How many samples do you need to be “95% confident that at least 95%—or even 99%—of your product is good?

The answer depends on the type of response variable you are using, categorical or continuous. The type of response will dictate whether you 'll use:

1. Attribute Sampling: Determine the sample size for a categorical response that classifies each unit as Good or Bad (or, perhaps, In-spec or Out-of-spec).
    
2. Variables Sampling: Determine the sample size for a continuous measurement that follows a Normal distribution.

The attribute sampling approach is valid regardless of the underlying distribution of the data. The variables sampling approach has a strict normality assumption, but requires fewer samples.

In this blog post, I'll focus on the attribute approach.

A simple formula gives you the sample size required to make a 95% confidence statement about the probability an item will be in-spec when your sample of size n has zero defects.

, where the reliability is the probability of an in-spec item.

For a reliability of 0.95 or 95%,

For a reliability of 0.99 or 99%,  

Of course, if you don't feel like calculating this manually, you can use the Stat > Basic Statistics > 1 Proportion dialog box in Minitab to see the reliability levels for different sample sizes.  

These two sampling plans are really just C=0 Acceptance Sampling plans with an infinite lot size. The same sample sizes can be generated using Stat > Quality Tools > Acceptance Sampling by Attributes by:

1. Setting RQL at 5% for 95% reliability or 1% for 99% reliability.
2. Setting the Consumer’s Risk (β) at 0.05, which results in a 95% confidence level.
3. Setting AQL at an arbitrary value lower than the RQL, such as 0.1%.
4. Setting Producer’s Risk (α) at an arbitrary high value, such as 0.5 (note, α must be less than 1-β to run).

By changing RQL to 1%, the following C=0 plan can be obtained:

If you want to make the same confidence statements while allowing 1 or more defects in your sample, the sample size required will be larger. For example, allowing 1 defect in the sample will require a sample size of 93 for the 95% reliability statement. This is a C=1 sampling plan. It can be generated, in this case, by lowering the Producer’s risk to 0.05.

As you can see, the sample size for an acceptance number of 0 is much smaller—in this case, raising the acceptance number from 0 to 1 has raised the sample size from 59 to 93.

Check out this post for more information about acceptance sampling. 

If you have a process that isn’t meeting specifications, using the Monte Carlo simulation and optimization tools in Companion by Minitab can help. Here’s how you, as an engineer in the medical device industry, could use Companion to improve a packaging process and help ensure patient safety.

Your product line at AlphaGamma Medical Devices is shipped in heat-sealed packages with a minimum seal strength requirement of 13.5 Newtons per square millimeter (N/mm2). Meeting this specification is critical, because when a seal fails the product inside is no longer sterile and puts patients at risk.

Seal strength depends on the temperature of the sealing device and the sealing time. The relationship between the two factors is expressed by the model:

Seal Strength= 9.64 + 0.003*Temp + 4.0021*Time + 0.000145 Temp*Time

Currently, your packages are sealed at an average temperature of 120 degrees Celsius, with a standard deviation of 25.34. The mean sealing time is 1.5 seconds, with a standard deviation of 0.5. Both parameters follow a normal distribution.

To assess the process capability under the current conditions, you can enter the parameter, transfer function, and specification limits into Companion’s straightforward interface, verify that the model diagram matches your process, and then instantly simulate 50,000 package seals using your current input conditions.

The process performance measurement (Cpk) for your process is 0.42, far less than the minimum standard of 1.33. Under the current conditions, more than 10% of your seals will fail to meet specification.

Companion’s intuitive workflow guides you to the next step: optimizing your inputs.

You set the goal—in this case, minimizing the percent out of spec—and enter high and low values for your inputs. Companion does the rest.

After finding the optimal input settings in the ranges you specified, Companion presents the simulated results of the recommended process changes.

The simulation shows the new settings would reduce the amount of faulty seals to less than 1% with a Ppk of 0.78—an improvement, but still shy of the 1.33 Ppk standard.

To further improve the package sealing process, Companion then suggests that you perform a sensitivity analysis.

Companion’s unique graphic presentation of the sensitivity analysis provides you with insight into how the variation of your inputs influences seal strength.

The blue line representing time indicates that this input’s variability has more of an impact on percent of spec than temperature. The blue line also indicates how much of an impact you can expect to see: in this case, reducing the variability in time by 50% will reduce the percent out of spec to about 0 percent. Based on these results, you run another simulation to visualize the strength of your seals using the 50% variation reduction in time.

The simulation shows that reducing the variability will result in a Ppk of 1.55, with 0% of your seals out of spec, and you’ve just helped AlphaGamma Medical Devices bolster its reputation for excellent quality.

Figuring out how to improve a process is easier when you have the right tool to do it. With Monte Carlo simulation to assess process capability, Parameter Optimization to identify optimal settings, and Sensitivity Analysis to pinpoint exactly where to reduce variation,  Companion can help you get there.

To try the Monte Carlo simulation tool, as well as Companion's more than 100 other tools for executing and reporting quality projects, learn more and get the free 30-day trial version for you and your team at companionbyminitab,com.

There may not be a situation more perilous than being a character on Game of Thrones. Warden of the North, Hand of the King, and apparent protagonist of the entire series? Off with your head before the end of the first season! Last male heir of a royal bloodline? Here, have a pot of molten gold poured on your head! Invited to a wedding? Well, you probably know what happens at weddings in the show. 

So what do all these gruesome deaths have to do with statistics? They are data that come from a Poisson distribution.

Data from a Poisson distribution describe the number of times an even occurs in a finite observation space. For example, a Poisson distribution can describe the number of defects in the mechanical system of an airplane, the number of calls to a call center, or in our case it can describe the number of deaths in an episode of Game of Thrones.

If you're not certain whether your data follow a Poisson distribution, you can use Minitab Statistical Software to perform a goodness-of-fit test. If you don't already use Minitab and you'd like to follow along with this analysis, download the free 30-day trial.

I collected the number of deaths for each episode of Game of Thrones (as of this writing, 57 episodes have aired), and put them in a Minitab worksheet. Then I went to Stat > Basic Statistics > Goodness-of-Fit Test for Poisson to determine whether the data follow a Poisson distribution. You can get the data I used here. 

Before we interpret the p-value, we see that we have a problem. Three of the categories have an expected value less than 5. If the expected value for any category is less than 5, the results of the test may not be valid. To fix our problem, we can combine categories to achieve the minimum expected count. In fact, we see that Minitab actually already started doing this by combining all episodes with 7 or more deaths.

So we'll just continue by making the highest category 6 or more deaths, and the lowest category 1 or 0 deaths. To do this, I created a new column with the categories 1, 2, 3, 4, 5 and 6. Then I made a frequency column that contained the number of occurrences for each category. For example, the "1" category is a combination of episodes with 0 deaths and 1 death, so there were 15 occurrences. Then I ran the analysis again with the new categories.

Now that all of our categories have expected counts greater than 5, we can examine the p-value. If the p-value is less than the significance level (usually 0.05 works well), you can conclude that the data do not follow a Poisson distribution. But in this case the p-value is 0.228, which is greater than 0.05. Therefore, we cannot conclude that the data do not follow the Poisson distribution, and can continue with analyses that assume the data follow a Poisson distribution. 

When you have data that come from a Poisson distribution, you can use Stat > Basic Statistics > 1-Sample Poisson Rate to get a rate of occurrence and calculate a range of values that is likely to include the population rate of occurrence. We'll perform the analysis on our data.

The rate of occurrence tells us that on average there are about 3.2 deaths per episode on Game of Thrones. If our 57 episodes were a sample from a much larger population of Game of Thrones episodes, the confidence interval would tell us that we can be 95% confident that the population rate of deaths per episode is between 2.8 and 3.7.

The length of observation lets you specify a value to represent the rate of occurrence in a more useful form. For example, suppose instead of deaths per episode,  you want to determine the number of deaths per season. There are 10 episodes per season. So because an individual episode represents 1/10 of a season, 0.1 is the value we will use for the length of observation. 

With a different length of observation, we see that there are about 32 deaths per season with a confidence interval ranging from 28 to 37.

The last thing we'll do with our Poisson data is perform a regression analysis. In Minitab, go to Stat > Regression > Poisson Regression > Fit Poisson Model to perform a Poisson regression analysis. We'll look at whether we can use the episode number (1 through 10) to predict how many deaths there will be in that episode.

The first thing we'll look at is the p-value for the predictor (episode). The p-value is 0.042, which is less than 0.05, so we can conclude that there is a statistically significant association between the episode number and the number of deaths. However, the Deviance R-Squared value is only 18.14%, which means that the episode number explains only 18.14% of the variation in the number of deaths per episode. So while an association exists, it's not very strong. Even so, we can use the coefficients to determine how the episode number affects the number of deaths. 

The episode number was entered as a categorical variable, so the coefficients show how each episode number affects the number of deaths relative to episode number 1. A positive coefficient indicates that episode number is likely to have more deaths than episode 1. A negative coefficient indicates that episode number is likely to have fewer deaths than episode 1.

We see that the start of each season usually starts slow, as  7 of the 9 episode numbers have positive coefficients. Episodes 8, 9, and 10 have the highest coefficients, meaning relative to the first episode of the season they have the greatest number of deaths. So even though our model won't be great at predicting the exact number of deaths for each episode, it's clear that the show ends each season with a bang.

So, if you're a Game of Thrones viewer you should brace yourself, because death is coming. Or, as they would say in Essos:

Valar morghulis.

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

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

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

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

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

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

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

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

All processes have variation, some of which is inherent in the process, and isn't a reason for concern. But when processes show unusual variation, it may indicate a change or a "special cause" that requires your attention. 

Control charts are the primary tool quality practitioners use to detect special cause variation and distinguish it from natural, inherent process variation. These charts graph process data against an upper and a lower control limit. To put it simply, when a data point goes beyond the limits on the control chart, investigation is probably warranted.

It seems so straightforward. But veteran control chart users can tell you about “false alarms,” instances where data points went outside the control limits, and those limits fell close to the mean—even though the process was in statistical control.

The attribute charts, the traditional P and U control charts we use monitor defectives and defects, are particularly prone to false alarms due to a phenomenon known as overdispersion. That problem had been known for decades, until quality engineer David Laney solved it by devising P' and U' charts. 

The P' and U' charts avoid false alarms so only important process deviations are detected. In contrast to the traditional charts, which assume a defective or defect rate remains constant, P' and U' charts assume that no process has a truly constant rate, and accounts for that when calculating control limits.

That's why the P' and U' charts deliver a more reliable indication of whether the process is really in control, or not.

Minitab's control chart capabilities include P' and U' charts, and the software also includes a diagnostic tool that identifies situations where you need to use them. When you choose the right chart, you can be confident any special-cause variation you're observing truly exists.

When you have too much variation, or “overdispersion,” in your process data, false alarms can result—especially with data collected in large subgroups. The larger the subgroups, the narrower the control limits on a traditional P or U chart. But those artificially tight control limits can make points on a traditional P chart appear out of control, even if they aren't.

However, too little variation, or “underdispersion,” in your process data also can lead to problems. Underdispersion can result in artificially wide control limits on a traditional P chart or U chart. Under that scenario, some points that appear to be in control could well be ones you should be concerned about.

If your data is affected by overdispersion or underdispersion, you need to use a P' or U' chart will to reliably distinguish common-cause from special-cause variation. 

If you aren't sure whether or not your process data has over- or underdispersion, the P Chart or U Chart Diagnostic in Minitab can test it and tell you if you need to use a Laney P' or U' chart.

Choose Stat > Control Charts > Attributes Charts > P Chart Diagnostic or Stat > Control Charts > Attributes Charts > U Chart Diagnostic. 

The following dialog appears:

Enter the worksheet column that contains the number of defectives under "Variables." If all of your samples were collected using the same subgroup size, enter that number in Subgroup sizes. Alternatively, identify the appropriate column if your subgroup sizes varied.

Let’s run this test on the DefectiveRecords.MTW from Minitab's sample data sets. This data set features very large subgroups, each having an average of about 2,500 observations.

The diagnostic for the P chart gives the following output:

Below the plot, check out the ratio of observed to expected variation. If the ratio is greater than the 95% upper limit that appears below it, your data are affected by overdispersion. Underdispersion is a concern if the ratio is less than 60%. Either way, a Laney P' chart will be a more reliable option than the traditional P chart.

To make a P' chart, go to Stat > Control Charts > Attributes Charts > Laney P'.  Minitab will generate the following chart for the Defectives.MTW data. 

This P' chart shows a stable process with no out-of-control points. But create a traditional P chart with this data, and several of the subgroups appear to be out of control, thanks to the artificially narrow limits caused by the overdispersion.  

So why do these data points appear to be out of control on the P but not the P' chart? It’s the way each defines and calculates process variation. The Laney P' chart control limits account for the overdispersion when calculating the variation and eliminate these false alarms.

The Laney P' chart calculations include within-subgroup variation as well as the variation between subgroups to adjust for overdispersion or underdispersion.

If over- or underdispersion is not a problem, the P' chart compares to a traditional P chart. But the P' chart expands the control limits where overdispersion exists, ensuring that only important deviations are identified as out of control. And in the case of underdispersion, the P' chart calculations result in narrower control limits.

To learn more about the statistical foundation underlying the Laney P' and U' charts, read On the Charts: A Conversation with David Laney.

We had solar panels fitted on our property in 2011. Last year, we had a few problems with the equipment. It was shutting down at various times throughout the day, typically when it was very sunny, resulting in no electricity being generated.

Can you trust your data? 

That's the very first question we need to ask when we perform a statistical analysis. If the data's no good, it doesn't matter what statistical methods we employ, nor how much expertise we have in analyzing data. If we start with bad data, we'll end up with unreliable results. Garbage in, garbage out, as they say.

So, can you trust your data? Are you positive? Because, let's admit it, many of us forget to ask that question altogether, or respond too quickly and confidently.

You can’t just assume we have good data—you need to know you do. That may require a little bit more work up front, but the energy you spend getting good data will pay off in the form of better decisions and bigger improvements.

Here are 3 critical actions you can take to maximize your chance of getting data that will lead to correct conclusions. 

Failing to plan is a great way to get unreliable data. That’s because a solid plan is the key to successful data collection. Asking why you’re gathering data at the very start of a project will help you pinpoint the data you really need. A data collection plan should clarify:

• What data will be collected
• Who will collect it
• When it will be collected
• Where it will be collected
• How it will be collected

Answering these questions in advance will put you well on your way to getting meaningful data.

Many quality improvement projects require measurement data for factors like weight, diameter, or length and width. Not verifying the accuracy of your measurements practically guarantees that your data—and thus your results—are not reliable.

A branch of statistics called Measurement System Analysis lets you quickly assess and improve your measurement system so you can be sure you’re collecting data that is accurate and precise.

When gathering quantitative data, Gage Repeatability and Reproducibility (R&R) analysis confirms that instruments and operators are measuring parts consistently.

If you’re grading parts or identifying defects, an Attribute Agreement Analysis verifies that different
evaluators are making judgments consistent with each other and with established standards.

If you do not examine your measurement system, you’re much more likely to add variation and
inconsistency to your data that can wind up clouding your analysis.

As you collect data, be careful to avoid introducing unintended and unaccounted-for variables. These “lurking” variables can make even the most carefully collected data unreliable—and such hidden factors often are insidiously difficult to detect.

A well-known example involves World War II-era bombing runs. Analysis showed that accuracy increased when bombers encountered enemy fighters, confounding all expectations. But a key variable hadn’t been factored in: weather conditions. On cloudy days, accuracy was terrible
because the bombers couldn’t spot landmarks, and the enemy didn’t bother scrambling fighters.

Suppose that data for your company’s key product shows a much larger defect rate for items made by the second shift than items made by the first.

Given only this information, your boss might suggest a training program for the second shift, or perhaps even more drastic action.

But could something else be going on? Your raw materials come from three different suppliers.

What does the defect rate data look like if you include the supplier along with the shift?

Now you can see that defect rates for both shifts are higher when using supplier 2’s materials. Not
accounting for this confounding factor almost led to an expensive “solution” that probably would do little to reduce the overall defect rate.

Nobody sets out to waste time or sabotage their efforts by not collecting good data. But it’s all too easy to get problem data even when you’re being careful! When you collect data, be sure to spend
the little bit of time it takes to make sure your data is truly trustworthy. 

In April 2017, overbooking of flight seats hit the headlines when a United Airlines customer was dragged off a flight. A TED talk by Nina Klietsch gives a good, but simplistic explanation of why overbooking is so attractive to airlines.

Overbooking is not new to the airlines; these strategies were officially sanctioned by The American Civil Aeronautics Board in 1965, and since that time complex statistical models have been researched and developed to set the ticket pricing and overbooking strategies to deliver maximum revenue to the airlines.  

Six Sigma is a quality improvement method that businesses have used for decades—because it gets results. A Six Sigma project follows a clearly defined series of steps, and companies in every industry in every country around the world have used this method to resolve problems. Along the way, they've saved billions of dollars.

But Six Sigma relies heavily on statistics and data analysis, and many people new to quality improvement feel intimidated by the statistical aspects.

You needn't be intimidated. While it's true that data analysis is critical in improving quality, the majority of analyses in Six Sigma are not hard to understand, even if you’re not very knowledgeable about statistics.

Familiarizing yourself with these tools is a great place to start. This post briefly explains 5 statistical tools used in Six Sigma, what they do, and why they’re important.

The Pareto Chart stems from an idea called the Pareto Principle, which asserts that about 80% of outcomes result from 20% of the causes. It's easy to think of examples even in our personal lives. For instance, you may wear 20% of your clothes 80% of the time, or listen to 20% of the music in your library 80% of the time.

The Pareto chart helps you visualize how this principle applies to data you've collected. It is a specialized type of bar chart designed to distinguish the “critical few” causes from the “trivial many” enabling you to focus on the most important issues. For example, if you collect data about defect types each time one occurs, a Pareto chart reveals which types are most frequent, so you can focus energy on solving the most pressing problems. 

A histogram is a graphical snapshot of numeric, continuous data. Histo­grams enable you to quickly identify the center and spread of your data. It shows you where most of the data fall, as well as the minimum and maximum values. A histogram also reveals if your data are bell-shaped or not, and can help you find unusual data points and outliers that may need further investigation. 

Accurate measurements are critical. Would you want to weigh yourself with a scale you know is unre­liable? Would you keep using a thermometer that never shows the right temperature? If you can't measure a process accurately, you can't improve it, which is where Gage R&R comes in. This tool helps you determine if your continuous numeric measurements—such as weight, diameter, and pressure—are both repeatable and reproducible, both when the same person repeatedly measures the same part, and when different operators measure the same part.

Another tool for making sure you can trust your data is attribute agreement analysis. Where Gage R&R assesses the reliability and reproducibility of numeric measurements, attribute agree­ment analysis assess categorical assessments, such as Pass or Fail. This tool shows whether people rating these categories agree with a known standard, with other appraisers, and with themselves. 

Nearly every process has an acceptable lower and/or upper bound. For example, a supplier's parts can’t be too large or too small, wait times can’t extend beyond an acceptable threshold, fill weights need to exceed a specified minimum. Capability analysis shows you how well your process meets specifications and provides insight into how you can improve a poor process. Frequently cited capability metrics include Cpk, Ppk, defects per million opportunities (DPMO), and Sigma level. 

Six Sigma can bring significant benefits to any business, but reaping those benefits requires the collection and analysis of data so you can understand opportunities for improvement and make significant and sustainable changes.

The success of Six Sigma projects often depends on practitioners who are highly skilled experts in many fields, but not statistics. But with a basic understanding of the most commonly used Six Sigma statistics and easy-to-use statistical software, you can handle the statistical tasks associated with improving quality, and analyze your data with confidence. 

The Six Sigma quality improvement methodology has lasted for decades because it gets results. Companies in every country around the world, and in every industry, have used this logical, step-by-step method to improve the quality of their processes, products, and services. And they've saved billions of dollars along the way.

However, Six Sigma involves a good deal of statistics and data analysis, which makes many people uneasy. Individuals who are new to quality improvement often feel intimidated by the statistical aspects.

Don't be intimidated. Data analysis may be a critical component of improving quality, but the good news is that most of the analyses we use in Six Sigma aren't hard to understand, even if statistics isn't something you're comfortable with.

Just getting familiar with the tools used in Six Sigma is a good way to get started on your quality journey. In my last post, I offered a rundown of 5 tools that crop up in most Six Sigma projects. In this post, I'll review 5 more common statistical tools, and explain what they do and why they’re important in Six Sigma.

We use t-tests to compare the average of a sample to a target value, or to the average of another sample. For example, a company that sells beverages in 16-oz. containers can use a 1-sample t-test to determine if the production line’s average fill is on or off target. If you buy flavored syrup from two suppliers and want to determine if there’s a difference in the average volume of their respective shipments, you can use a 2-sample t-test to compare the two suppliers. 

Where t-tests compare a mean to a target, or two means to each other, ANOVA—which is short for Analysis of Variance—lets you compare more than two means. For example, ANOVA can show you if average production volumes across 3 shifts are equal. You can also use ANOVA to analyze means for more than 1 variable. For example, you can simultaneously compare the means for 3 shifts and the means for 2 manufacturing locations. 

Regression helps you determine whether there's a relationship between an output and one or more input factors. For instance, you can use regression to examine if there is a relationship between a company’s marketing expenditures and its sales revenue. When a relationship between the variables exists, you can use the regression equation to describe that relationship and predict future output values for given input values.

Regression and ANOVA are most often used for data that’s already been collected. In contrast, Design of Experiments (DOE) gives you an efficient strategy for collecting your data. It permits you to change or adjust multiple factors simultaneously to identify if relationships exist between inputs and outputs. Once you collect the data and identify the important inputs, you can then use DOE to determine the optimal settings for each factor. 

Every process has some natural, inherent variation, but a stable (and therefore predictable) process is a hallmark of quality products and services. It's important to know when a process goes beyond the normal, natural variation, because it can indicate a problem that needs to be resolved. A control chart distinguishes “special-cause” variation from acceptable, natural variation. These charts graph data over time and flag out-of-control data points, so you can detect unusual variability and take action when necessary. Control charts also help you ensure that you sustain process improvements into the future. 

Conclusion

Any organization can benefit from Six Sigma projects, and those benefits are based on data analysis.  However, many Six Sigma projects are completed by practitioners who are highly skilled, but not expert statisticians. But a basic understanding of common Six Sigma statistics, combined with easy-to-use statistical software, will let you handle these statistical tasks and analyze your data with confidence. 

by Matthew Barsalou, guest blogger

Once upon a time, in the Kingdom of Wetzlar, there was a farm with over a thousand chickens, two pigs, and a cow. The chickens were well treated, but a few rabble-rousers among them got the rest of the chickens worked up. These trouble-making chickens looked almost like the other chickens, but in fact they were evil chickens. 

By HerbertT - Eigenproduktion, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=962579

Hidden among the good chickens and the evil chickens was Sid. Sid was not like other chickens. He was a secret spy for The Swan of the Lahn, who ruled Wetzlar and was concerned about the infiltration of evil chickens. Sid was also a duck. That's right, a duck disguised as a chicken. Sid knew who the evil chickens were, and sent regular reports on their activities back to Wetzler.

Mallard drake by Bert de tilly

One stormy and dark night, an evil chicken snuck out with an enormous basket of beautiful hand-painted eggs to throw at the two pigs and the cow. Sid snuck out into the pouring rain and took a sample of 18 of the eggs. The intrepid duck spy was familiar with a previous study of 157 eggs, which showed that the mean of those eggs was 57.079 grams with a standard deviation of 2.30 grams. Sid was determined to find out if the mean of his current samples had a statistically significant difference from the mean of the previous study.

By Jan Kameníček - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=732984

If you'd like to recreate Sid's analysis, download his data set and, if you need it, the free trial of Minitab 18 Statistical Software. We will need to use summarized data since we only have actual values for the sample from the study and not the full data set. Go to Stat > Basic Statistics > Display Descriptive Statistics... and select the column containing the data as the Variable. Click on Graphs and select Individual value plot to view a graph of the data.

Click OK twice and Minitab will create an individual value plot of the data and the mean and standard deviation will appear in the session window with the rest of the descriptive statistics.

We can see that the sample mean is 57.315 and the standard deviation is 2.439 so now we can perform a 2 sample t-test to compare the means by going to Stat > Basic Statistics > 2-Sample t... and selecting Summarized data in the drop down menu. Enter the sample size of 18, sample mean of 57.315 and standard deviation of 2.439 under Sample 1 and enter the sample size of 157, mean of 57.079, and the population standard deviation of 2.30 under Sample 2.

Then click OK.

The p -value is greater than 0.05 so we can conclude there is no statistically significant difference between the means of the eggs the evil chickens planned to throw and the eggs in the previous study.

Unfortunately, Sid made a critical mistake. The first step in an analysis is to ask the right question. Sid's statistics were correct, but he asked the wrong question: “Is the mean of the second sample different from the mean of the first sample with an alpha of 0.05?” 

What he should have asked was, “What will happen when the pigs and the cow get hit by eggs?” The weight of the eggs was irrelevant; what mattered was the consequences of the pigs and cow being pummeled with eggs.

If Sid had prepared a report for The Swan of the Lahn that only said the eggs collected by the evil chickens weighed the same as eggs in the earlier study, the Swan would conclude that the process had not changed. But had the right question been answered, the correct conclusion would have been, “Trouble may be brewing.”

By Dick Daniels (http://carolinabirds.org/) - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=11053305

Trouble did indeed result when the evil chickens put their egg-throwing plan into action. As darkness fell, first the cow and then the pigs were bombarded by egg after messy egg.

The cow simply ate the eggs. But the pigs, holding all the chickens to be responsible, were outraged. They rampaged and terrorized the poor chickens all that night. By midnight, the muddy fields were full of pig prints and feathers were ruffled in the chicken coop. 

One of the evil chickens seized on the traumatized crowd's passions, and demanded of the others, “How can we live like this?" The evil chickens soon convinced the others that they would all be happier if they moved to the high-walled village of Wetzlar beside the Lahn River. The chickens began to march into the stormy night.

Continued in Part 2

About the Guest Blogger

by Matthew Barsalou, guest blogger

At the end of the first part of this story, a group of evil trouble-making chickens had convinced all of their fellow chickens to march on the walled city of Wetzlar, where, said the evil chickens, they all would be much happier than they were on the farm. 

By Krusto - self made by Krusto, CC BY 2.0 de, https://commons.wikimedia.org/w/index.php?curid=1712222

"No," said the Swan of the Lahn, the ruler of Wetzlar.

The chickens spent the day trying to force open the gates of Wetzlar. One chicken snuck off to meet with a goose known for dealing in antiques such as lamps, chairs, and main battle tanks. The chicken returned by early evening driving a slightly used T-55 tank.

By Sandstein - Own work, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=5069466

Sid, the undercover duck who'd infiltrated the flock to spy on the evil chickens for the Swan of the Lahn, realized he needed to do something, and fast. So he looked up the amount of fuel used for the distance driven for 47 T-55s, then performed a regression analysis to determine how far this one could go if it had full fuel tanks.

To recreate Sid's analysis in Minitab, download his data set (and the trial version of Minitab 18, if you need it) and go to Stat > Regression > Fit Regression Model..., and select Distance as the Response and Fuel as the Continuous predictor. Then click on Graphs and select Four in one.

Click OK twice, and Minitab produces the following output: 

We can see that there is a statistically significant relationship between fuel used and distance traveled. The R-squared statistics indicates the amount of fuel used explains 95.28% of the variability in distance traveled. There seems to be something odd with the order of the data, as seen in the Four in One graph. The Session window output shows three unusual values, two of which have large residuals. This is an indication that these data are not perfect for a regression analysis.

However, better data would not matter in this case, since Sid the Duck once again investigated the wrong question. Predicting how far the tank could travel was irrelevant, given it had already arrived at the town. The real question Sid should have asked was, “Can a T-55 round penetrate the gates of Wetzlar?”

By Peter Haas /, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=28496059

The chicken inside the tank huffed and puffed and fired the main gun directly at the gates of Wetzlar, but the round simply bounced off. He fired again and again, but the rounds just bounced off again and again. Eventually, the T-55 broke down—as they are known to do—so the chickens gathered in force and attempted to knock the gates down by running into them.

But a gate that can survive a tank’s main gun round will not budge when rammed by chickens, no matter how determined they are.

The Swan of Wetzlar had had enough by this time, so boiling chicken soup with noodles and vegetables was poured onto the chickens. This was too much for the chickens, so they fled.

Chicken Noodle Soup by Hoyabird8 at English Wikipedia

Unfortunately, the road they had followed had washed out in the heavy rains so the only route home was through the Bird Mountains...the terribly misnamed Bird Mountains, which could more accurately be called the Hungry Foxes Everywhere Mountains. 

Von Pulv - Eigenes Werk, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=15979924

The chickens fled into the forests of the Bird Mountains. Knowing the dangers of these forests, the evil chickens let the other chickens lead so that they would encounter the foxes first. However, the evil chickens failed to consider that foxes are, as they say, sly as foxes. The foxes of the woefully misnamed Bird Mountains waited till the chickens were well into their territory, and then fell upon those in the rear—the evil chickens.

Evil chickens nonetheless taste like chicken, and the foxes feasted.

By Foto: Jonn Leffmann, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=21536441

Upon returning with the flock to the farm, Sid suspected the evil chickens had been decimated, so he did a survey. Originally, 647 out of the population of 1,541 chickens were evil, so he randomly sampled 175 chickens and found only 22 of these chickens were evil. Sid wanted to know if the new proportion of evil chickens was less than the older portion, so so he did a one-tailed two proportion test.

To do this in Minitab, go to Stat > Basic Statistics > 2 Proportions... and select Summarized data in the drop down menu. Enter 22 for the number of events and 175 for the number of trials under Sample 1 and enter 647 for the number of events and 1,541 for the number of trials under Sample 2. Click on options and select Difference < hypothesized difference...

Then click OK twice. 

The resulting p-value is less than 0.05, so Sid can conclude there is a statically significant difference between the samples.

After returning home, the chickens were confused. Just days earlier all had been well, and suddenly they had found themselves in such an adventure. The remaining evil chickens weren't confused, since they had instigated it all. But, they were understandably upset with how things and ended and they began to argue and blame each other for the failure. Both recriminations and feathers flew.

Evil chickens started turning each other into the farmer which resulted in a weight gain for the farmer and an even greater reduction in the proportion of evil chickens in the flock. Sid surreptitiously arranged a few “accidents” to dispatch the remaining evil chickens.

The pigs eventually forgave the innocent chickens for the egg-throwing incident, and the remaining chickens lived happily ever after. The cow spent the rest of her life hoping for another dinner of eggs.

As for Sid, his next assignment was the infiltration of a rabbit den. How a duck disguised himself as a rabbit is a tale for another time.

By Brammers - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=7862100

There is a moral to this story: If you need help with statistics, call a statistician...not a duck.

About the Guest Blogger

In statistics, as in life, absolute certainty is rare. That's why statisticians often can't provide a result that is as specific as we might like; instead, they provide the results of an analysis as a range, within which the data suggest the true answer lies.

Most of us are familiar with "confidence intervals," but that's just of several different kinds of intervals we can use to characterize the results of an analysis. Sometimes, confidence intervals are not the best option. Let's look at the characteristics of some different types of intervals, and consider when and where they should be used. Specifically, we'll look at confidence intervals, prediction intervals, and tolerance intervals. 

A confidence interval refers to a range of values that is likely to contain the value of an unknown population parameter, such as the mean, based on data sampled from that population.

Collected randomly, two samples from a given population are unlikely to have identical confidence intervals. But if the population is sampled again and again, a certain percentage of those confidence intervals will contain the unknown population parameter. The percentage of these confidence intervals that contain this parameter is the confidence level of the interval.

Confidence intervals are most frequently used to express the population mean or standard deviation, but they also can be calculated for proportions, regression coefficients, occurrence rates (Poisson), and for the differences between populations in hypothesis tests.

If we measured the life of a random sample of light bulbs and Minitab calculates 1230 - 1265 hours as the 95% confidence interval, that means we can be 95% confident the mean for the population of bulbs falls between 1230 and 1265 hours.

In relation to the parameter of interest, confidence intervals only assess sampling error—the inherent error in estimating a population characteristic from a sample. Larger sample sizes will decrease the sampling error, and result in smaller (narrower) confidence intervals. If you could sample the entire population, the confidence interval would have a width of 0: there would be no sampling error, since you have obtained the actual parameter for the entire population! 

In addition, confidence intervals only provide information about the mean, standard deviation, or whatever your parameter of interest happens to be. It tells you nothing about how the individual values are distributed.

What does that mean in practical terms? It means that the confidence interval has some serious limitations. In this example, we can be 95% confident that the mean of the light bulbs will fall between 1230 and 1265 hours. But that 95% confidence interval does not indicate that 95% of the bulbs will fall in that range. To draw a conclusion like that requires a different type of interval...

A prediction interval is a confidence interval for predictions derived from linear and nonlinear regression models. There are two types of prediction intervals.

Given specified settings of the predictors in a model, the confidence interval of the prediction is a range likely to contain the mean response. Like regular confidence intervals, the confidence interval of the prediction represents a range for the mean, not the distribution of individual data points.

With respect to the light bulbs, we could test how different manufacturing techniques (Slow or Quick) and filaments (A or B) affect bulb life. After fitting a model, we can use statistical software to forecast the life of bulbs made using filament A under the Quick method.

If the confidence interval of the prediction is 1400–1450 hours, we can be 95% confident that the mean life for bulbs made under those conditions falls within that range. However, this interval doesn't tell us anything about how the lives of individual bulbs are distributed. 

A prediction interval is a range that is likely to contain the response value of an individual new observation under specified settings of your predictors.

If Minitab calculates a prediction interval of 1350–1500 hours for a bulb produced under the conditions described above, we can be 95% confident that the lifetime of a new bulb produced with those settings will fall within that range.

You'll note the prediction interval is wider than the confidence interval of the prediction. This will always be true, because additional uncertainty is involved when we want to predict a single response rather than a mean response.

A tolerance interval is a range likely to contain a defined proportion of a population. To calculate tolerance intervals, you must stipulate the proportion of the population and the desired confidence level—the probability that the named proportion is actually included in the interval. This is easier to understand when you look at an example.

To assess how long their bulbs last, the light bulb company samples 100 bulbs randomly and records how long they last in this worksheet.

To use this data to calculate tolerance intervals, go to Stat > Quality Tools > Tolerance Intervals in Minitab. (If you don't already have it, download the free 30-day trial of Minitab and follow along!) Under Data, choose Samples in columns. In the text box, enter Hours. Then click OK.  

The normality test indicates that these data follow the normal distribution, so we can use the Normal interval (1060 1435). The bulb company can be 95% confident that at least 95% of all bulbs will last between 1060 to 1435 hours. 

As we mentioned earlier, the width of a confidence interval depends entirely on sampling error. The closer the sample comes to including the entire population, the smaller the width of the confidence interval, until it approaches zero.

But a tolerance interval's width is based not only on sampling error, but also variance in the population. As the sample size approaches the entire population, the sampling error diminishes and the estimated percentiles approach the true population percentiles.

Minitab calculates the data values that correspond to the estimated 2.5th and 97.5th percentiles (97.5 - 2.5 = 95) to determine the interval in which 95% of the population falls. You can get more details about percentiles and population proportions here for more information about percentiles and population proportions.

Of course, because we are using a sample, the percentile estimates will have error. Since we can't say that a tolerance interval truly contains the specified proportion with 100% confidence, tolerance intervals have a confidence level, too.

Tolerance intervals are very useful when you want to predict a range of likely outcomes based on sampled data.

In quality improvement, practitioners generally require that a process output (such as the life of a light bulb) falls within spec limits. By comparing client requirements to tolerance limits that cover a specified proportion of the population, tolerance intervals can detect excessive variation. A tolerance interval wider than the client's requirements may indicate that product variation is too high.

Minitab statistical software makes obtaining these intervals easy, regardless of which one you need to use for your data.

Control charts take data about your process and plot it so you can distinguish between common-cause and special-cause variation. Knowing the difference is important because it permits you to address potential problems without over-controlling your process.  

Control charts are fantastic for assessing the stability of a process. Is the process mean unstable, too low, or too high? Is observed variability a natural part of the process, or could it be caused by specific sources? By answering these questions, control charts let you dedicate your actions to where you can make the most impact.

Assessing whether your process is stable is valuable in itself, but it is also a necessary first step in capability analysis. Your process has to be stable before you can measure its capability. You can predict the performance of a stable process and therefore improve its capability. If your process is unstable, by definition it is unpredictable.

Control charts are commonly applied to business processes, but they have great benefits beyond Six Sigma and statistical process control (SPC). In fact, control charts can reveal information that would otherwise be very difficult to uncover.

Let's consider processes beyond those we encounter in business. Instability and excessive variation can cause problems in many other kinds of processes. 

• A test process that causes subjects to experience an impact of 6 times their body weight.
• A teacher's process to help students learn. the material as measured by test scores.
• A diabetic's process for maintaining blood sugar levels.

The first example stems from a colleague's research. The researchers had middle-school students jump 30 times from 24-inch steps every other school day to see if it increased their bone density. Treatment was defined as the subjects experiencing an impact of 6 body weights, but the research team didn't quite hit the mark.

My colleague conducted a pilot study and graphed the results in an Xbar-S chart.

The fact that the S chart (on the bottom) is in control means each subject has a consistent landing style with impacts of a consistent magnitude—the variability is in control.

But the Xbar chart (at the top) is clearly out of control, indicating that even though the overall mean (6.141) exceeds the target, individual subjects have very different means. Some are consistently hard landers while others are consistently soft landers. The control chart suggests that the variability is not natural process variation (common cause) but rather due to differences among the participants (special cause variation).

The researchers addressed this by training the subjects how to land. They also had a nurse observe all future jumping sessions. These actions reduced the variability to the point that impacts were consistently greater than 6 body weights.

Control charts can verify that a process is stable, as required for capability analysis. But control charts can be used similarly to test assumptions for hypothesis tests.

Specifically, the measurements used in a hypothesis test are assumed to be stable, though this assumption is often overlooked. This assumption parallels the requirement for stability in capability analysis: if your measurements are not stable, inferences based on those measurements will not be reliable.

Let’s assume that we’re comparing test scores between group A and group B. We’ll use this data set to perform a 2-sample t-test as shown below.

The results indicate that group A has a higher mean and that the difference is statistically significant. We’re not assuming equal variances, so it's not a problem that Group B has a slightly higher standard deviation. We also have enough observations per group that normality is not a concern. Concluding that group A has a higher mean than group B seems safe. 

But wait a minute...let's look at each group in an I-MR chart. 

Group A's chart shows stable scores. But group B's chart indicates that the scores are unstable, with multiple out-of-control points and a clear negative trend. Even though these data satisfy the other assumptions, we can make a valid comparison between stable and an unstable groups! 

This is not the only type of problem you can detect with control charts. They also can test for a variety of patterns in your data, and for out-of-control variability.

An I-MR chart can assess process stability when your data don’t have subgroups. The XBar-S chart, the first one in this post, assesses process stability when your data does have subgroups.

Other control charts are ideal for other types of data. For example, the U Chart and Laney U’ Chart use the Poisson distribution. The P Chart and Laney P’ Chart use the binomial distribution. 

In Minitab Statistical Software, you can get step-by-step guidance in control chart selection by going to Assistant > Control Charts.  The Assistant will help you with everything from determining your data type, to ensuring it meets assumptions, to interpreting your results.

Maybe you're just getting started with analyzing data. Maybe you're reasonably knowledgeable about statistics, but it's been a long time since you did a particular analysis and you feel a little bit rusty. In either case, the Assistant menu in Minitab Statistical Software gives you an interactive guide from start to finish. It will help you choose the right tool quickly, analyze your data properly, and even interpret the results appropriately. 

One type of analysis many practitioners struggle with is multiple regression analysis, particularly an analysis that aims to optimize a response by finding the best levels for different variables. In this post, we'll use the Assistant to complete a multiple regression analysis and optimize the response.

In our example, we'll use a data set based on some solar energy research. Scientists found the position of focal points could be used to predict total heat flux. The goal of our analysis will be to use the Assistant to find the ideal position for these focal points. 

When you select Assistant > Regression in Minitab, the software presents you with an interactive decision tree. If you need more explanation about a decision point, just click on the diamonds to see detailed information and examples.

This data set has three X variables, or predictors, and we're looking to fit a model and optimize the response. For this goal, the tree leads to the Optimize Response button located at the bottom right. Clicking that button brings up a simple dialog box to complete.

HeatFlux is the response variable. The X variables are the focal points located in each direction, East, West, North, and South. Based on previous knowledge, we know we should use 234 as the target heat flux value of 234, but we could also ask the Assistant to maximize or minimize the response. Because we checked the box labeled "Fit 2-way interactions and quadratic terms," the Assistant also will check for curvature and interactions.

When we press "OK," the Assistant quickly generates a regression model for the X variables using stepwise regression. It presents the results in a series of reports written in plain, easy-to-follow language. 

This Summary Report delivers the "big picture" about the analysis and its results. With a p-value less than 0.001, this report shows that the regression model is statistically significant, with an R-squared value of 96.15%! The comments window shows which X variables the model includes: East, South, and North, as well as interaction terms. To model curvature, the model also includes several polynomial terms.

The Effects Report shows all of the interaction and main effects included in the model. The presence of curved lines indicates the Assistant used a polynomial term to fit a curve.

In this report, the East*South interaction is significant. This means the effect of one variable on heat flux varies based on the other variable. If South has a low setting (31.84), heat flux is reduced by increasing East. But if South is set high (40.55), the heat flux increases as East gets higher.

The Diagnostic Report shows you the plot of residuals versus fitted values, and indicates any unusual points that ought to be investigated. This report has flagged two points, but these are not necessarily problematic, since based on the criteria for large residuals we'd expect roughly 5% of the observations to be flagged. The report also identifies two points that had unusual X values; clicking the points reveals which worksheet row they are in.

The Model Building Report details how the Assistant arrived at the final regression model. It also contains the regression equation, identifies the variables that contribute the most information, and indicates whether the X variables are correlated. In this model, North contributes the most information. Even though East is not significant, since it is part of a higher-order term the Assistant includes it.

This is a good opportunity to point out how The Assistant helps ensure that an analysis is done in the best way. For example, the Assistant uses standardized X variables to create the regression model. That's because standardizing the X variables removes most of the correlation between linear and higher-order terms, which reduces the chance of adding these terms to your model if they aren't needed. However, the Assistant still displays the final model in natural (unstandardized) units.

The Assistant's Prediction and Optimization Report provides solutions for obtaining the targeted heat flux value of 234. The optimal settings for the focal points have been identified as East 37.82, South 31.84, and North 16.01. The model predicts that these settings will deliver a heat flux of 234, with a prediction interval of 216 to 252. But the Assistant provides alternate solutions you may want to consider, particularly in cases where specialized subject area expertise might be critical.

Finally, the Report Card prevents you from missing potential problems that could make your results unreliable. In this case, the report suggests collecting a larger sample and investigating the unusual residuals. It also shows that normality is not an issue for these data. Finally, it provides a helpful reminder to validate the model's optimal values by doing confirmation runs.

The Assistant's methods are based on established statistical practice, guidelines in the literature, and simulations performed by Minitab's statisticians. You can read the technical white paper for Multiple Regression in the Assistant if you would like all the details.

Overfitting a model is a real problem you need to beware of when performing regression analysis. An overfit model result in misleading regression coefficients, p-values, and R-squared statistics. Nobody wants that, so let's examine what overfit models are, and how to avoid falling into the overfitting trap.

Put simply, an overfit model is too complex for the data you're analyzing. Rather than reflecting the entire population, an overfit regression model is perfectly suited to the noise, anomalies, and random features of the specific sample you've collected. When that happens, the overfit model is unlikely to fit another random sample drawn from the same population, which would have its own quirks.

A good model should fit not just the sample you have, but any new samples you collect from the same population. 

For an example of the dangers of overfitting regression models, take a look at this fitted line plot:

Wildfires in California have killed at least 40 people and burned more than 217,000 acres in the past few weeks. Nearly 8,000 firefighters are trying to contain the blazes with the aid of more than 800 firetrucks, 70 helicopters and 30 planes.

In remote areas difficult to access by firetruck, smokejumpers may be needed to parachute in to fight the fires. But danger looms before a smokejumper even confronts a fire.

In statistics, we ask: “Can you trust your data?”

For a smokejumper, the critical initial question is: “Can you trust your parachute?”

When they’re not battling wildfires, many smokejumpers pursue advanced studies in fields like fire management, ecology, forestry, and engineering in the off-season.

At Washington Institute, smokejumpers and other students in the Technical Fire Management (TFM) program apply quantitative methods — often using Minitab Statistical Software — to evaluate alternative solutions to fire management problems.

“The students in this program are mid-career wildland firefighters who want to become fire managers, i.e., transition from a technical career path to a professional path in the federal government,” said Dr. Robert Loveless, a statistics instructor for TFM.

As part of the program, the students must complete a project in wildland fire management. One primary analysis tool for these projects is statistics.

“Many students have no, or a limited, background in any college-level coursework,” Loveless noted. "So, teaching stats can be a real challenge."

Minitab often helps students overcome that challenge.

“Most students find using Minitab to be easy and intuitive,” Dr. Loveless said. That helps them focus on their research objectives without getting lost in tedious calculations or a complex software interface.

For his TFM project, Steve Stroud used Minitab to evaluate the relationship between the age, the number of jumps used, and the permeability of a smokejumper’s parachute.

The permeability of a parachute is a key measure of its performance. Repeated use and handling cause the nylon fabric to degrade, increasing its permeability. If permeability becomes too high, the chute opens more slowly, the speed of descent increases, and the chute becomes less responsive to steering maneuvers.

Not things you want to happen when you’re skydiving over the hot zone of raging wildfire.

Stroud sampled 70 smokejumper parachutes and recorded their age, number of jumps, and the permeability of cells within each parachute.

Permeability is measured as the airflow through the fabric in cubic feet of air per one square foot per minute (CFM). For a new parachute, industry standards dictate that the CFM should be less than 3.0 CFM. The chute can be safely used until its average permeability exceeds 12.0 CFM, at which time it’s considered unsafe and should be removed from service.

Using the descriptive statistics command in Minitab, the study determined:

• Smokejumpers could be 99% confident that the mean permeability in unused parachutes (0-10 years old, with no jumps) was between 1.99 and 2.31 CFM, well within industry standards.
• Only one unused parachute, an outlier, had a cell with a CFM greater than 3.0 (3.11). Although never used in jumps, this parachute was 10 years old and had been packed and repacked repeatedly.
• For used parachutes (0-10 years old, with between 1-140 jumps), smokejumpers could be 99% confident that the mean permeability of the parachutes was between 4.23 and 4.61 CFM. The maximum value in the sample, 9.88, was also well below the upper limit of 12.0 CFM.

The service life for the smokejumper parachutes was 10 years at that time. However, this duration was based on a purchase schedule used by the U.S. military for a different type of parachute with different usage. Smokejumpers use a special rectangular parachute made of pressurized fabric airfoil.

Stroud wanted to determine a working service life appropriate for the expected use and wear of smokejumper chutes. Using Minitab’s regression analysis, he developed a model to predict permeability of smokejumper parachutes based on number of jumps and age (in years). (A logarithmic transformation was used to stabilize the unconstant variance shown by the Minitab residual plots.)

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

The regression equation is logPerm = 0.388 + 0.198 logJumps + 0.170 logAge

S = 0.196473    R-Sq = 76.6%    R-Sq(adj) = 76.4%

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

Both predictors, the number of jumps (log) and age of the parachute (log), were statistically significant (P = 0.000). The coefficient for LogJumps (0.19708) was greater than logAge (0.17021), indicating that number of jumps is a stronger predictor of the permeability of a parachute than its age. The R-Sq value indicates the model explains approximately 75% of the variation in parachute permeability.

Using the fitted model, the permeability of the chutes can be predicted for a given number of jumps and age. Based on 99% prediction intervals for new observations, the study concluded that the service life of chutes could be extended safely to 20 years and/or 300 jumps before the permeability of any single parachute cell reached an upper prediction limit of 12 CFM.

By adopting this extended service life, Stroud estimated they could save over $700,000 in budget costs over a period of 20 years, while still ensuring the safety of the chutes.

Stroud’s TFM student research project, completed in 2010, provided the impetus for further investigation and potential policy change in two federal agencies.

Photo credits: Smokejumper photos courtesy of Mike McMillan

Want to experience what it feels like to smokejump over a wildfire while remaining safely at your seat? Check out the incredible smokejumping video clips at http://www.spotfireimages.com/

On the heels of Healthcare Quality Week last week, we wanted to share our conversation with Dr. Sandy Fogel, the surgical quality officer at Carilion Clinic in Roanoke, VA.

Although he has been a practicing surgeon for nearly 40 years, Dr. Fogel’s enthusiasm for quality improvement makes it sound as if he is just getting started. After medical school at Washington University in St. Louis and surgical training at the University of Rochester School of Medicine, Fogel practiced surgery in Baltimore, at Johns Hopkins for 15 years and then at Greater Baltimore Medical Center for another decade.

While at Greater Baltimore, Fogel became involved with the National Surgical Quality Improvement Program (NSQIP), an outcomes-based program developed by the American College of Surgeons to measure and improve the quality of surgical care in hospitals. That involvement deepened in 2008, when he became the NSQIP Surgeon Champion at the Carilion Clinic in Roanoke, Virginia. Today, after a variety of projects that have transformed the clinic’s surgical experience, Fogel is a committed advocate for healthcare quality improvement—publishing papers and speaking about quality to healthcare organizations, all while continuing to work on projects at Carilion and practicing surgery himself.

How did you get involved in quality improvement?

Back in my residency, I promised myself that I would work to prevent unnecessary harm to patients by their surgeons — that I would be part of the solution. I was active in peer review, and in my career I’d been involved in hundreds of event reviews — including four major reviews that involved removing privileges or suspending licenses. I’ve had a longstanding interest in keeping the field of surgery properly evaluated, so when Greater Baltimore Medical Center adopted the National Surgical Quality Improvement Program in 2006, I was very intrigued. I became friends with a nurse who was responsible for NSQIP, and I became an ex-officio NSQIP surgeon champion. When I joined Carilion in 2008, the chair gave me 30 percent protected time to be the NSQIP surgeon champion, so it was not something I had to do as a sideline with two to five percent of my time — I could really work on it. So my role didn’t evolve as a result of my passion for quality improvement. It was the other way around — my passion for quality improvement has evolved as my role in NSQIP grew.

What kinds of projects have you been involved in as NSQIP surgeon champion?

When I arrived at Carilion Clinic, we decided to attack the problem of surgical site infection (SSI) first. When we looked NSQIP’s best practices to reduce SSI, there were many we were not doing at all. We were doing others, but not very well. So we started with a few small projects, which enabled us to take care of some low-hanging fruit at the same time we were developing a system to do more involved projects.

Our first project looked at normothermia — keeping patients warm. We collected some data and found that 35 percent of our patients came to the recovery room warm, with a temperature of 36.8 degrees Celsius or above. That meant 65 percent were below the temperature where the SSI rate starts going up. In our project to address normothermia we got that percentage up to over 90 percent of the patients coming into the recovery room warm. The literature also said that surgery patients should shower with an antiseptic soap before coming into the hospital, and they weren’t. We arranged to give patients a bottle of antiseptic for a shower before coming in.

Then we took on a much bigger project to improve post-operative glucose control in our critically ill patients. That was my trial-by-fire, where I really learned what implementing a quality improvement project involves. Carilion has eight intensive care units, and getting them all to agree to do this was very difficult. I had to gather large numbers of people in a room, define the barriers, find the brick walls we needed to go around, and figure out how to work the system and get people on my side. It took about a year and a half from the time I conceived the project to the point where it was properly up and running and having good compliance.

Tell us about the current project you’re working on.

It’s called enhanced recovery and it’s the largest single project I’ve done to date. We started in 2014 and it’s still rolling out and getting bigger. After an abdominal operation, it typically takes six to eight weeks before people get back to themselves. But with enhanced recovery protocols we can cut their hospital stay by several days, and their overall recovery by several weeks. It costs us about $500 per patient to employ the enhanced recovery protocol — but just the shorter length of stay in the hospital pays for itself many times over. These protocols are multifaceted, and they affect what takes place before surgery – in the hospital, in the operating room, in the recovery room, and even when patients go home.

These protocols completely change how I learned to treat a post-abdominal operation patient as a resident, and it’s difficult to get everyone on board and get people to learn all these things. Every single hospital that does surgery is talking about enhanced recovery, but less than 5% have been able to successfully pull it off.

What makes improving surgical quality so difficult?

Many hospitals know what to do, and why they’re doing it, and they even have someone who’ll pay for it — but actually implementing the process is very difficult, and that’s where hospitals fail.

They don’t do the legwork. They don’t have somebody who has the time, to be a surgeon champion pushing it. They don’t involve a senior nurse who knows the ropes of making a hospital work. And they don’t have someone who understands surgical quality improvement and how to implement it. So they’re missing one or more key links, because it’s very difficult to get other nurses and clinicians to buy into the idea of quality improvement. Surgeons get very good at what they do, and they don’t want to change. Deming said, “Every system is perfectly designed to get the results that it gets”. So if you have a system and you’re very good at your system, you will get consistent results. That doesn’t mean they’re quality results, though. As soon as we ask someone to change, what happens to efficiency and effectiveness and reliability? The outcomes change. Eventually it gets better, because you’ve now become good at your new system. But the fear of short-term reliability and efficiency problems are a barrier.

How do you overcome that resistance?

Persistence and data. Persistence and data. Persistence and data.

With the current project, we need to show patients who do have shorter stays, less mortality, lower pneumonia rates, etc. compared to those who do not go through the enhanced recovery protocol. It’s just much better care! But getting buy-in takes lots and lots of one-on-one conversations. It takes group conversations. E-mails. Posters. Letters. Talks. You can’t make it obnoxious, but over the course of months you’ve got to use every kind of tool you have, and keep it up. We’ve converted some non-believers. When we started this, there were probably only two of us on the surgical team who really believed this could work. Now it’s more like half the surgical team, and I’d say two-thirds of us are doing most of the components, so we’re getting there.

You talked about your passion for quality improvement. Where does that come from?

When I was in Baltimore, I operated on one patient at a time, and I made an impact in one person’s life at a time. But I’ve realized that I can do better than I did for 24 years in Baltimore. I’m not working with just one patient at a time now. The entire surgical experience at this hospital is improved.

Quality improvement is something we have to do. This isn’t a game we’re playing. It’s not just doing statistics. It’s not just collecting data. It’s not just doing projects — it really has meaning.

When we reduced our mortality rate, we calculated that it meant about 300 more patients left the hospital through the front door rather than through the morgue. Those are our spouses, parents, children, friends, and family members, who — instead of leaving in a hearse — are leaving in a wheelchair, getting in a car, and going home. Now that is making an impact.

Are you going to be a witch today? Batman? Jedi? You're not alone according to National Retail Federation statistics on top costumes and Halloween spending trends. Last-minute candy shopping? Check out kidzworld.com’s list of the top 10 candies for Halloween.

And of course, you have to plan your daily candy consumption to match the limits on free sugar recommended by the World Health Organization (WHO).

The guideline says that no more than 10 percent of your calories should come from free sugars and that you can achieve increased health benefits by keeping the number below five percent. If you’re a good nutrition tracker, that should be no problem for you. For those of you looking for more general suggestions, we’re going to make a scatterplot in Minitab that should provide a helpful reference.

The USDA last published their dietary guidelines in 2015. Appendix 2 contains calorie estimates based on age, gender and activity level, rounded to the nearest 200 calories. Multiply those levels by 0.05 to get an estimate of your recommended sugar limit in calories. To change that into grams that you can find on candy labels, we’ll assume that sugar has 4 calories per gram.

Now, if we create the default graph in Minitab we get something a bit like this. Note the symbols crammed together along each line:

Let’s be honest, pushing all those symbols together to show a line with no variation looks a bit silly. But select those symbols and a clearer graph is only a right-click away:

Without the symbols on the graph, the lines and the differences between them are clearer, especially when the lines are closest together during the early phase when people grow rapidly.

Much has been made of the fact that the five percent WHO guideline is less than the sugar in a can of soda, so Halloween can be a treacherous time for someone who wants to limit their sugar intake.

So what do you do if your ghost or goblin brings home more candy than you want? Natalie Silverstein offers some suggestions about how to make your candy do some good for others.

The image of mixed candy is by Steven Depolo and appears under this Creative Commons license.