# Understanding statistical control charts

Control charts are all the rage at my company these days. Our "data analysts" (quotes are on purpose) are putting control limits on almost all graphs that they produce. The other day we were looking at some graphs of calls into our support call center over the last year. We have measurements of the number of calls for each week that are plotted on a line graph. Recently, the analysts who prepare these graphs have started putting lines on the graphs indicating "control limits".

The question I have is about how they are calculating where these control limits should be. When I asked how they determined the limits here is the answer I got:

First we find a section of the graph where the points look somewhat stable (meaning small variance) then calculate the mean of those points. Then we calculate the standard error of those points and draw the control limits at +/- 3 standard errors from the mean that we calculated.

Is it correct to calculate the mean this way? It seems like we should be using ALL the points to calculate the mean instead of throwing out anything that "looks" like it varies too much.

Does it even make sense to put control limits on a measurement like this (the number of calls coming into a call center)? As long as the measurements are within the control limits we consider the week normal. If the measurement for a week's call volume is outside these control limits it is deemed significant and warrants further analysis. Something just seems artificial about all this to me.

• Did they really mean "standard error" (of the mean) and not "standard deviation"? (Regardless, there is nothing right about the procedures you describe; none of this has any justification in theory or practice; and you are correct to be suspicious of every single element you have mentioned.)
– whuber
Sep 9 '11 at 15:09
• Agree w/ whuber. Is the goal to estimate the 0.135th and 99.865th percentiles of the calls-per-week distribution? (If the dist'n were normal, and you meant std dev, and they didn't throw out data, then that seems to be the target estimate.) With <1000 observations (from the same dist'n), it's hard, but possible, e.g. Dekkers & de Haan (1989). So: I agree w/ whuber the current method is unjustified; wondering if you were asking for positive suggestions, too, or just confirmation of your (correct) critical thoughts. Sep 9 '11 at 15:45
• I'm really trying to understand if control charts are appropriate for these measurements and if they are then how are they supposed to be implemented. Sep 9 '11 at 16:23
• As for the question about standard deviation vs. standard estimate they are calling it standard deviation but their calculation looks like standard error (sd/sqrt(N)). I do confess to not fully grasping which is which though. Sep 9 '11 at 16:24
• The standard deviation tells us how much the values will typically differ from their mean. The standard error tells us how much the sample mean is likely to differ from the true mean. Note the $1/\sqrt(N)$: this implies that the control limits will narrow as larger historical periods are used, demonstrating that the limits depend on this arbitrary choice. The estimate of the SD, on the other hand, will stabilize (to the population SD) as $N$ grows large.
– whuber
Sep 9 '11 at 19:35

The purpose of a control chart is to identify, as quickly as possible, when something fixable is going wrong. For it to work well, it must not identify random or uncontrollable changes as being "out of control."

The problems with the procedure described are manifold. They include

• The "stable" section of the graph is not typical. By definition, it is less variable than usual. By underestimating variability of the in-control situation, it will cause the chart incorrectly to identify many changes as out of control.

• Using standard errors is simply mistaken. A standard error estimates the sampling variability of the mean weekly call rate, not the variability of the call rates themselves.

• Setting the limits at $\pm 3$ standard deviations might or might not be effective. It is based on a rule of thumb applicable for normally distributed data that are not serially correlated. Call rates will not be normally distributed unless they are moderately large (around 100+ per week, approximately). They might or might not be serially correlated.

• The procedure assumes the underlying process has an unvarying rate over time. But you're not making widgets; you're responding to a market that--hopefully--is (a) increasing in size yet (b) decreasing its call rate over time. Temporal trends are expected. Sooner or later any trends will cause the data to look consistently out of control.

• People tend to undergo annual cycles of activity corresponding to seasons, the academic calendar, holidays, and so on. These cycles act like trends to cause predictable (but meaningless) out-of-control events.

A simulated dataset illustrates these principles and problems.

The simulation procedure creates a realistic series of data that are in control: relative to a predictable underlying pattern, it includes no out-of-control excursions that can be assigned a cause. This plot is a typical outcome of the simulation.

These data a drawn from Poisson distributions, a reasonable model for call rates. They start at a baseline of 100 per week, trending upward linearly by 13 per week per year. Superimposed on this trend is a sinusoidal annual cycle with an amplitude of eight calls per week (traced by the dashed gray curve). This is a modest trend and a relatively small seasonality, I believe.

The red dots (around weeks 12 - 37) were identified as the 26-week period of lowest standard deviation encountered during the first 1.5 years of this two year chart. The thin red and blue lines are set at $\pm 3$ standard errors around this period's mean. (Obviously they are useless.) The thick gold and green lines are set at $\pm 3$ standard deviations around the mean.

(One doesn't usually project control lines backwards in time, but I have done that here for visual reference. It's usually meaningless to apply controls retroactively: they're intended to identify future changes.)

Note how the secular trend and the seasonal variations drive the system into apparent out-of-control conditions between weeks 40-65 (an annual high) and after week 85 (an annual high plus over one year's cumulative trend). Anybody attempting to use this as a control chart would be mistakenly looking for nonexistent causes most of the time. In practice, this system would be hated and soon ignored by everyone. (I have seen companies where every office door and all the hallway walls were covered in control charts that nobody bothered to read, because they all knew better.)

The right way to proceed begins by asking the basic questions, such as how do you measure quality? What influences can you have over it? How, despite your best efforts, are these measures likely to fluctuate? What would extreme fluctuations tell you (what could their controllable causes be)? Then, you need to perform a statistical analysis of the past data. What is their distribution? Are they temporally correlated? Are there trends? Seasonal components? Evidence of past excursions that might have indicated out of control situations?

Having done all this, it may then be possible to create an effective control chart (or other statistical monitoring) system. The literature is large, so if this company is serious about using quantitative methods to improve quality, there is ample information about how to do so. But ignoring these statistical principles (whether through lack of time or lack of knowledge) practically guarantees that the effort will fail.

• +1: Amazing reply. I don't suppose you have some references to some of the canonical literature you mentioned? May 17 '13 at 18:05
• @SnOrfus I wish I could give you reliable references but I am not up-to-date on the literature. This post is based on what I read a quarter century ago and have learned in applying that information. (I wrote and sold specialized control charting software back then, and consequently had opportunities to look at--quite literally-several million control charts of customers' data and think about the adequacy of those charts for their decision making processes.)
– whuber
May 17 '13 at 18:13
• Completely understandable. Thank you very much either way. May 17 '13 at 18:28

The general idea of control charts is to distinguish between common cause variation and special cause variation. The idea is that the process is fairly stable and generates data from a given distribution (though the Poisson makes more sense for number of calls than the normal). One big advantage of control charts is that they limit overreacting to natural variation while still allowing for finding when the process has changed.

Choosing a set of observations because they have small variation would almost guarantee that the limits are too narrow and therefore increase the inappropriate reactions to normal variation. Using all the data makes a lot more sense, and using a Poisson C chart might be better than an x-bar chart. But, it also seems that a call center would expect differences due to holidays or season (depending on what is being supported), so the underlying assumptions may not even be appropriate here.

It sounds like they are doing something because they can rather than because it answers a meaningful question.