Employing the Rules of Statistics – Not Mathematics – Will Provide More Accurate Metric Measurement in Emergency 
Medicine
By Phillip Stephens, DHSc, PA-C
“Math” and “fun” rarely appear in the same sentence, and bridging math and medicine is especially challenging. With that said, understanding numeracy is essential to emergency medicine. So let’s attempt to have some fun with math to understand these challenges and see if we can bridge the gap.
Do you ever wonder why the numbers that guide emergency medicine practice never seem to equate to reality? After all, we need emergency department metrics to guide our practice. Numbers are assigned to productivity, length of stay and wait times to measure and accomplish important goals. These models are a good thing.
But emergency physicians, managers and hospital administrators intuitively feel something isn’t quite right. We are continually chasing numbers but never seem to master many.
Other industries have mastery over their metrics. NASCAR has precise measures of fuel usage, tire wear and even track temperature that create accurate projections down to the millisecond. The metrics have utility. From NASA to restaurants, there are dozens of industries that maximize mathematical potential with precision. Why can’t emergency medicine?
Mathematical and critical thought is a beautiful thing, yet it’s often quite deceptive in practice. There are times in many industries when the numbers can’t be relied upon, especially if used incorrectly. If there is anything worse than no data, it’s bad data, and that is the very issue.
Harvard professor of biology Stephen Jay Gould alluded to this problem by pointing out that our culture tends to ignore variation. Instead, we focus on central tendency. The result is that if you plan based on average assumptions, on average you will be wrong. Let’s do a simple math problem to illustrate this based on Stanford University’s Sam Savage’s attempt to describe Gould’s observation. It’s fun and illuminating. And math.
According to Savage’s model, let’s say you and your wife work in different parts of the city. The commute from work for each of you is 30 minutes. You both leave work at 5 p.m., meaning you should both arrive home at the same time of 5:30 p.m.
On one particular evening, you both must attend a reception at 6 p.m. The reception is 30 minutes from your home in the opposite direction of your places of employment. The plan for this evening is for each of you to leave work at 5, meet at home at 5:30, and drive together to the reception, arriving at 6.You figure that mathematically, it works out. But intuitively something doesn’t seem quite right. Will you be late? Or more realistically, do you have any chance at all of being on time?
Although mathematical concepts are used in statistics, there is a difference between the two. We won’t examine the differences in detail so you’ll have to trust me on that one. The point is that when viewed mathematically, the trip certainly should work. However, statistically it won’t. Here’s why:
Savage’s classic model assumes you have a 50/50 chance that each of you will make it home by 5:30. That way each trip home is like a statistical coin toss where heads equals arriving by 5:30, and tails equals arriving after 5:30. Four combinations are possible about the trip home for the two of you (a coin toss for you and a coin toss for your spouse on whether you make it on time):
- Heads / Tails = You are home by 5:30, but your spouse isn’t.
- Tails / Heads = Your spouse is home by 5:30, but you aren’t.
- Heads / Heads = Both of you are home by 5:30
- Tails / Tails = Neither of you makes it home by 5:30 .
The only way you both arrive home by 5:30 p.m. is if you both flip heads. Your chance of arriving on time at the reception is not 100 percent just because the commutes work out mathematically. In fact, it isn’t even 50 percent. Statistically, there is only one chance in four that you both arrive at home on time. Mathematics deals with numbers, relationships and patterns. Statistics is systematic and analyzes data. There is a distinct difference in the limitations and utility of each.
Now suppose grandma is going to ride with you and must drive 30 minutes to meet at your home by 5:30 as well. Your chances of leaving on time just dropped to one in eight. Then imagine you are going to take a van with a half dozen friends who are going to join you, and each of them has a trip averaging 30 minutes. That is now like flipping heads 9 times in a row. You have dropped the odds of leaving on time to less than 1 in 500–and once you leave, you still have only a 50/50 chance that your drive to the reception will take only 30 minutes!
Doing simple math correctly resulted in an expected value. But professors like Gould and Savage understand this is a mistake. It’s a mistake that permeates business activities — including medicine. Savage calls it the Flaw of Averages.
Cartoonist Jeff Danziger illustrates the concept by depicting a man who drowns crossing a stream that is supposed to be, on average, only 3-feet deep. In actuality, the stream is 1-foot deep with a 6-foot hole mid-stream. Mathematically, it is on average only 3 feet deep. Now you see the problem. Let’s apply these concepts to emergency medicine flow now that we have a bit of understanding regarding the power of statistical analysis.
Consider a hypothetical meeting of representatives from registration, nursing, triage, radiology, lab, ancillary services such as ECG, blood gas, IV teams and physicians who sit to work on a project. It’s a single flow project, but up to a dozen subroutines or processes must be developed in parallel. It could be patient flow, throughput or anything requiring parallel tasks.
The leader asks each participant how long it will take each to conduct his or her part. Each provides a range, like 10 to 20 minutes to complete their individual function. But a range will not suffice, as the leader needs a specific goal. Math is a precise thing and so are management functions. So each provides the average or reasonable time it takes to complete their task.
The durations of all of these subroutines are not fixed like an assembly line. Each is uncertain and independent. But with an average time for each, the leader feels he can mathematically come up with an expected value of how long the patient care experience will take based on these calculations. Do you see where this is going? You’d think if you added up these processes or averaged them out you’d come up with an accurate projection, right?
Just like our statistical coin toss, the more parallel processes involved, the more times you must flip heads in a row. If more than 10 processes must simultaneously occur, you may have a 1 in 1,000 chance that your calculated projection will occur as calculated. And we wonder why we never hit our metrics!
Focusing on simple math and central tendency is irresistible, but it can be disastrous. And we haven’t even begun to inject uncertainties such as margins of error or probability ranges that create even more misjudgments.
Statisticians refer to uncertain levels of demand as input probability distributions. We don’t know how to generate these equations in medicine nor do we know what to do with them. Generating them can be terribly complex even in a static environment, and emergency medicine is far from static. Our industry is clearly unpredictable, dynamic and more biologic, defying inorganic methodologies.
Consider an emergency department that on two independent days had a volume of 200 patients. Mathematically, it’s enticing to assume central tendencies for each day. In other words, the two days are comparable, as the same number of patients was seen both days. It’s a seductive thought.
Mathematically we try to discern why the wait times, throughput, walkout rates or productivity differed, perhaps dramatically, on days when the volume was identical. But we forget Gould’s admonition that our culture fails to account for variation.
We don’t consider variations of patient acuity, arrival times, resources, staffing, productivity potential, turnover, boarding, space limitations or even weather conditions that account for myriad input probability distributions that occur independently each day with uncertain and unpredictable variability. We simply fail to take into account statistical variation.
To properly understand and describe the complexity of things from productivity to patient flow requires the order of differential equations (dx/dt = m sin t + nt3) not simple math (1+1=2). It’s simply the nature of our industry.
Managers are doing the math correctly. But have you ever looked at physician productivity, throughput times or any other calculated models and said to yourself, “The numbers look right, but that sure doesn’t seem to be how it actually works.” The math and reality just don’t seem to match, and we wonder why.
Our culture tends to ignore variation, so we do mathematics instead of statistics. It’s that simple… and that’s the reason why.
Phillip Stephens, DHSc, PA-C,is the associate practitioner site director for Emergency Medical Associates at Southeastern Regional Medical Center, Lumberton, N.C. He is adjunct faculty at A.T. Still University in Mesa, Ariz., where he teaches Research Methodology and has practiced as an emergency medicine physician assistant for 25 years.
