Using the phrase “voice of the process,” what is Figure 2 trying to tell us?
The single signal in these data is the red-circled point way above the upper limit, which was found to be a special order for a large cookout; hence, the detectably different, and higher, number of sales. While special orders are desired and great for business, the voice of the process tells us that it’s valid to consider them as nonroutine events. As such, it’s reasonable to discount this high sales value in the computation of the limits. Figure 4 shows an updated chart, noting that the high point is still displayed but was excluded from the calculation of the limits. (For those with a keen eye, you’ll notice that the updated limits in Figure 4 are a little narrower.) See a note on this in the postscript.
Our “noise filter” for these weekly averages is represented by the upper and lower control limits in Figure 10 of 61.94 to 68.09. We do find signals of detectably higher and lower resting heart rate: • Higher heart rate: Weeks 7, 15, and 39 in 2023 • Lower heart rate: Weeks 41, 42, and 43 in 2023
The learnings from Figure 5 offer different options for further exploration, one of which is to use process limits that are based exclusively on the “no inhaler” data (weeks 14–27). The chart based on this thought process is seen in Figure 6. (If a reader would like further guidance on the essential properties of an average and range chart, or details on how these limits are calculated, put a note in the comments to this article.)
One “watch out” for the Fitbit data is therefore the occurrence of many consecutive plot points having identical values, which brings into scope the problem of “chunky data.” An example of a simple way to create chunky data would be to measure different people’s height to the nearest yard. In doing so, we’d mistakenly think many people were the same height, e.g., two yards. The resolution here would be to measure height to the nearest inch.
The daily values—morning values at around 7 a.m.—are shown in Figure 11 in the postscript. To allow for fair and valid comparison, the values from Week 28 onward came from measurements before use of the inhaler.
Some years back, a patient was prescribed an inhaler due to asthma-related breathing difficulties. In addition to a confirmed cat allergy, another indication came from concerningly low peak-flow values in the range 300–350 L/min. (Peak flow data indicate how open the airways in the lungs are, with values about 600 L/min expected for men.)
For the signals in weeks 7 and 15, some explanations were put forward—a holiday in Week 15, which is something nonroutine—but explanations as definitive as Covid-19 or a heart attack weren’t forthcoming. One weakness in the data around the time of the signals in weeks 7 and 15 was missing data. Another weakness was the long time lag—trying to find causes for events that happened close to one year earlier—which helps to explain the first rule of thumb mentioned above.
In this article we’ve illustrated and discussed the relevance of SPC outside of its more well-known uses in manufacturing. While our four examples touched on planning and healthcare, things don’t stop there. If you have data, and you want to learn from those data and apply those learnings to improve your processes, why not give SPC a go? Please share your thoughts and questions in the comments section.
Postscript
We can’t predict an exact number of sales per day but rather a range of possible sales. With a view to estimating this range, daily sales for close to one and one-half months were plotted on a control chart of individual values, as seen in Figure 3.
Listening to the voice of the process provides a lot more than “statistics,” which to some might seem of little consequence when it comes to what really matters in business. This voice provides insight to enable smarter work, not harder work.
Importantly—and this is valid only because predictable behavior is demonstrated—the average number of daily sales can be trusted and relied upon (at least until a signal of unpredictability, or a lack of consistency, is found).
Important context is that we have close to 500 data values and are looking back over one and one-half years. With so many data, we decided to group them by calendar week and use the weekly averages as individual values. The control chart using this approach is shown below in Figure 10. If we’d had less data, it’s unlikely we’d have taken this route, which again illustrates the importance of context and the need for careful thought.
Two rules of thumb to consider when it comes to following up on control chart signals are: • Start with the most recent signals first (because events are fresher in the mind). • Focus on the biggest signals first (because they tend to offer the greatest payback).
Figure 5: Average and range chart of the peak flow data
Figure 4: Control chart of the daily sales data (limits computed without the point above the upper limit)
If two people are assigned to handle support requests, we see that most of the time—at least two in every three weeks, which is Part 1 of the empirical rule—things ought to run smoothly with eight or fewer requests expected. Thus, high-quality customer support is seemingly ensured.
Notes: • Another reason why so much was being made when Scott got involved was because the manager liked to make one prediction in the morning, make the product, clean the equipment, and that was it. • Question: Similar to the situation described in Example 1, how would you go about deciding whether “things stay the same” during the weeks and months ahead? Also, why were the limits narrower in Figure 4?
Example 3: Asthma management
Following success with an inhaler, as well as avoiding cats and other risk factors, it was decided a couple of years later to run an inhaler-free period while continuing to measure peak flow daily. Without daily use of an inhaler, no specific symptoms of deterioration were felt (and cats were avoided). The only indicator of a potential worsening of the situation was a little drop in the peak flow data—the values were a little lower than a year earlier. In consultation with the doctor, it was decided to restart use of an inhaler to decide if, long-term, an inhaler was appropriate. A follow-up appointment was arranged for two to three months later. In the meantime, daily peak values continued to be collected.
Referring again to the “voice of the process,” what are these data trying to tell us? Excluding the special order, this process, or system, of daily sales displays predictable behavior, meaning that, unless things change, we can plan for: • Daily sales to average about 57 items • Routine sales for any single day to be anywhere in the range 12 to 101
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Published: Wednesday, January 24, 2024 – 12:03
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What can be learned by listening to the voice of the process?
By characterizing the system as predictable up to Week 50, we learn that any single week having up to, and including, 14 requests is “normal.” This insight debunks the idea that two people can satisfactorily handle all requests in each and every week of the year (assuming, as stated above, that one person can be expected to successfully manage up to four or five requests in a given week).
Figure 9: Illustration of successive values being equal
Here in Part 7 we move away from manufacturing and discuss SPC’s continued relevance, and potential, in areas such as planning and healthcare. In the examples that follow, we also aim to reinforce the importance of three key elements inherent in SPC: 1. Aim: What do you want to achieve? (Which questions should the data be helping you to answer?) 2. Context: You need to know what your data represent (i.e., when collected, how collected, conditions when collected, what might have influenced the results you got). 3. Thought: How to organize, use, and analyze the data to extract the needed insight and maximum information from them.
Example 1: Resource planning
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The high point at the right of the chart (Week 39) was the beginning of a Covid-19 illness. This ties in with what we were told: “Interestingly, according to a recent paper, resting heart rate tends to increase at the beginning of Covid, then drops down low, then eventually returns to normal.” Although weeks 40 and 41 are consistent with this hypothesis, we didn’t get the chance to see if resting heart rate could return to normal because the patient had a small heart attack, which coincides with the last two points on the chart (weeks 42 and 43).
As Scott told us, the workers had no idea of how many packages to produce daily. One of the workers who’d been there for 30 years routinely made more than 300 packages per day, yet, as we see in Figure 4, we can scarcely expect to sell more than 100 packages on one day unless something out of the ordinary happens (such as a special order, which is something nonroutine to specially plan for).
We got these data by email and were asked to look at them as described: “…I wonder if you could look at some data (mine), just for the heck of it, and see if you can detect any signals.”
Questions: How would you know if the number of weekly requests remains the same in the coming year? And why is the lower limit in Figure 2 set at zero? See our answers in the postscript.
Example 2: How many units?
You take over a new group and would like to plan your team’s resources for the next year to most effectively respond to technical support requests. How many of your group’s resources should be assigned weekly to ensure high-quality support? If too few people are assigned, you likely won’t fulfill the customers’ support needs. If too many people are assigned, you risk leaving some resources sitting idle.
Health Care
SPC Outside of Manufacturing
Part 7 of our series on statistical process control in the digital era
Figure 1: Support request data per week
Having demonstrated that there is a difference when using an inhaler, we can estimate how big this is. The average increase in peak flow of 29 L/min is easily calculated and can also be shown visually in a histogram (Figure 7):
Figure 10: Control chart of the heart rate data using weekly averages as individual values
Range chart: • Behavior? Consistent and predictable • Signals? None—no signals on the range chart
We now look at Fitbit data—a measurement of resting heart rate in beats per minute (bpm)—to see how such data can be effectively analyzed with an SPC “way of thinking.” Each value represents the average resting heart rate per day. The data cover roughly one and one-half years (May 2022 to October 2023). Of 527 days of potential measurement, there are 48 missing values; the reason for this is that the smart watch that provides the daily Fitbit data wasn’t worn on these days.
Having listened to the voice of the process, Scott and the team opted to go close to the average, with 60 packages per day as the “nominal target.” Depending on the stock level at the start of a new day, more or less units would be made to hit this nominal target.
To finalize a resource plan for the next year, we can use the empirical rule as explained by Donald J. Wheeler. With an average of close to 5 and a standard deviation of 3, we get the data below. (In Figure 2, the central line, which is the data average, is of value 5.0, and the 3-sigma distance is 14.12 – 5.0 = 9.12; hence, standard deviation = 9.12/3 ~ 3.)
Although some data have an obvious way of being organized for a control chart, this isn’t always the case. Hence, a good idea is to start with a time-series plot of the data for a first interpretation (see Figure 8).
Occasionally, however, things could get stretched—see the upper intervals for parts 2 and 3 of the empirical rule with values at 11 and 14, respectively—and customer support could fall below its usual high quality. This might happen something like two to three weeks per year. The “voice of the process” tells us that these “high” weeks must be anticipated, but when in the year isn’t foreseeable (i.e., random variation).
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Returning to the request we got to try to detect signals in the data, we indeed detected them and have again shown that control chart signals tell you about events, or changes, in your process that are worth knowing about. We’ve also highlighted an important “watch out”—chunky data. And, we’ve demonstrated that effective control charting can’t be automated because of the data-specific issues discussed above.
Figure 7: Histograms of the with- and without-inhaler peak flow data
Figure 8: Time-series plot of the average resting heart rate data
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When the data are grouped by week, each subgroup consists of seven values. The traditional chart to use in this case is the average and range chart, which for the peak flow data, is shown in Figure 5. On the upper chart we find the weekly averages, and on the lower chart the weekly range (range = highest value – lowest value). The red limits bracket the range of anticipated “routine variation.”
Figure 1 adds more insight to the “about five requests per week” because the occurrence of requests as high as seven and eight during weeks 2–11 tells you straightaway that one person per week is unable to satisfy your customers’ needs and expectations.
We also see that, about half the time, weeks will be “easy” with five or fewer requests to be expected.
Example 2: • As with Example 1, to examine whether things stay the same or change, we’d collect new data and monitor them against the limits for expected routine variation found in Figure 4. • The process limits in Figure 4 were narrower than those in Figure 3 because, for Figure 3’s limits, the high value—the special order—was associated with two high moving range values in the computation of the process standard deviation (or sigma). If further details are needed, put a note in the comments.
Moving into the next year, the agreed plan became:
Figure 11: Peak flow data
In a first discussion you’re told 1) to expect about five requests per week; and 2) that one person can manage up to four or five requests in a given week. Naturally, you wonder whether one person on duty per week might be sufficient. Nonetheless, you decide to take a look at some data before moving toward any final decision. You request data from the last calendar year, which are shown in Figure 1. (No requests were handled in weeks 1 and 52.)
In Figure 8, we see a kind of “wavy” pattern—also indicative of autocorrelation for those familiar with the term, and for these data the lag one autocorrelation value is 0.76—because there are many cases of successive values being equal. We illustrate this in Figure 9 using a subset of the data; the highlighted points are those that have the same value as the previous point (25 of the 75 points are highlighted).
For the Fitbit data, one option to explore would be to express the average heart rate to a first decimal and not to a whole number (as we see in Figures 8 and 9).
Waste was thought to come primarily from having regularly made too many units. How could the company create a better match between what was made and what was sold? Not all excess units would be thrown away, but for those that could be sold, the packages would have to be rewrapped, reduced in price by 20% or more, and frozen. Moreover, manpower would be needed—another source of waste—for all this extra handling.
Figure 2: Control chart of the weekly request data
While the voice of the process tells us that up to 101 units can be needed on a given “normal day,” the approach taken by Scott prefers running low on items and perhaps on the occasional day in the year even running out (see Part 3 of the empirical rule discussed in Example 1). Scott’s safeguard was a guy on duty until closing who had access to a small, easy-to-clean grinder to make the odd unit as needed.
Finally, we’re not advocating that future data should continue to be grouped and analyzed as weekly averages. This takes us back to Part 3, where we wrote “…sampling plans are meant to evolve.” The thought process here should consider at least the following point: How quickly do signals—changes in resting heart rate—need to be discovered? If waiting a week is too long, a change in the way of organizing and looking at the data would have to be considered.
Wrap-up
Where we struggle with a time-series plot is when filtering out the noise in the data to detect the signals. Control charts do this, and that is why they are so valuable. A signal in the Fitbit data means we find a detectably higher or lower heart rate that has a “findable cause.” In Figure 8, yes, we do see fluctuations in the data, but aside from the reasonably obvious downward trend at the very end of the record, when can we confidently state that resting heart rate is differently “high” and/or “low”?
Following some preliminary analysis, including plots of the data using the individual values, it was decided to group the data by calendar week. (In Figure 11, the first column is calendar week.) In SPC, this takes us into the topic of rational subgrouping which “…has to do with organizing your data so that the chart will answer the important questions regarding your process.”
Figure 6: Annotated average and range chart with limits based exclusively on the “no inhaler” data
In this example we’ve seen that SPC techniques can play a useful role in healthcare, contributing to better decisions while also involving the patient in the treatment process. For those wishing to learn more, one of many examples that discuss SPC techniques in healthcare is a paper by Boggs, P. B. et al.,“Using Statistical Process Control Charts for the Continual Improvement of Asthma Care in the Journal on Quality Improvement (Vol. 25(4), pp. 163–181, 1999).
Example 4: Fitbit data
Figure 3: Control chart of the daily sales data
Example 3: Peak flow data: Weeks 14 to 27 show peak flow without use of an inhaler. Weeks 28 and on show peak flow with the daily use of an inhaler.
Figure 6 leaves no doubt: When using an inhaler, the result is higher peak flow values. This means that the airways in the lungs are open to a greater, and better, extent. Note that the “inhaler” data—right side of Figure 6—display predictable behavior (chart not shown), meaning that the upward shift was consistently sustained over weeks 28 to 36.
This worker may indeed have worked hard, but for what benefit? Most of his work would end up being reworked! Thanks to charts like Figures 3 or 4, we could explain to the worker what makes most sense—i.e., producing fewer items—and discuss alternative tasks that might be needed to satisfy his hardworking mentality.
Aside from Week 51, the year can be characterized as predictable due to the consistency displayed by the data (i.e., there are no outliers, runs, trends, or shifts that characterize inconsistent behavior).
So far in this series our focus has remained on statistical process control (SPC) in manufacturing. We’ve alternated between more traditional uses of SPC that remain relevant in this digital era and discussing uses of SPC and its related techniques that are enabled by the marvels of modern technology.
Next, you plot the data in time order on a control chart for individual values—also known as a process behavior chart—to gain yet more insight; more insight = better decisions. (See Figure 2.) You do this to examine: 1. Process behavior: Do the data indicate predictability, or consistency, in the weekly requests? 2. Range of variation: How many requests can be expected under “routine” conditions?
A key question was: Is there a difference in peak flow when using the inhaler? As with most data, these can be looked at and analyzed in different ways (for example, as individual daily values or grouped by week). The thinking process alluded to at the start of this article focused on finding the simplest way to best answer the question of interest, with emphasis on an effective communication of the outcome. This points us in the direction of a “good graph.”
This example is courtesy of Allen Scott, and it uses data on the daily sales of a meat product. Scott’s focus was on daily planning, i.e., How many units are needed on a daily basis? The aim was to plan better to minimize waste.
Our questions
• There is one, and only one, signal in these 50 data, which is the count of 15 requests from week 51, the final working week of the year. • The data themselves place an upper limit on “routine requests” at 14 (see the value of 14.12 for the upper limit, labeled UCL, which stands for upper control limit). • This limit of 14 comes exclusively from the data and is an estimate of the highest number of requests to expect, or plan for, in a single week when things are “normal.” • If, in any single week, 15 or more requests come in, the data give you a signal of something different having happened. • The week 51 signal means that a cause, or reason, to explain this point above the limit should be identifiable: You learn that the last week of the year is said to be different because, for the customers who make the requests, it represents the final opportunity to close out all requests in the calendar year to meet deadlines and objectives.
Average chart: • Behavior? Definitely not predictable • Signals? Yes, the points out of the limits at weeks 14, 15, 29, 32 and 34 • Dig deeper and investigate? Yes, seek to identify when the process change/s occurred. • When? A big, and sustained, upward shift occurs between weeks 27 and 28, which is the transition from no inhaler to daily use of an inhaler (and use of an inhaler is expected to increase peak flow).
We also discussed and illustrated chunky-like data in Part 3 of our series when looking at high-frequency data. We explored the option of averaging values over 15-minute intervals and then control-charting these averages as individual values.
Why is the lower limit in Figure 2 set at zero? The technically correct lower limit is –4.1, but the number of requests in a given week can’t be negative.
Example 1: We asked, How would you know if the number of weekly requests remains the same in the coming year? We’d plot the count of weekly requests in the coming year on a control chart having the same limits as Figure 2. We’d check for a lack of consistency vs. the range of routine variation, which is counts between 0 and 14 and an average of 5. If a lack of consistency were found, the “resource plan” might need to change.
