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.

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.

As we start off 2018, our eyes are on the winter weather, specifically low temperatures and snowfall. After 2015-2016's warmest winter on record and Chicago breaking records in 2017 with no snow sticking to the ground in January or February, our luck might have run out. We shall see, though. The Old Farmer's Almanac is reporting that 2017-2018 winter temperatures will be colder than last winter.

If you live in the United States, you might know the winter of 2014-2015 was one for the record books. In fact, more than 90 inches of snow fell in Boston in the winter of 2015! Have you ever wondered how likely of an occurrence this was?

Dr. Diane Evans, Six Sigma Black Belt and professor of engineering management at Rose-Hulman Institute of Technology, and Thomas Foulkes, National Science Foundation Graduate Research Fellow in the electrical and computer engineering department at the University of Illinois at Urbana-Champaign, also wondered. They set out to explore the rarity of the 2015 Boston snowfall by examining University of Oklahoma meteorologist Sam Lillo’s estimate of the likelihood of this event occurring. Below I’ll outline some points from their article, A Statistical Analysis of Boston’s 2015 Record Snowfall.

Following this historic snowfall of 94.4 inches in a 30-day period in 2015 Lillo analyzed historical weather data from the Boston area from as far back as 1938 in order to determine the rarity of this event.

Lillo developed a simulated set of one million hypothetical Boston winters by sampling with replacement snowfall amounts gathered over 30-day periods. Eric Holthaus, a journalist with The Washington Post, reported that Lillo’s results indicated that winters like the 30 days of consecutive snowfall from January 24 to February 22, 2015 should “only occur approximately once every 26,315 years” in Boston:

To assess Lilo’s findings, Evans and Foulkes obtained snowfall amounts in a specified Boston location from 1891 to 2015 via the National Oceanic and Atmospheric Administration (NOAA) for comparison with his simulated data.

On March 15, 2015, the cumulative Boston snowfall of 108.6 inches surpassed the previous Boston record of 107.6 inches set in the winter of 1996. In the figure below, a graphical display of Boston snow statistics from 1938 to 2015 illustrates the quick rise in snowfall amounts in 2015 as compared to record setting snowfalls in years 1996, 1994, and 1948:

Also included in the figure is the annual average Boston snowfall through early June. The final tally on Boston’s brutal snowfall in 2015 clocked in at 110 inches!

The dashed rectangular region inserted in the graphic highlights the 30 days of snowfall from January 24 to February 22, 2015, which resulted in 94.4 inches of snow. In order to obtain hypothetical 30-day Boston snowfall amounts, Lillo first generated one million resampled winters by:

... stitching together days sampled from past winters. A three-day period was chosen, to represent the typical timescale of synoptic weather systems. In addition, to account for the effect of long-term pattern forcing, the random selection of 3-day periods was weighted by the correlation between consecutive periods. Anomalies tended to persist across multiple periods, such that there’s a better chance that a snowier than normal three days would follow a similarly snowy three days. This is well observed (and in extreme cases, like this year), so it’s important to include in the simulation.

After generating the one million resampled winters, Lillo recorded the snowiest 10-period stretches, i.e., 30 days, from each winter. Percentile ranges of the resampled distribution were compared to the distribution of observed winters to check the validity of the simulated data. In simulating the winters’ snowfalls in this manner, Lillo had to assume that consecutive winters and winter snow patterns within a particular year were independent and identically distributed (IID). Evans and Foulkes recognize that these assumptions are not necessarily valid.

Since they were unable to obtain Lillo’s simulated data and used actual historical data for their own Sigma Level calculations, Evans and Foulkes used a digitizer and the graphical display of Boston snow statistics above to simply create a “copy” of his data for further analysis.

Once they had data values for “Maximum 30-day snowfall (inches)” and “Number of winters,” they added them to Minitab and created histograms of the snowfall amounts with overlaid probability plots to offer reasonable distributions to fit the data:

For more on how they used Minitab distributions to fit the snowfall data and how they determined Sigma levels for the 2015 Boston snowfall using Lilo’s data, check out the full article here.