11.6 Control Charts: The Lane Guidance System of Process Management

If you have ever driven a modern car with advanced safety systems, then you’d know exactly what Lane Guidance System does: it monitors where your car is positioned in your lane at all times. Even if your car is not perfectly centered in the middle of the lane, minor drifts to your left or right would not set off the system. However, the moment your car gets a little too close to either side of the lane, possibly even crossing over without a turn signal, your Lane Guidance System would give you either an audible alarm or a gentle tug on your steering wheel to let you know that you might be accidentally drifting into another lane and risk either falling into a ditch or colliding with another car on the road.

Control charts work the same way! All the work you have put into the other phases of DMAIC have resulted in a process that should be capable of producing high quality products or providing exceptional customer service. In fact, when you learned capability index, you were shown that every process, whether production or service, has a target as well as variations away from the target. These two concepts, which can be gathered from randomly sampled data, form the basis of a control chart:

1. Target. Every process is designed to produce something with a specific trait. Every bottle of soft drink is labeled with a certain fluid ounce. Every bag of chips is labeled with a certain weight. Those are examples of targets, similar to how when you are driving, you should aim to stay perfectly middle in the lane. Your upper and lower acceptable limits are built around this target, similar to the lane markings.

2. Variability. Of course, no one ever stays perfectly middle in their lane when they drive. Natural variations tend to cause you to drift closer to the left or right of the center of your lane. The same thing happens with a process. In other words, sometimes a bottle of soft drink is either slightly overfilled or under-filled. With a large enough sample, you can estimate the statistical variation your process tends to yield in the long run, which in turn determines how far your upper and lower limits stray from your target.

3. Confidence. Unlike target and variability, confidence is something you choose. Being able to select a certain level of confidence is also the reason why we must take random samples of KPIV or KPOV. Typically, we want to have our upper and lower limits to cover a minimum of 3 standard deviations away from the target, which would allow us to cover 99.74% of all process variations. Anything above or below 3 standard deviations would be considered a process being “out of control,” in which case you will need to investigate potential causes and perhaps invoke your response plan.

For example, 24% of every bag of M&Ms should be blue. That is your target. Of course, natural variations of bag filling process might result in this proportion to slightly deviate from 24%. By taking random samples, you are able to obtain both the sample average and their standard deviation, with which you may then put together a control chart:

As seen above, samples 1 through 5 are all within three standard deviations of the target. Sample 6, however, falls below 21.7%. You may infer that something caused this particular sample to underfill with blue M&Ms, because it is not within the expected range of variation. Note, an observation falling outside the control limit simply provides you with a clue that something may not be going as planned. In other words, control charts do not tell you that something is definitely going wrong; they simply keep track of how well your process is performing relative to your target and specification limits, and to provide you with potential points of investigation to make sure that things continue to do well, much like a Lane Guidance System. In fact, you can put together control charts in three different ways according to how you measure your process. Let’s go over the calculations for each one.

Constructing a P-Chart

P-Charts are used to track targets that are based on percentages, or proportions of either a desirable or undesirable attribute. For instance, the above M&M example is a P-Chart, in which your target was 24% of a bag of M&Ms being blue. Other examples include percent of pizzas delivered on time, customers who report highly satisfied, hospital patients readmitted after release, and more. P-Chart is best used for attribute-based measures: An M&M is either blue or it is not; a customer is either highly satisfied or not; newly-released hospital patients are either readmitted or not. Here’s how to put together a P-Chart.

Step 1: Sampling Plan

There are several components to a sampling plan. First, you need to refer to your Control Plan to see what you are measuring and how you are measuring it. Is it through visual inspection? Is it through a precision instrument? Regardless of how you are taking your measurements, the “garbage in garbage out” principle applies: if you incorrectly measure your attribute, then any analytical outcome you derive from it cannot be trusted.

Second, you need to make sure that your sampling process is truly random. That is because random sampling allows you to make sure that your measurements aren’t being systematically biased. For example, if you determine that you would only take measurements on Mondays, then you are effectively capturing only what happens on Mondays and not any other days of the week.

Lastly, you need to determine the right sample size. In general, larger samples tend to be more trustworthy. However, depending on the overall level of difficulty and cost involved with measurement, you might be limited to a small sample. Tradeoffs there must be determined on a case-by-case basis. As a general good practice, each sample should be the same size.

Step 2: Calculating the Center Line

Next you will need to calculate your center line: the overall sample proportion of interest. To do this, you need to add together all your cases of interest (Side Effect) and divide the total by the total responses (p̄). In this case, 219/1000 gives us 21.9%. On average, 21.9% of vaccine recipients over the course of your study period reported side effects that interfered with their daily activities. However, are there specific samples that need a closer look?

Step 3: Calculating Variability Limits

To calculate variability, or the standard deviation of sample proportions:

Going back to our Lane Guidance System analogy. We have calculated what the center of the lane looks like (average sample proportion), our potential to drift out of lane (standard deviation). Next, we will need to establish the width of our lane by calculating the upper and lower control limits. For this calculation, we typically use z-value of 3.

Based on the control limits we established, any sample where more than 34.308% or less than 9.492% of trial participants reported significant side effects would be considered “out of control,” or crossing over the lane. In fact, if we graph this in an actual control chart, this is what it would look like:

As we can see from the above, all of our sample observations are well within the upper and lower control limits. However, that doesn’t mean that nothing is out of the ordinary! In fact, you can see that samples 6 to 10 all reported below-average proportions, suggesting that there was a prolonged period during which side effects from the vaccine were less problematic than earlier samples. Is this definitive? No. It does provide a starting point for investigation as to whether there was a cause.

As you learned previously, the nature of your data can be either attributes or continuous variables. While attributes can be tracked through P-charts, continuous variables can be tracked in two other charts: the x̄ and R-chart. Let’s briefly go over them with sample questions below:

Constructing a Mean (x̄) Chart

Now, let’s build a mean (x̄) chart. Consider the following scenario.

In this case, you will apply the same steps as laid out before. Having collected your sample, up next you will need to calculate your target and your variability. For x̄ chart, we need to first calculate the grand mean (x̿), which is simply the average of all sample averages. Next, variability is simply the standard deviation of sample averages (σ_x̅ ). To establish your control limits, use the following calculation:

Based on what you have calculated so far, the restaurant typically takes about 29.05 minutes to deliver an order. However, the restaurant’s delivery process appears to vary somewhat. 99.73% of their orders are in between 19.713 minutes and 48.763 minutes. Let’s plot all sample averages to our control chart to see how things look:

Based on the control chart, it appears that the process overall does not exhibit any systematic biases, maybe except for the last four samples that are below average.

Constructing a Range Chart

Based on the mean chart, it would appear that the restaurant manager has nothing to worry about, right? Yet, it doesn’t change the fact that customers are complaining! In fact, if you think about driving, for instance, if you tend to have a heavy foot and step on the gas pedal only to slow down later, you might not arrive at your destination much faster than if you simply kept a steady speed. The difference, though, is that the heavy-footed approach to driving ultimately consumes much more gasoline and inflicts higher wear on brake pads. This is why a range chart can help. Range is simply the difference between the highest and the lowest values in a sample. Going with the above example,

The Mean Range (R̅) again represents the middle of the lane. To establish the lane’s width, we now need to calculate the upper and lower limit using the following equations:

Where D₄ and D₃ can be obtained by looking up their values as predetermined in the table:

Plugging in the appropriate numbers (sample size = 4) would give us UCL = 2.28(15.5) = 35.34, LCL = 0(15.5) = 0. Plotting this out in to the range chart would give us

Now we begin to see part of the reason why the restaurant is experiencing more customer complaints! Samples 1 and 4 both have ranges that are above the upper control limit. What this means is that the restaurant manager will need to investigate these two samples to see what might be causing such high variability in the restaurant’s delivery process.

Wrapping Up Control Charts

As you can see from the three different types of control charts, they all follow the same principles of first identifying a center line across all randomly-collected samples: average sample proportion for P-chart, grand mean for X̅-chart, and average range for R-chart. Hopefully, the grand average is close to your target specification. For instance, the restaurant’s target delivery time was 30 minutes. The average delivery time across all samples was actually 29.05 minutes. Second, all three types of control charts require you to calculate some measure of variation. Typically this is calculated as simply the standard deviation across samples. Constructing the confidence interval, while typically uses the z-value of 3 for P-chart and X̅-chart, requires you to refer to a predetermined table of D values for R-chart.

The end results are all similar: you are constructing a horizontal lane to chart your sample values over time to spot patterns. More specifically, you are looking for these patterns below:

• A single point plots outside the control limits.

• Two out of three successive points are on the same side of the centerline and farther than 2 σ from it.

• Four out of five successive points are on the same side of the centerline and farther than 1 σ from it.

• A run of eight in a row are on the same side of the centerline.

• Any consistent or persistent pattern tells you that your process includes systematic variation.

Similarities aside, each chart also provides its unique perspective. How they should be used depends mostly on the process. For attributes, such as whether a patient reports serious side effects from a vaccine, P-chart allows you to keep track of the overall percentage of an event occurring as an outcome of a process. For continuous variables, X̅-Chart allows you to keep track of fluctuations in average outcome over time. Yet, as we saw above with the China House example, averages tend to mask serious variations in process outcomes, for which R-chart is the appropriate choice. It’s important to remember that a diversity of perspectives tend to portray a more accurate picture of how well your process is working, and how to keep it working as intended.