## What Is the Theory Behind Control Charts

The term control chart is short for statistical process control (SPC) chart. What control charts reveal is whether observed variation in a process is consistent with random variation due to innumerable common causes. When variation occurs within control limits, the process is said to be in statis tical control, free from special cause effects or assignable cause variation. The process is subject only to chance factors or random cause variation.

As mentioned earlier, calculated upper and lower control limits (or, more correctly labeled, statistical process control limits) are added to a run chart to make it into a control chart. Control limits are calculated on the basis of statistical theory describing distribution of values within any population of data.

Figure 22.9 shows a typical normal distribution (called normal because it applies to so much of the data that we all typically encounter). It is also known as a Gaussian

FIGURE 22.9. NORMAL DISTRIBUTION, AKA THE BELL CURVE.

3-Sigma

3-Sigma (after Gauss, the person who first described it) or bell (for the obvious reason) curve. The normal distribution is formed by showing graphically the spread of results from observing a variety of samples (a sample might be, for example, a group of patients) drawn from a universe of possible samples. What the normal distribution shows is that samples tend to have average values that center on the true average value but that most samples are either above or below the true average value by a little bit, by a medium amount, or by a large amount. The likelihood that a sample's value will be close to the true average value is greater than the likelihood that it will be far away. The mathematics for normal distributions uses the concept of standard deviations of values from the true center value of the distribution. The way this works is that about 68 percent of sample values will fall within 1 standard deviation (1 SD, or 1 sigma) and that about 95 percent of sample values will fall within 2 SD, or 2 sigma, and that more than 99 percent will fall within 3 SD, or 3 sigma.

Upper and lower control limits are generally calculated so that values that fall outside 3 SD are displayed as falling outside the control limit. Those data points that fall outside 3 SD, or 3 sigma, are very unlikely to happen by chance. The normal distribution tells us that the vast majority of data points (99.73 percent) will fall within the upper and lower range (within 3 SD) and thus it is likely that the process is experiencing special causes when a point falls outside these calculated statistical process control limits. Figure 22.10 illustrates this concept by juxtaposing a control chart and the normal distribution. The reason for setting your limits

FIGURE 22.10. CONTROL CHART IN RELATION TO NORMAL DISTRIBUTION. at 3 standard deviations rather than 2, as is customary in much scientific writing, is based on Walter Shewhart's observation that it is generally more effective, from a practical, economic point of view, to restrict investigation for special causes to situations in which the data are quite different from what one would expect, not just somewhat different (Shewhart, 1931/1980, 1939/1986). 