**6. What happens when the mean shift is transient?**

This issue was studied in a paper published in 2021 [42]. We investigated the impact of transient shifts on the operational characteristics of ShCCs. Traditionally, all guides and standards tell the readers about the efficiency of the X-bar chart to detect the shift of the process mean. This efficiency notably surpasses the efficiency of x-chart, and the superiority grows when the sample size *n* increases. However, this is true but only for the so-called sustained shift. When the shift becomes transient, the situation may change radically, which most practitioners simply do not know because this issue is rarely discussed in the literature. In [42], we discussed an elementary model shown in **Figure 3**.

In order to compare the efficiency of different types of ShCCs one needs to calculate the so-called power function (PF) for each type of chart. In Ref. [42] there was

#### **Figure 3.**

*(a) – sustained shift, and (b) – transient shift, n – sample size, here n = 3 The number of subgroups lying entirely inside the step we denoted as m. Here m = 1.*

considered the case when the transient shift starts between subgroups and ends after *m*-shifted subgroups within the (m + 1)th. In other words, we assumed that the X-bar chart with subgroups of size *n* was being built and an assignable cause of variation emerged between subgroups and lasted so that it covered *m* subgroups totally and *r* points within the last (m + 1)th. In this case, it was obtained for the average of shifted subgroups:

$$\overline{X} = \frac{\left[nm\left(\mu\_0 + k\sigma\right) + r\left(\mu\_0 + k\sigma\right) + \left(n - r\right)\mu\_0\right]}{n\left(m + 1\right)} = \mu\_0 + \frac{m}{m + 1}k\sigma + \frac{rk\sigma}{n\left(m + 1\right)}\tag{7}$$

Clearly, the bias of the average will be approaching the sustained value when *m* → ∞ , and the impact of the transient shift will be maximal for m = 0. In this case the average is equal to

$$\overline{X} = \frac{\left[r\left(\mu\_0 + k\sigma\right) + \left(n - r\right)\mu\_0\right]}{n} = \mu\_0 + \frac{rk\sigma}{n} \tag{8}$$

Obviously, the shift of the average will be decreasing and therefore the probability of detecting it will be decreasing as well. The value of *SD* is assumed to be constant (at least at the first approximation) and equal to σ/ *n* . Thus, one can obtain:

*Shewhart Control Chart: Long-Term Data Analysis Tool with High Development Capacity DOI: http://dx.doi.org/10.5772/intechopen.113991*

$$PF = \mathbf{1} - \Phi\left(\mathbf{3} - \frac{kr}{\sqrt{n}}\right) \tag{9}$$

Using of traditional assumptions of data normality and of being *i.i.d.* (identically independently distributed) was implied. The results of calculations on the base of (9) for different values of *n* and *r* are shown in **Figure 3** of [42]. One example of those pictures is given in **Figure 4**.

As one could expect, when *r* is equal to *n* the PF coincides with the traditional one given in many books (see, for example, Figure 10.1 in [9]).

As soon as not all the subgroup's points fall into the changed process, the PF starts to decrease, and for some value of *r* – in [42], we called this value boundary, *r*b – the PF becomes less than the corresponding PF for *n = 1* (the PF of the chart for the individual values). It is quite easy to find out the value of *r*b for which the PF of X-bar chart becomes less than the PF of the chart for individuals:

$$r\_b \le \sqrt{n} \tag{10}$$

This means that the traditional conclusion about the superiority of X-bar chart over an X-chart does not work, at least when the duration of the process jump is shorter than the time to gather a subgroup. It all depends on the amount of points falling into the changed part of the subgroup. This example displays that even the simplest ShCCs are not as easy in practice as they may seem. And even for the values of *r > r*b, the resulting values of probability to reveal the signal about the shift may be essentially less than what is traditionally mentioned in SPC textbooks. For example, for an X-bar chart with *n* = 10, the probability of detecting the mean's shift of one sigma falls down from 0.564 when the shift is sustained to 0.078 when only five points are within the changed process, and five points belong to the unchanged process.

**Figure 4.** *An example of PF for sample of five with different values of r.*

#### **7. ShCCs and 2 M's: metrology and management**

In the final section of this chapter, we would like to briefly discuss two areas of people's activity where ShCCs must be used most often and where they are not practically being used at all. These fields are metrology and management.

Metrology's primary concern is variation of measurement, so it seems very logical that all metrology-responsible people have to know the basics of SPC and have to be able to construct and interpret ShCCs. Unfortunately, reality looks quite the opposite [43, 44].

Management's primary concern is variation between people and its impact on the organization. Again, the knowledge of ShCCs looks like it leaves no alternative for managers. Once more, the dark side of reality has beaten us to it [45]. Dr. Deming wrote the honest truth in the foreword to [23]: "The fact is that some of the greatest contributions from control charts lie in areas that are only partly explored so far, such as applications to supervision, management, and systems of measurements, including the standardization and use of instruments, test panels, and standard samples of chemicals and compounds" ([23], *ii*). A detailed discussion of these issues would require writing another chapter that would most likely be much longer than this one. So we decided to confine ourselves to these ultrashort comments in order to put these topics to the next discussion.

#### **8. Conclusions**

We live in a World full of complexity, human irrationality, variability, and, consequently, unpredictability [45]. One of the main features of this world is the ubiquitous presence of variations. Variations differ from each other, so the question arises if one needs to react to them or leave them unnoticed. The ability to answer this question is not something innate to the human being. People should be taught, especially taking into account that knowledge about variability and how to cope against it emerged only after Shewhart's works of 1931 and 1939 [23, 24]. The tool invented by him – control chart – is the only way to understand if the process is controlled/stable/predictable or not. Without this knowledge, there are no chances to improve any system. Ironically, ShCCs turned out to be very simple technically and rather difficult to use profitably. The reason – contextual knowledge is vital for a genuine understanding of what the charts are trying to tell us. ShCC is a communication tool between a system where a process is going and the process owner. Its construction cannot be totally algorithmized [46]. In other words, everybody in any organization should be taught the art of constructing and interpreting ShCCs. In fact, the reality looks completely different. We started this chapter with the quote of Hoyer, Ellis about quality professionals who do not know the basics of SPC4 , and here we will continue citing their phrase "And why should they? Our review of a very large number of SPC textbooks reveals page after page of "cookbook" discussions of practically everything under the sun—with very little discussion on the foundation of SPC ([14], *63*)". Unfortunately, the basics of SPC is a set of rules and recipes which are essentially more narrow than wants of practice. We tried to widen this set in this chapter. To this end, we discussed some uncertainties in the interpretation of assignable causes of variation, and proposed to

<sup>4</sup> We are sure that engineers, managers, CEOs, teachers, physicians, etc. should be added to quality professionals.

*Shewhart Control Chart: Long-Term Data Analysis Tool with High Development Capacity DOI: http://dx.doi.org/10.5772/intechopen.113991*

differentiate between assignable causes that change the system, and those, that do not influence it. Then, we considered the impact of data order on ShCCs and suggested using the average moving range as an indicator of nonrandomness. The next part of the chapter was devoted to the influence of data asymmetry on the limits of ShCC. Recent results for some asymmetrical DFs were delivered. Further, we discussed a simple case of transient jump of process mean and demonstrated that some traditional and established rules should be essentially changed. At last, at the very end of the chapter, we formulated two important issues that should be discussed carefully in the near future. They relate to the link between ShCCs and 2 M's: management and metrology.
