldn't be that odd). Or both.

As it turns out, the very elements that make Levene's test robust to departures from normality also leave the test susceptible to paradoxical dissociations like this one. You see, Levene's test isn't actually based on the standard deviation. Instead, the test is based on a statistic called the mean absolute deviation from the median, or MADM. The MADM is much less affected by nonnormality and outliers than is the standard deviation. And even though the MADM and the standard deviation of a sample can be very different, the ratio of MADM1/MADM2 is nevertheless a good approximation for the ratio of Ps1/Ps2. 

However, in extreme cases, outliers can affect the sample standard deviations so much that s1/s2 can fall completely outside of Levene's CI. And that's when you're left with an awkward and confusing case of paradoxical dissociation. 

Fortunately (and this may be the first and last time that you'll ever hear this next phrase), our statisticians have made things a lot less awkward. One of the brave folks in Minitab's R&D department toiled against all odds, and at considerable personal peril to solve this enigma. The result, which has been incorporated into Minitab 17, is an effective, elegant, and non-enigmatic test that we call Bonett's test.

Like Levene's test, Bonett's test can be used with nonnormal data. But unlike Levene's test, Bonett's test is actually based on the actual standard deviations of the actual samples. Which means that Bonett's test is not subject to the same awkward and confusing paradoxical dissociations that can accompany Levene's test. And I don't know about you, but I try to avoid paradoxical dissociations whenever I can. (Especially as I get older, ... I just don't bounce back the way I used to.) 

When you compare two standard deviations in Minitab 17, you get a handy graphical report that quickly and clearly summarizes the results of your test, including the point estimate and the CI from Bonett's test. Which means no more awkward and confusing paradoxical dissociations.

------------------------------------------------------------

1 So, that bit about the name of the F-test—I kind of made that up. Fortunately, there is a better source of information for the genuinely curious. Our white paper, Bonett's Method, includes all kinds of details about these tests and comparisons between the CIs calculated with each. Enjoy.

 
return to text of post

by Laerte de Araujo Lima, guest blogger

The NBA's 2015-16 season will be one for the history books. Not only was it the last season of Kobe Bryan, who scored 60 points in his final game, but the Golden State Warriors set a new wins record, beating the previous record set by 1995-96 Chicago Bulls.

The Warriors seem likely to take this season's NBA title, in large part thanks to the performance of point guard Stephen Curry. A lot of my friends are even saying Curry's skill and performance make him the best point guard ever in NBA history—but it is true? Curry’s performance is amazing, and he's the key element of Warriors’ success, but it seems a little early to define him as the best NBA point guard ever. But in the meantime, we can use data to answer another question:

Has any other point guard in NBA history matched Stephen Curry’s performance during their initial seven seasons?

As a fan of both basketball and Six Sigma, I set out to answer this question methodically, following these steps:

ESPN recently published their list of the 10 best NBA point guards, which puts Magic Johnson first and Curry fourth. ESPN considers both objective factors (NBA titles, MVP nominations, etc.) and subjective parameters (player vision, charisma, team engagement, etc.) to compare players. In keeping with Six Sigma, I want my analysis to be based on figure and facts; however, ESPN's list makes a good starting point. Here are their rankings:

1. Magic Johnson
2. Oscar Robertson
3. John Stockton
4. Stephen Curry
5. Isiah Thomas
6. Chris Paul
7. Steve Nash
8. Jason Kidd
9. Walt Frazier
10. Bob Cousy

This is the easiest part of the job. The NBA web site is a rich source of data, so we are going to use it to check the regular-season performances of each player in ESPN's list. This makes the data average well balanced among all players, because we are going to use the same number of matches per player per season.

In my opinion, the following CTQ factors (based on NBA standards criteria) best characterize point guard performance and how they add value to the team's main target—winning a game:

CTQ

CTQ Definition

Rationale

PTS

Average points per game

Impact of the player on the overall score makes a positive contribution to winning the game.

FG%

Percentage of successful field goals

Player efficiency in shooting makes a positive contribution to winning the game.

3P%

Percentage of successful 3-point field goals

Player efficiency in the 3-point line shoot makes a positive contribution to winning the game.

FT%

Percentage of successful free-throw field goals

Player efficiency in the free throw makes a positive contribution to winning the game.

AST

Average assistance per game

Assisting teammates makes a positive contribution to winning the game.

STL

Average steal per game

New ball possession and counterattacks make a positive contribution to winning the game.

MIN

Average minutes player per game

Player's strategic importance to the team.
Positive contribution to team strategy.

GS

Games per season where player is part of the initial 5.

Initial starts indicate importance in terms of strategy, as well as fewer injuries.

With the players, critical factors, and the source of data defined, let's dig into the analysis.

When I opened Minitab Statistical Software to begin looking at each player's average for each CTQ factor, I faced the first challenge in the analysis. Some players did not have the same CTQ measurements in the NBA database. They had played in the NBA’s early years, and the statistics for all CTQ factors weren't available (for example, the 3-point shot didn't exist at the time some players were active). Consequently, I decided to exclude those players from the analysis to avoid discrepancy in the data. That leaves us with this short list:

1. Magic Johnson
2. John Stockton
3. Stephen Curry
4. Isiah Thomas
5. Chris Paul
6. Steve Nash
7. Jason Kidd

For this analysis, I used the Assistant in Minitab to perform One-Way ANOVA analysis. To access this tool, select Assistant > Hypothesis Tests... and choose One-Way ANOVA.

By performing one-way ANOVA for each of the factors, I can position the players based on the average values of their CTQ variables during each of their first seven seasons. After compiling all results, I deployed a Decision Matrix (another Six Sigma tool) to assess all the players, based on the ANOVA results. The ultimate goal is to determine if Curry’s average performance is superior, inferior, or equal to that of the other players.

Let's take a look at the results of the ANOVA results for the individual CTQ factors.

The Assistant's output is designed to be very easy to understand. The blue bar at the top left answers the bottom-line question, "Do the means differ?" The p-value (0,001) is less than the threshold (< 0.05), telling us that there is a statistically significant difference in means. The intervals displayed on the Means Comparison Chart indicate that Curry and Nash both had huge variation in their average points-per-game in the first 7 years. Statistically speaking, the only player with a average PPG performance that was significantly different from Curry’s is Kidd; all the others had similar performance in their first 7 seasons.

As in the previous analysis, the p-value (0,001) is less than the threshold (< 0.05), telling us that there is a difference in means. However, the interpretation of analysis is clearer. In terms of statistical significance, Curry’s performance is better than Kidd's (again), but not better than Magic's, and it is similar to that of the all other players.

Again, we see that Nash has tremendous variation in his field-goal percentage, and Kidd exhibits the worst average FG% among these players.

To my surprise, based on this comparison chart Magic has the worst performance—and the most variation— among the players for this factor. On the other hand, Curry has an extremely high average performance, with small variation, and this is what we see in the Warriors games.

If we take a closer look at the three highest performers in this category, Nash, Stockton, and Curry, we see that Nash and Curry’s performances are slightly different. Interestingly, the variation in Stockton's data prevents us from being able to conclude that statistically significant difference exists between his average and those of Curry or Nash.

As happens in many Six Sigma projects, the results of this factor contradict conventional wisdom: how could Magic Johnson have the lowest average for this factor? I decided to dig a little bit deeper into Magic’s data using the Assistant's Diagnostic Report, which offers a better view of the data's distribution. we can see an outlier in Magic's data. According to this analysis, he actually had a season with 0% of 3-point field goals!

I could not believe this, so I double-checked the data at the source. To my surprise, it was correct:

In the free throw analysis, Curry's performance is similar to that of Nash and Paul, all of whom performed better than the other players. Once again, Kidd (whom I have nothing against!) has the worst performance.

For this factor, both Nash and Curry are at the end of the queue with similar performance. For this factor, it's also clear that while Stockton has both the highest average and small variation in his performance, he's still comparable with Isiah and Magic.

Again, the p-value (0,001) is less than the threshold (< 0.05), telling us that there is a statistically significant difference in means. It is clear clear that Nash is not a big “stealer” when compared with the other players. It's interesting to see that Curry’s mean performance is better than Nash's and worse than Paul's, but is not statistically significantly different from the mean performance of the remaining players.

For the first time, the ANOVA results have a p-value (0.075) greater than the threshold (< 0.05), telling us that there is no statistically significant difference in means. It is clear that Nash's performance has huge variation, indicating that his contribution was very irregular in the first 7 season (perhaps due to injuries, adaptation, etc.). The amount of variation in Curry's performance follows Nash's.

For this final CTQ, we can see that the p-value (0.006) is less than the threshold (< 0.05), indicating that the means are different. In this case, Stockton and Kidd's means differ. Curry’s presence in the initial 5 in the first 7 season is not statistically significantly different from that of any other other palyers.

Let's take a look at the Diagnostic Report. We can see that Stockton's performance in this CTQ is incredible—he started all seasons' games in the initial 5, showing his importance to the team

Based on the analyses of these criteria, we now have a final have the final outlook based purely on the data. We can use Minitab's conditional formatting to highlight the differences between players for the different factors (> means "better than", < means "worse than", and  = means similar).

• Considering all of the CTQs, Curry’s overall performance is not better than any other point guard in the study, although he does stand out for some individual factors.
• Curry’s PTS is superior only to Kidd's.
• In terms of shot efficiency, Curry’s FG% is better than Kidd's but inferior to Magic's, and at the same level as all other players.
• Curry’s 3-point performance is amazing, but this analysis shows Stockton’s at the same level.
• On the other hand, Curry's FT% is better than that of all the other players, except Paul and Nash.
• Curry’s assistance per season is inferior to all other point guards, except Nash.
• For steals, Curry’s mean performance is better than Nash's, worse than Paul's, and not statistically significantly different from the remaining players.
• In terms of MIN and GS, Curry's performance is similar to that of the other players.
• If we just compare points-per-game (PTS) and shot efficiency (FG%,FT%,3P%) separately, Curry’s overall performance is better than Kidd's, for sure. But if we compare the other CTQ (AST, STL, MIN,GS) factors in the same way, Chris Paul has better performance than Curry.

Based on this analysis, perhaps we need a few more seasons' worth of data to compare these players overall performance and reach a more certain conclusion.

About the Guest Blogger: 

Laerte de Araujo Lima is a Supplier Development Manager for Airbus (France). He has previously worked as product quality engineer for Ford (Brazil), a Project Manager in MGI Coutier (Spain), and Quality Manager in IKF-Imerys (Spain). He earned a bachelor's degree in mechanical engineering from the University of Campina Grande (Brazil) and a master's degree in energy and sustainability from the Vigo University (Spain). He has 10 years of experience in applying Lean Six Sigma to product and process development/improvement. To get in touch with Laerte, please follow him on Twitter @laertelima or on LinkedIn.

Photo of Stephen Curry by Keith Allison, used under Creative Commons 2.0.

Among the most underutilized statistical tools in Minitab, and I think in general, are multivariate tools. Minitab offers a number of different multivariate tools, including principal component analysis, factor analysis, clustering, and more. In this post, my goal is to give you a better understanding of the multivariate tool called discriminant analysis, and how it can be used.

Discriminant analysis is used to classify observations into two or more groups if you have a sample with known groups. Essentially, it's a way to handle a classification problem, where two or more groups, clusters, populations are known up front, and one or more new observations are placed into one of these known classifications based on the measured characteristics. Discriminant analysis can also used to investigate how variables contribute to group separation.

An area where this is especially useful is species classification. We'll use that as an example to explore how this all works. If you want to follow along and you don't already have Minitab, you can get it free for 30 days. 

Analysis of variance (ANOVA) can determine whether the means of three or more groups are different. ANOVA uses F-tests to statistically test the equality of means. In this post, I’ll show you how ANOVA and F-tests work using a one-way ANOVA example.

But wait a minute...have you ever stopped to wonder why you’d use an analysis of variance to determine whether means are different? I'll also show how variances provide information about means.

As in my posts about understanding t-tests, I’ll focus on concepts and graphs rather than equations to explain ANOVA F-tests.

F-tests are named after its test statistic, F, which was named in honor of Sir Ronald Fisher. The F-statistic is simply a ratio of two variances. Variances are a measure of dispersion, or how far the data are scattered from the mean. Larger values represent greater dispersion.

Variance is the square of the standard deviation. For us humans, standard deviations are easier to understand than variances because they’re in the same units as the data rather than squared units. However, many analyses actually use variances in the calculations.

F-statistics are based on the ratio of mean squares. The term “mean squares” may sound confusing but it is simply an estimate of population variance that accounts for the degrees of freedom (DF) used to calculate that estimate.

Despite being a ratio of variances, F-tests are used in a wide variety of situations. Unsurprisingly, the F-test can assess the equality of variances. However, by changing the variances that are included in the ratio, the F-test becomes a very flexible test that you can use for a variety of purposes. For example, you can use F-statistics and F-tests to test the overall significance for a regression model, to compare the fits of different models, to test specific regression terms, and to test the equality of means.

To use the F-test to determine whether group means are equal, it’s just a matter of including the correct variances in the ratio. In one-way ANOVA, the F-statistic is this ratio:

F = variation between sample means / variation within the samples

The best way to understand this ratio is to walk through a one-way ANOVA example.

We’ll analyze four samples of plastic to determine whether they have different mean strengths. You can download the sample data if you want to follow along. (If you don't have Minitab, you can download a free 30-day trial.) I'll refer back to the one-way ANOVA output as I explain the concepts.

In Minitab, choose Stat > ANOVA > One-Way ANOVA... In the dialog box, choose "Strength" as the response, and "Sample" as the factor. Press OK, and Minitab's Session Window displays the following output: 

One-way ANOVA has calculated a mean for each of the four samples of plastic. The group means are: 11.203, 8.938, 10.683, and 8.838. These group means are distributed around the overall mean for all 40 observations, which is 9.915. If the group means are clustered close to the overall mean, their variance is low. However, if the group means are spread out further from the overall mean, their variance is higher.

Clearly, if we want to show that the group means are different, it helps if the means are further apart from each other. In other words, we want higher variability among the means.

Imagine that we perform two different one-way ANOVAs where each analysis has four groups. The graph below shows the spread of the means. Each dot represents the mean of an entire group. The further the dots are spread out, the higher the value of the variability in the numerator of the F-statistic.

What value do we use to measure the variance between sample means for the plastic strength example? In the one-way ANOVA output, we’ll use the adjusted mean square (Adj MS) for Factor, which is 14.540. Don’t try to interpret this number because it won’t make sense. It’s the sum of the squared deviations divided by the factor DF. Just keep in mind that the further apart the group means are, the larger this number becomes.

We also need an estimate of the variability within each sample. To calculate this variance, we need to calculate how far each observation is from its group mean for all 40 observations. Technically, it is the sum of the squared deviations of each observation from its group mean divided by the error DF.

If the observations for each group are close to the group mean, the variance within the samples is low. However, if the observations for each group are further from the group mean, the variance within the samples is higher.

In the graph, the panel on the left shows low variation in the samples while the panel on the right shows high variation. The more spread out the observations are from their group mean, the higher the value in the denominator of the F-statistic.

If we’re hoping to show that the means are different, it's good when the within-group variance is low. You can think of the within-group variance as the background noise that can obscure a difference between means.

For this one-way ANOVA example, the value that we’ll use for the variance within samples is the Adj MS for Error, which is 4.402. It is considered “error” because it is the variability that is not explained by the factor.

The F-statistic is the test statistic for F-tests. In general, an F-statistic is a ratio of two quantities that are expected to be roughly equal under the null hypothesis, which produces an F-statistic of approximately 1.

The F-statistic incorporates both measures of variability discussed above. Let's take a look at how these measures can work together to produce low and high F-values. Look at the graphs below and compare the width of the spread of the group means to the width of the spread within each group.

The low F-value graph shows a case where the group means are close together (low variability) relative to the variability within each group. The high F-value graph shows a case where the variability of group means is large relative to the within group variability. In order to reject the null hypothesis that the group means are equal, we need a high F-value.

For our plastic strength example, we'll use the Factor Adj MS for the numerator (14.540) and the Error Adj MS for the denominator (4.402), which gives us an F-value of 3.30.

Is our F-value high enough? A single F-value is hard to interpret on its own. We need to place our F-value into a larger context before we can interpret it. To do that, we’ll use the F-distribution to calculate probabilities.

For one-way ANOVA, the ratio of the between-group variability to the within-group variability follows an F-distribution when the null hypothesis is true.

When you perform a one-way ANOVA for a single study, you obtain a single F-value. However, if we drew multiple random samples of the same size from the same population and performed the same one-way ANOVA, we would obtain many F-values and we could plot a distribution of all of them. This type of distribution is known as a sampling distribution.

Because the F-distribution assumes that the null hypothesis is true, we can place the F-value from our study in the F-distribution to determine how consistent our results are with the null hypothesis and to calculate probabilities.

The probability that we want to calculate is the probability of observing an F-statistic that is at least as high as the value that our study obtained. That probability allows us to determine how common or rare our F-value is under the assumption that the null hypothesis is true. If the probability is low enough, we can conclude that our data is inconsistent with the null hypothesis. The evidence in the sample data is strong enough to reject the null hypothesis for the entire population.

This probability that we’re calculating is also known as the p-value!

To plot the F-distribution for our plastic strength example, I’ll use Minitab’s probability distribution plots. In order to graph the F-distribution that is appropriate for our specific design and sample size, we'll need to specify the correct number of DF. Looking at our one-way ANOVA output, we can see that we have 3 DF for the numerator and 36 DF for the denominator.

The graph displays the distribution of F-values that we'd obtain if the null hypothesis is true and we repeat our study many times. The shaded area represents the probability of observing an F-value that is at least as large as the F-value our study obtained. The F-value falls within this shaded region about 3.1% of the time when the null hypothesis is true. This probability is low enough to reject the null hypothesis using the common significance level of 0.05. We can conclude that not all the group means are equal.

Learn how to correctly interpret the p-value.

ANOVA uses the F-test to determine whether the variability between group means is larger than the variability of the observations within the groups. If that ratio is sufficiently large, you can conclude that not all the means are equal.

This brings us back to why we analyze variation to make judgments about means. Think about the question: "Are the group means different?" You are implicitly asking about the variability of the means. After all, if the group means don't vary, or don't vary by more than random chance allows, then you can't say the means are different. And that's why you use analysis of variance to test the means.

For hundreds of years, people having been improving their situation by pulling themselves up by their bootstraps. Well, now you can improve your statistical knowledge by pulling yourself up by your bootstraps. Minitab Express has 7 different bootstrapping analyses that can help you better understand the sampling distribution of your data. 

A sampling distribution describes the likelihood of obtaining each possible value of a statistic from a random sample of a population—in other words, what proportion of all random samples of that size will give that value. Bootstrapping is a method that estimates the sampling distribution by taking multiple samples with replacement from a single random sample. These repeated samples are called resamples. Each resample is the same size as the original sample.

The original sample represents the population from which it was drawn. Therefore, the resamples from this original sample represent what we would get if we took many samples from the population. The bootstrap distribution of a statistic, based on the resamples, represents the sampling distribution of the statistic.

For example, let’s estimate the sampling distribution of the number of yards per carry for Penn State’s star running back Saquon Barkley. Going through all 182 of his carries from last season seems daunting, so instead I took a random sample of 49 carries and recorded the number of yards he gained for each one. If you want to follow along, you can get the data I used here.

Repeated sampling with replacement from these 49 samples mimics what the population might look like. To take a resample, one of the carries is randomly selected from the original sample, the number of yards gained is recorded, and then then that observation is put back into the sample. This is done 49 times (the size of the original sample) to complete a single resample.

To obtain a single resample, in Minitab Express go to STATISTICS > Resampling > Bootstrapping > 1-Sample Mean. Enter the column of data in Sample, and enter 1 for number of resamples. The following individual plot represents a single bootstrap sample taken from the original sample.

Note: Because Minitab Express randomly selects the bootstrap sample, your results will be different.

The resample is done by sampling with replacement, so the bootstrap sample will usually not be the same as the original sample. To create a bootstrap distribution, you take many resamples. The following histogram shows the bootstrap distribution for 1,000 resamples or our original sample of 49 carries.

The bootstrap distribution is centered at approximately 5.5, which is an estimate of the population mean for Barkley’s yards per carry. The middle 95% of values from the bootstrapping distribution provide a 95% confidence interval for the population mean. The red reference lines represent the interval, so we can be 95% confident the population mean of Barkley’s yards per carry is between approximately 3.4 and 7.8.

The central limit theorem is a fundamental theorem of probability and statistics. The theorem states that the distribution of the mean of a random sample from a population with finite variance is approximately normally distributed when the sample size is large, regardless of the shape of the population's distribution. Bootstrapping can be used to easily understand how the central limit theorem works.

For example, consider the distribution of the data for Saquon Barkley’s yards per carry.

It’s pretty obvious that the data are nonnormal. But now we’ll create a bootstrap distribution of the means of 10 resamples.  

The distribution of the means is very different from the distribution of the original data. It looks much closer to a normal distribution. This resemblance increases as the number of resamples increases. With 1,000 resamples, the distribution of the mean of the resamples is approximately normal.

Note: Bootstrapping is only available in Minitab Express, which is an introductory statistics package meant for students and university professors.

While many Six Sigma practitioners and other quality improvement professionals like to use the Fishbone diagram in Quality Companion for brainstorming because of its ease of use and integration with other Quality Companion tools, some Minitab users find an infrequent need for a Fishbone diagram. For the more casual user of the Fishbone diagram, Minitab has the right tool to get the job done.

Minitab’s Fishbone (or Cause-and-Effect) diagram can be accessed from the Quality Tools menu:

There are two ways to complete the dialog box and create a Fishbone diagram in Minitab:

1. By typing the information directly into the Cause-and-Effect dialog window, or

2. By entering the information in the worksheet first and then using the worksheet data to complete the Cause-and-Effect dialog box.

In this post, I’ll walk through examples of how to create a Fishbone diagram using both options, starting with the first option above. Because I’m a baking aficionado, I’ll be using an example related to brainstorming the choice of factors in a cake-baking experiment (where the response is the moisture after baking the cake).

First, we’ll start by using the drop-down lists on the left side to tell Minitab that our information is in Constants (meaning we will type the information into this dialog box, versus having the data already typed into the worksheet). 

For this example, I’ll have four branches in the Fishbone, so I’ve selected Constants next to Branch 1, 2, 3 and 4 below, and then I’ve typed the name of each branch on the right side, under Label:

As we work through this, we can always click OK to see our progress.  So far we have:

To go back to the last dialog to keep entering information, press Ctrl+E on the keyboard.

Next, I’ve entered the causes in the empty column in the middle. Note that any individual cause that includes multiple words (for example, Day of Week) must be included in double-quotes: “Day of Week.”  Without the double-quotes, Minitab will assign each individual word as a cause. Multiple causes for the same branch are entered with a space between the causes. For example, to enter Ambient Temperature and Ambient Moisture as causes, I’ll enter:

“Ambient Temperature” “Ambient Moisture”

After completing the dialog like in the example below, we can click OK again to see our progress:

Now I’ve used Ctrl+E on my keyboard again to return to the dialog box.  As a final step, I’m going to add sub-branches to some of my causes. For this example, two of the causes in the ‘Held constant factors’ branch have sub-branches. To add my sub-branches, I’ll click the Sub… button below for that particular branch:

This will bring up the Sub-Branches dialog.  Here the names of each of my causes are automatically listed in the Labels column. All I need to do is (1) choose Constants from the drop-down list and (2) type in the sub-branch labels. Note that the same double-quote rule for sub-braches with multiple words applies here:

After completing the dialog above and clicking OK in each window, we can see our final graph:

As a first step, I”ll type in my branch labels, effect, and title for my fishbone diagram:

Now I’ll click OK to see my progress and go to the worksheet to type in my data like in the example below:

Notice that here we don’t need to include double-quotes for any causes or sub-branches that are described with multiple words. Also, note that the branch titles are still typed into the dialog (so the column titles in the columns above are just for my own reference, because Minitab does not use these column titles).

After entering the data in the worksheet, I can use Ctrl + E to go back to the dialog box. This time I’ll leave the default option for the Causes (‘In column’) and I’ll select the columns I want to use for each cause:

Now I can click OK in each dialog box to show the fishbone diagram, which looks just like the one we generated using the first method:

For one reason or another, the response variable in a regression analysis might not satisfy one or more of the assumptions of ordinary least squares regression. The residuals might follow a skewed distribution or the residuals might curve as the predictions increase. A common solution when problems arise with the assumptions of ordinary least squares regression is to transform the response variable so that the data do meet the assumptions. Minitab makes the transformation simple by including the Box-Cox button. Try it for yourself and see how easy it is!

The government in Queensland, Australia shares data about the number of complaints about its public transportation service. 

I’m going to use the data set titled “Patronage and Complaints.” I’ll analyze the data a bit more thoroughly later, but for now I want to focus on the transformation. The variables in this data set are the date, the number of passenger trips, the number of complaints about a frequent rider card, and the number of other customer complaints. I'm using the range of the data from the week ending July 7th, 2012 to December 22nd 2013.  I’m excluding the data for the last week of 2012 because ridership is so much lower compared to other weeks.

If you want to follow along, you can download my Minitab data sheet. If you don't already have it, you can download Minitab and use it free for 30 days. 

Let’s say that we want to use the number of complaints about the frequent rider card as the response variable. The number of other complaints and the date are the predictors. The resulting normal probability plot of the residuals shows an s-curve.

Because we see this pattern, we’d like to go ahead and do the Box-Cox transformation. Try this:

1. Choose Stat > Regression > Regression > Fit Regression Model.
2. In Responses, enter the column with the number of complaints on the go card.
3. In Continuous Predictors, enter the columns that contain the other customer complaints and the date.
4. Click Options.
5. Under Box-Cox transformation, select Optimal λ.
6. Click OK.
7. Click Graphs.
8. Select Individual plots and check Normal plot of residuals.
9. Click OK twice.

The probability plot that results is more linear, although it still shows outlying observations where the number of complaints in the response are very high or very low relative to the number of other complaints. You'll still want to check the other regression assumptions, such as homoscedasticity.

So there it is, everything that you need to know to use a Box-Cox transformation on the response in a regression model. Easy, right? Ready for some more? Check out more of the analysis steps that Minitab makes easy.

Do you recall my “putting the cart before the horse” analogy in part 1 of this blog series? The comparison is simple.

We all, at times, put the cart before the horse in relatively innocuous ways, such as eating your dessert before you’ve eaten your dinner, or deciding what to wear before you’ve been invited to the party. But performing some tasks in the wrong order, such as running a statistical analysis before you’ve prepared your data, might result in more serious consequences.

Eating your dessert first might merely spoil your appetite for dinner, but performing a statistical analysis on dirty data could have much more serious repercussions—including misleading results, mistaken decisions, or, if you’re lucky enough to catch your mistake before it's too late, costly rework.

Spending quality time with your data up front can prevent you from wasting time and energy on an analysis that either can’t work or can’t be trusted. We began exploring this idea in Part 1 of this best practices series, where I offered some tips for cleaning your data before you import it into Minitab. The biggest takeaway from Part 1 is that cleaning your data before you begin an analysis can save time by preventing rework, such as reformatting data or correcting data entry errors, after you’ve already begun the analysis.

So, once our data is clean, what comes next?

You can use Minitab’s worksheet visualization tools to explore your data. Conditional formatting in particular brings color to your worksheet and can be used to highlight aspects of your data that you’d like to call attention to quickly.

In our data set, recall that we’ve recorded the amount of time a machine was out of operation, the reason for the machine being down, the shift number during which the machine went down, and the speed of the machine when it went down. Suppose you wish to identify frequently occurring values or points that are out-of-spec or out-of-control. You can use formatting rules to do just that!

In this example, I’ve used one of the statistical rules available in Minitab’s conditional formatting to identify values that are not within spec. Highlighting these values may indicate either a data entry error or be valid cause for investigation, and can help you better understand where to focus your exploration and visualization efforts moving forward.

With a simple right-click directly in the Minitab worksheet, you can identify out-of-spec values you may wish to investigate before you begin your analysis.

You can also use Cell Properties (available by right-clicking in the worksheet) to highlight individual cells or rows, and add cell comments to draw attention to data that need further investigation, such as out-of-control points, unusual observations, or other data of interest. Rather than removing questionable data right away, you can take note of the data, perhaps by commenting on the cell as a reminder to follow-up. Doing this will keep you from committing the statistically unsound practice of cherry-picking data, and will ensure you handle the data correctly when it comes time to analyze it.

In the Minitab worksheet, you can highlight an entire row to easily visualize all variables associated with particular data, or add a cell comment to an out-of-control point for future reference.

Finally, data subsets are a good way to visualize only the data that is relevant to answering your questions. Minitab 17 makes it really easy to subset by right-clicking in the worksheet, and allows you to create subsets based on the data you’ve explored and highlighted with conditional formatting.

For example, suppose you want to understand why machines are experiencing downtime so you can address productivity problems. You can use conditional formatting to identify the most frequent reason for a machine’s downtime, and then subset your data based on those formatted rows to understand the relationship the most frequent cause of machine downtime has with other variables.

It’s easy to subset your data in Minitab by right-clicking directly within the worksheet.

All of the data cleaning and exploration you’ve seen in the worksheet is just the beginning—but consider how much insight you’ve drawn from your data before you’ve visualized it graphically or formally analyzed it!

Taking the time to clean and explore your data before you begin an analysis is well worth the investment. Doing so will help you better understand and answer key questions about your process, lead to a more efficient analysis as you tackle only the most relevant data for answering your questions, and ultimately yield results you can trust.