History Is Written by the Winners. Reflections on Control Charts for Process Control

1. Introduction

Since 1989, the author (FG) tried to inform the Scientific Community about the flaws in the use of (“wrong”) quality methods for making Quality [1] and in 1999 about the GIQA (Golden Integral Quality Approach) showing how to manage Quality during all the activities of the Product and Process Development in a Company [2] , including the Process Management and Control Charts (CC) for Process Control.

Here we show how to deal correctly with I-CC (Individual Control Charts) by analysing a literature case [3] comprising the famous data-set on coal-mining disasters of Jarrett (1979); these data are considered and analysed in [4,5].

We found very interesting the statements in the Excerpt 1:

In the recent paper “Misguided Statistical Process Monitoring Approaches” by W. Woodall, N. Saleh, M. Mahmoud, V. Tercero-Gómez, and S. Knoth, published in Advanced Statistical Methods in Process Monitoring, Finance, and Environmental Science, 2023, We read in the Abstract: Hundreds of papers on flawed statistical process monitoring (SPM) methods have appeared in the literature over the past decade or so. The presence of so many ill-advised methods, and so much incorrect theory, adversely affects the SPM research field. Critiques of some of the various misguided, and/or misrepresented, approaches have been published in the past 2 years in an effort to stem this tide. These critiques are briefly reviewed here. References…

Excerpt 1 . From the paper “Misguided Statistical Process Monitoring Approaches”.

We agree with the authors in the excerpt 1, but, nevertheless, they did not realise the problem that we are giving here: wrong Control Limits in CCs for Rare Events, with data exponentially or Weibull distributed. See References…

Using the data in [3–5] with good statistical methods [6–33] we give our “reflections on Control Charts (CCs)”.

We will try to state that several papers (that are not cited here, but you can find in the “Garden of flowers” [24] and some in the Appendix C ) compute in an a-scientific way (see the formulae in the Appendix C ) the Control Limits of CCs for “Individual Measures or Exponential, Weibull and Gamma distributed data”, indicated as I-CC (Individual Control Charts); we dare to show, to the Scientific Community, how to compute the True Control Limits. If the author is right, then all the decisions, taken up today, have been very costly to the Companies using those Control Limits; therefore, “Corrective Actions” are needed, according to the Quality Principles, because NO “Preventive Actions” were taken [1,2,27–36] : this is shown through the suggested published papers. Humbly, given our strong commitment to Quality [34–57] , we would dare to provide the “truth”: Truth makes you free [hen (“hic et nunc”=here and now)].

On 22^nd of February 2024, we found the paper “Publishing an applied statistics paper: Guidance and advice from editors” published in Quality and Reliability Engineering International (QREI-2024, 1-17) [by C. M. Anderson-Cook, Lu, R. B. Gramacy, L. A. Jones-Farmer, D. C. Montgomery, W. H. Woodall; the authors have important qualifications and Awards]; since I-CC is a part of “applied statistics” we think that their hints will help: the authors’ sentence “Like all decisions made in the face of uncertainty, Type I (good papers rejected) and Type II (flawed papers accepted) errors happen since the peer review process is not infallible.” is very important for this paper: the interested readers can see [34–57] .

By reading [24] , the readers are confronted with this type of practical problem: we have a warehouse with two departments

1)

in the 1^st of them, we have a sample (the “The Garden of flowers… in [24]”) of “products (papers)” produced by various production lines (authors)

2)

while, in the other, we have some few products produced by the same production line (same author)

3)

several inspectors (Peer Reviewers, PRs) analyse the “quality of the products” in the two departments; the PRs can be the same (but we do not know) for both the departments

4)

The final result, according to the judgment of the inspectors (PRs), is the following: the products stored in the 1^st dept. are good, while the products in the 2^nd dept. are defective. It is a very clear situation, as one can guess by the following statement of a PR: “Our limits [in the 1^st dept.] are calculated using standard mathematical statistical results/methods as is typical in the vast literature of similar papers [4,5,24].” See the standard mathematical statistical results/methods in theFigures A1, A2, A3, of the Appendix A and meditate (see the formulae there)!

Hence, the problem becomes “…the standard … methods as is typical … ”: are those standards typical methods (in the “The Garden … in [24] ” and in the Appendix C ) scientific?

The practical problem becomes hence a Theoretical one [1–57] (all references and Figure 1 ): we show here, immediately, the wrong formulae (either using the parameter $\theta ={\theta }_0$  or its estimate ${\overline{\mathit{t}}}_0$ , with $\alpha =0.0027$ ) in the formula (1)

$$LCL={\theta }_{0}ln\left(1-\alpha /2\right)=0.00135 {\overline{\mathit{t}}}_0 UCL={\theta }_{0}ln\left(\alpha /2\right)=6.6077 {\overline{\mathit{t}}}_0$$

(1)

Figure 1. Theoretical Difference between L^------U and LCL^------UCL.

Figure 1. Theoretical Difference between L^------U and LCL^------UCL.

In the formulae (1), in the (named) interval LCL^------UCL (Control Interval), the LCL must be L and the UCL must be U, vertical interval L^------U (Figure 1); the actual interval LCL^------UCL is the horizontal one in the Figure 1, which is not that of the formulae (1). Since the errors have been continuing for at least 25 years, we dare to say that this paper is an Education Advance for all the Scholars, for the software sellers and the users: they should study the books and papers in [1–57].

The readers could think that the I-CCs are well known and well dealt in the scientific literature about Quality. We have some doubt about that: we will show that, at least in one field, the I-CC_TBE (with TBE, Time Between Event data) usage, it is not so: there are several published papers, in “scientific magazines and Journals (well appreciated by the Scholars)” with wrong Control Limits; a sample of the involved papers (from 1994 to January 2024) can be found in [23,24] ”. Therefore, those authors do not extract the maximum information from the data in the Process Control. “The Garden…” [24] and the excerpts 1, with the Deming’s statements, constitute the Literature Review.

Excerpt 2 . Some statements of Deming about Knowledge and Theory (Deming 1986, 1997).

We hope that the Deming statements about knowledge will interest the Readers (Excerpt 2).

A preliminary case is shown in Appendix A .

2. Materials and Methods

2.1. A Reduced Background of Statistical Concepts

This section is essential to understand the “problems related to I-CC” as we found in the literature. We suggest it for the formulae given and for the difference between the concepts of PI (Probability Interval, NOT “Prediction Interval”) and CI (Confidence Interval): this is overlooked in “The Garden … [24] ” (a sample is in the Appendix C ).

See a first case in the appendix A. Therefore, we humbly ask the reader to carefully meditate on the content.

Engineering Analysis is related to the investigation of phenomena underlying products and processes; the analyst can communicate with the phenomena only through the observed data, collected with sound experiments (designed for the purpose): any phenomenon, in an experiment, can be considered as a measurement-generating process [MGP, a black box that we do not know] that provides us with information about its behaviour through a measurement process [MP, known and managed by the experimenter], giving us the observed data (the “message”).

It is a law of nature that the data are variable, even in conditions considered fixed, due to many unknown causes.

MGP and MP form the Communication Channel from the phenomenon to the experimenter.

The information, necessarily incomplete, contained in the data, has to be extracted using sound statistical methods (the best possible, if we can). To do that, we consider a statistical model F(x| θ ) associated with a random variable (RV) X giving rise to the measurements, the “determinations” {x₁, x₂, …, x_n}=D of the RV, constituting the “observed sample” D; n is the sample size. Notice the function F(x| θ ) [a function of real numbers, whose form we assume we know] with the symbol θ accounting for an unknown quantity (or some unknown quantities) that we want to estimate (assess) by suitably analysing the sample D.

We indicate by $f\left(x|\theta \right)=dF\left(x|\theta \right)/dx$  the pdf (probability density function) and by $F\left(x|\theta \right)$  the Cumulative Function, where $\theta$  is the set of the parameters of the functions.

When $\theta =\left\{\mu ,{\sigma }^2\right\}$  we have the Normal model, written as $n$ (x| $\mu ,{\sigma }^2$ ), with (parameters) mean E[X]= μ and variance Var[X]= σ ²

$$f\left(\mathrm{x}\right|\mu ,{\sigma }^2)=n(\mathrm{x}|\mu ,{\sigma }^2)={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}{e}^{-{(x-\mu )}^2/{(2\sigma }^2)}$$

(2)

When $\theta =\left\{\theta \right\}$  we have Exponential model, with (the single parameter) mean E[X]= $\theta =1/\lambda$  (variance Var[X]= $\theta$ ² $=1/{\lambda }^2$ ), written in two equivalent ways $f\left(x|\theta \right)=\left(1/\theta \right)e^{-x/\theta }=\lambda e^{-\lambda x}=f\left(x|\lambda \right)$ .

When we have the observed sample D={x₁, x₂, …, x_n}, our general problem is to estimate the value of the parameters of the model (representing the parent population) from the information given by the sample. We define some criteria which we require a "good" estimate to satisfy and see whether there exist any "best" estimates. We assume that the parent population is distributed in a form, the model, which is completely determinated but for the value θ ₀ of some parameter, e.g. unidimensional, θ , or bidimensional θ ={ μ , σ ² }; we consider only one or two parameters, for easiness.

We seek some function of θ , say τ ( θ ), named inference function, and we see if we can find a RV T which can have the following properties: unbiasedness, sufficiency, efficiency. Statistical Theory allows us the analysis of these properties of the estimators (RVs).

We use the symbols $\overline{X}$  and $S^2$  for the unbiased estimators T₁ and T₂ of the mean and the variance.

Luckily, we have that T₁, in the Exponential model $f\left(x|\theta \right)$ , is efficient [6–21,25–33] , and it extracts the total available information from any random sample, while the couple T₁ and T₂, in the Normal model, are jointly sufficient statistics for the inference function τ ( θ )=( μ , σ ² ), so extracting the maximum possible of the total available information from any random sample. The estimators (which are RVs) have their own “distribution” depending on the parent model F(x| θ ) and on the sample D: we use the symbol $\phi (t, \theta ,n)$  for that “distribution”. It is used to assess their properties. For a given (collected) sample D the estimator provides a value t (real number) named the estimate of τ ( θ ), unidimensional.

A way of finding the estimate is to compute the Likelihood Function $L\left(\theta \right|D)$  [LF] and to maximise it: the solution of the equation ${\displaystyle \frac{\partial }{\partial \theta }}L\left(\theta \right|D)$ =0 is termed Maximum Likelihood Estimate [MLE].

The LF is important because it allows us finding the MVB (Minimum Variance Bound, Cramer-Rao theorem) [1,2,6–16,26–36] of an unbiased RV T [related to the inference function τ ( θ )], such that

$$Var\left(T\right)\ge {\displaystyle \frac{{\left[\tau \left(\theta \right)\right]}^2}{E\left\{{\left[\partial lnL\left(\theta \right|D)/\partial \theta \right]}^2\right\}}}=MVB\left(T\right)$$

(3)

The inverse of the MVB(T) provides a measure of the total available amount of information in D, relevant to the inference function τ ( θ ) and to the statistical model F(x| θ ).

Naming I_T(T) the information extracted by the RV T we have that [6–21,26–36]

I_T(T)=1/MVB(T) ⇔ T is an Efficient Estimator.

If T is an Efficient Estimator there is no better estimator able to extract more information from D.

The estimates considered before were “point estimates” with their properties, looking for the “best” single value of the inference function τ ( θ ).

We must now introduce the concept of Confidence Interval (CI) and Confidence Level (CL) [6–21,26–36] .

The “interval estimates” comprise all the values between τ _L (Lower confidence limit) and τ _U (Upper confidence limit); the CI is defined by the numerical interval CI= {τ _L ^----- τ _U } , where τ _L and τ _U are two quantities computed from the observed sample D: when we make the statement that τ ( θ ) ∈ CI, we accept, before any computation, that, doing that, we can be right, in a long run of applications, (1- α )%=CL of the applications, BUT we cannot know IF we are right in the single application (CL=Confidence Level).

We know, before any computation, that we can be wrong α % of the times but we do not know when it happens.

The reader must be very careful to distinguish between the Probability Interval PI= { L^-----U } , where the endpoints L and U depends on the distribution $\phi (t, \theta ,n)$  of the estimator T (that we decide to use, which does not depend on the “observed sample” D) and, on the probability π =1- α (that we fix before any computation), as follows by the probabilistic statement (4) [se the Figure 1 for the exponential density, when n=1]

$$P\left[L\le T\le U\right]={\int }_L^U\phi (t, \theta ,n)dt=1-\alpha$$

(4)

and Confidence Interval CI= {τ _L ^----- τ _U } which depends on the “observed sample” D.

Notice that the Probability Interval PI= { L^-----U } does not depend on the data D: L and U are the Probability Limits. Notice that, on the contrary, the Confidence Interval CI= {τ _L ^----- τ _U } does depend on the data D.

Shewhart identified this approach, L and U, on page 275 of [19] where he states:

“For the most part, however, we never know $f_{\mathsf{\Theta }}(\mathsf{\Theta },n)$ [this is the symbols of Shewhart for our $\phi (t, \theta ,n)$] in sufficient detail to set up such limit… We usually chose a symmetrical range characterised by limits $\overline{\mathsf{\Theta }}\pm t{\sigma }_{\mathsf{\Theta }}$ symmetrically spaced in reference to $\mathsf{\Theta }$. Tchebycheff’s Theorem tells us that the probability P that an observed value of $\mathsf{\Theta }$ will lie within these symmetric limits so long as the quality standard is maintained satisfies the inequality P>1-1/t². We are still faced with the choice of t. Experience indicated that t=3 seems to be an acceptable economic value”. See the excerpts 3,…

The Tchebycheff Inequality : IF the RV X is arbitrary with density f(x) and finite variance ${\sigma }^2$  THEN we have the probability $P\left[\left|X-\mu \right|\ge k\sigma \right]\le 1/k^2$ , where $\mu =E\left[X\right]$ . This is a “Probabilistic Theorem”.

It can be transferred into Statistics. Let’s suppose that we want to determine experimentally the unknown mean $\mu$  within a “stated error ε ”. From the above (Probabilistic) Inequality we have $P\left[\mu -\epsilon <X<\mu +\epsilon \right]\ge 1-{\sigma }^2/{\epsilon }^2$ ; IF $\sigma \ll \epsilon$  THEN the event $\left\{\left|X-\mu \right|<\epsilon \right\}$  is “very probable” in an experiment: this means that the observed value $x$  of the RV X can be written as $\mu -\epsilon <x<\mu +\epsilon$  and hence $x-\epsilon <\mu <x+\epsilon$ . In other words, using $x$  as an estimate of $\mu$  we commit an error that “most likely” does not exceed $\epsilon$ . IF, on the contrary, $\sigma \nleqq \nleqq \epsilon$ , we need n data in order to write $P\left[\mu -\epsilon <\overline{X}<\mu +\epsilon \right]\ge 1-{\sigma }^2/\left(n{\epsilon }^2\right)$ , where $\overline{X}$  is the RV “mean”; hence we can derive $\overline{x}-\epsilon <\mu <\overline{x}+\epsilon$ ., where $\overline{x}$  is the “empirical mean” computed from the data. In other words, using $\overline{x}$  as an estimate of $\mu$  we commit an error that “most likely” does not exceed $\epsilon$ . See the excerpts 3, 3a, 3b.

Notice that, when we write $\overline{x}-\epsilon <\mu <\overline{x}+\epsilon$ , we consider the Confidence Interval CI [6–21,25–33] , and no longer the Probability Interval PI [6–21,25–33] .

These statistical concepts are very important for our purpose when we consider the Control Charts, especially the Individual CCs, I-CC.

Notice that the error made by several authors [4,5,24] is generated by lack of knowledge of the difference between PI and CI [6–21,25–33] : they think wrongly that CI=PI, a diffused disease [4,5,24] ! They should study some of the books/papers [6–21,25–33] and remember the Deming statements (excerpt 2).

The Deming statements are important for Quality. Managers, scholars; the professors must learn Logic, Design of Experiments and Statistical Thinking to draw good decisions. The authors must, as well. Quality must be their number one objective: they must learn Quality methods as well, using Intellectual Honesty [1,2,6–21,25–33] . Using (4), those authors do not extract the maximum information from the data in the Process Control. To extract the maximum information from the data one needs statistical valid Methods [1,2,6–21,25–33] .

As you can find in any good book or paper [6–21,25–33] there is a strict relationship between CI and Test Of Hypothesis, known also as Null Hypothesis Significance Testing Procedure (NHSTP). In Hypothesis Testing, the experimenter wants to assess if a “thought” value of a parameter of a distribution is confirmed (or rejected) by the collected data: for example, for the mean μ (parameter) of the Normal $n$(x|$\mu ,{\sigma }^2$) density, he sets the “null hypothesis” H₀={μ=μ₀} and the probability P=α of being wrong if he decides that the “null hypothesis” H₀ is true, when actually it is opposite: H₀ is wrong. We analyse the observed sample D={x₁, x₂, …, x_n} and we compute the empirical (observed) mean $\overline{x}$ and the empirical (observed) standard deviation $s;$hence, we define the Acceptance interval, which is the CI

$$\overline{x}-t_{1-\alpha /2}s/\sqrt{n}<\mu <\overline{x}+t_{1-\alpha /2}s/\sqrt{n}$$

(5)

Notice that the interval (for the Normal model) [see the appendix B]

$${{\mu }_0-t_{1-\alpha /2}{\sigma }_0/\sqrt{n}}^{\mathit{}\mathit{}------\mathit{}\mathit{}}{\mu }_0+t_{1-\alpha /2}{\sigma }_0/\sqrt{n}$$

(6)

is the Probability Interval such that $P\left[{\mu }_0-t_{1-\alpha /2}{\sigma }_0/\sqrt{n}<\overline{X}<{\mu }_0+t_{1-\alpha /2}{\sigma }_0/\sqrt{n}\mathrm{}\right]=1-\alpha$ .

A fundamental reflection is in order : the formulae (5) and (6) tempt the unwise guy to think that he can get the Acceptance interval, which is the CI [1–23] , by substituting the assumed values ${\mu }_0,{\sigma }_0$  of the parameters with the empirical (observed) mean $\overline{x}$  and standard deviation $s$ . This trick is valid only for the Normal distribution .

More ideas about this can be found in [34–57] .

In the field of Control Charts, following Shewhart, instead of the formula (5), we use (7)

$$\overline{\mathit{x}}-{\mathit{z}}_{1-\mathit{\alpha }/2}\mathit{s}/\left({\mathit{c}}_4\sqrt{\mathit{n}}\right)<\mathit{\mu }<\overline{\mathit{x}}+{\mathit{z}}_{1-\mathit{\alpha }/2}\mathit{s}/\left({\mathit{c}}_4\sqrt{\mathit{n}}\right)$$

(7)

where the value $t_{1-\alpha /2}$  of the t distribution is substituted by the value $z_{1-\alpha /2}$  of the Normal distribution, actually $z_{1-\alpha /2}$ =3, and a coefficient ${\mathit{c}}_4$  is used to make “unbiased” the estimate of the standard deviation, computed from the information given by the sample.

Actually, Shewhart does not use the coefficient ${\mathit{c}}_4$  is as you can see from page 294 of Shewhart book (1931), where $\overline{X}$  is the “Grand Mean”, computed from D [named here empirical (observed) mean $\overline{x}$ ], $\sigma$  is “estimated standard of each sample” (named here s, with sample size n=20, in excerpt 3)

Excerpt 3 . From Shewhart book (1931), on page 294.

The application of these ideas in the Individual CCs can be seen in the Appendix A , in the Figure A1 : the standard deviation is derived from the Mobile Range (which is exponentially distributed as the original UTI data). The formula in the excerpt 3 tells us that the process is OOC (Out Of Control).

2.2. ControlCharts for Process Management

Statistical Process Management (SPM) entails Statistical Theory and tools used for monitoring any type of processes, industrial or not. The Control Charts (CCs) are the tool used for monitoring a process, to assess its two states: the first, when the process, named IC (In Control), operates under the common causes of variation (variation is always naturally present in any phenomenon) and the second, named OOC (Out Of Control), when the process operates under some assignable causes of variation. The CCs, using the observed data, allow us to decide if the process is IC or OOC. CCs are a statistical test of hypothesis for the process null hypothesis H₀={IC} versus the alternative hypothesis H₁={OOC}. Control Charts were very considered by Deming [9,10] and Juran [12] after Shewhart invention [19,20] .

We start with Shewhart ideas (see the excerpts 3, 3a and 3b).

In the excerpts, $\overline{X}$ is the (experimental) “Grand Mean”, computed from D (we, on the contrary, use the symbol $\overline{x}$), $\sigma$ is the (experimental) “estimated standard of each sample” (we, on the contrary, use the symbol s, with sample size n=20, in excerpts 3a, 3b), $\overline{\sigma }$ is the “estimated mean standard deviation of all the samples” (we, on the contrary, use the symbol $\overline{s}$).

Excerpt 3a. From Shewhart book (1931), on page 89.

On page 95, he also states that

Excerpt 3b. From Shewhart book (1931), on page 294.

So, we clearly see that Shewhart, the inventor of the CCs, used the data to compute the Control Limits, LCL (Lower Control Limit) and UCL (Upper Control Limit) both for the mean ${\mu }_X$  (the 1^st parameter of the Normal pdf) and for ${\sigma }_X$  (the 2^nd parameter of the Normal pdf). They are considered the limits comprising 0.9973n of the observed data. Similar ideas can be found in [5–21,25–42] (with Rozanov, 1975, we see the idea that CCs can be viewed as a Stochastic Process).

We invite the readers to consider that if one assumes that the process is In Control (IC) and if he knows the parameters of the distribution he can test if the assumed known values of the parameters are confirmed or disproved by the data, then he does not need the Control Charts; it is sufficient to use NHSTP! (see App. B)

Remember the ideas in the previous section and compare Excerpts 3, 3a, 3b (where LCL, UCL depend on the data) with the following Excerpt 4 (where LCL, UCL depend on the Random Variables) and appreciate the profound “logic” difference: this is the cause of the many errors in the CCs for TBE [Time Between Events (see [4,5,24] ).

The same type of arguments are used in another paper [4] JQT , 2017 where the data are Erlang distributed with λ ₀ is the scale parameter and the Control Limits LCL and UCL are defined [copying Xie et al.] erroneously as

Excerpt 4 . From a paper in the “Garden… [24] ”. Notice that one of the authors wrote several papers….

The formulae, in the excerpt 4, LCL₁ and UCL₁ are actually the Probability Limits (L and U) of the Probability Interval PI in the formula (4), when $\phi \left(t, \theta ,n\right)$  is the pdf of the Estimator T, related to the Normal model F(x; μ , σ ² ). Using (4), those authors do not extract the maximum information from the data in the Process Control. From the Theory [6–36] we derive that the interval L= μ _Y -3 σ _Y ^------ μ _Y +3 σ _Y =U is the PI such that the RV Y= $\overline{X}$

$$P[{\mathsf{\mu }}_Y-3{\sigma }_Y\text{≤}\mathrm{Y}=\overline{X}\text{≤}{\mathsf{\mu }}_Y+3{\sigma }_Y]=0.9973$$

(7a)

and it is not the CI of the mean μ = μ _Y [as wrongly said in the Excerpt 4, where actually (LCL₁^-----UCL₁)=PI].

The same error is in other books and papers (not shown here but the reader can see in [21–24] ).

The data plotted in the CCs [6–21,25–36] (see the Figure 2 ) are the means $\overline{x}\left(t_i\right)$ , determinations of the RVs $\overline{X}\left(t_i\right)$ , i=1, 2, ..., n (n=number of the samples) computed from the collected data of the i-th sample D_i={x_ij, j=1, 2, ..., k} (k=sample size)}, determinations of the RVs $X\left(t_{ij}\right)$  at very close instants t_ij, j=1, 2, ..., k. In other applications I-CC (see the Figure 3 ), the data plotted are the Individual Data $x\left(t_i\right)$ , determinations of the Individual Random Variables $X\left(t_i\right)$ , i=1, 2, ..., n (n=number of the collected data), modelling the measurement process (MP) of the “Quality Characteristic” of the product: this model is very general because it is able to consider every distribution of the Random Process $X\left(t\right)$ , as we can see in the next section. From the excerpts 3, 3a, 3b and formula (5) it is clear that Shewhart was using the Normal distribution, as a consequence of the Central Limit Theorem (CLT) [6–20,26–36] . In fact, he wrote on page 289 of his book (1931) “… we saw that, no matter what the nature of the distribution function of the quality is, the distribution of the arithmetic mean approaches normality rapidly with increase in n (his n is our k), and in all cases the expected value of means of samples of n (our k) is the same as the expected value of the universe” (CLT in Excerpt 3, 3a, 3b).

 and “grand mean” $\stackrel{̿}{\mathrm{x}}.$

Figure 2. Control Limits LCL_X^----UCL_X=L^----U (Probability interval), for Normal data (Individuals x_ij, sample size k) “sample means” ${\overline{\mathrm{x}}}_{\mathrm{i}}$

Figure 2. Control Limits LCL_X^----UCL_X=L^----U (Probability interval), for Normal data (Individuals x_ij, sample size k) “sample means” ${\overline{\mathrm{x}}}_{\mathrm{i}}$

Figure 3. Individual Control Chart (sample size k=1). Control Limits LCL^----UCL=L^----U (Probability interval), for Normal data (Individuals x_i) and “grand mean” $\overline{\mathrm{x}}.$

Figure 3. Individual Control Chart (sample size k=1). Control Limits LCL^----UCL=L^----U (Probability interval), for Normal data (Individuals x_i) and “grand mean” $\overline{\mathrm{x}}.$

Let k be the sample size; the RVs $\overline{X}\left(t_i\right)$  are assumed to follow a normal distribution and uncorrelated; $\overline{X}\left(t_i\right)$  [i^th rational subgroup] is the mean of RVs IID $X\left(t_{ij}\right)$  j=1, 2, ..., k, (k data sampled, at very near times t_ij).

To show our way of dealing with CCs we consider the process as a “stand-by system whose transition times from a state to the subsequent one” are the collected data. The lifetime of “stand-by system” is the sum of the lifetimes of each unit. The process (modelled by a “stand-by …”) behaves as a Stochastic Process $X\left(t\right)$ [25–33], that we can manage by the Reliability Integral Theory (RIT): see the next section; this method is very general because it is able to consider every distribution of $X\left(t\right)$.

If we assume that $X\left(t\right)$  is distributed as f(x) [probability density function (pdf) of “transitions from a state to the subsequent state” of a stand-by subsystem] the pdf of the (RV) mean $\overline{X}\left(t_i\right)$  is, due the CLT (page 289 of 1931 Shewhart book), $\overline{X}\left(t_i\right)~N({\mu }_{\overline{X}\left(t_i\right)}, {\sigma }_{\overline{X}\left(t_i\right)}^2)$  [experimental mean $\overline{x}\left(t_i\right)$ ] with mean ${\mu }_{\overline{X}\left(t_i\right)}$  and variance ${\sigma }_{\overline{X}\left(t_i\right)}^2$ . $\stackrel{̿}{X}$  is the “grand” mean and ${\sigma }_{\stackrel{̿}{X}}^2$  is the “grand” variance: the pdf of the (RV) grand mean $\stackrel{̿}{X}~N({\mu }_{\stackrel{̿}{X}}, {\sigma }_{\stackrel{̿}{X}}^2)$  [experimental “grand” mean $\stackrel{̿}{x}$ ]. In Figure 2 we show the determinations of the RVs $\overline{X}\left(t_i\right)$  and of $\stackrel{̿}{X}$ .

When the process is Out Of Control (OOC, assignable causes of variation, some of the means ${\mu }_{\overline{X}\left(t_i\right)}$, estimated by the experimental means ${\overline{x}}_i=\overline{x}\left(t_i\right)$, are “statistically different)” from the others [6–21,25–36]. We can assess the OOC state of the process via the Confidence Interval (provided by the Control Limits) with CL=0.9973; see the Appendix B. Remember the trick valid only for the Normal Distribution ….; consider the PI, L=μ_Y-3σ_Y^------μ_Y+3σ_Y=U; putting $\stackrel{̿}{x}$ in place of ${\mu }_Y$ and $\overline{s}/\sqrt{k}$ in place of ${\sigma }_Y$ we get the CI of ${\mu }_{\stackrel{̿}{X}}$ when the sample size k is considered for each $\overline{X}\left(t_i\right)$, with CL=0.9973. The quantity $\overline{s}$ is the mean of the standard deviations of each sample. This allows us to compare each (subsystem) mean ${\mu }_{\overline{X}\left(t_q\right)}$, q=1,2, …, n, to any other (subsystem) mean ${\mu }_{\overline{X}\left(t_r\right)}$r=1,2, …, n, and to the (Stand-by system) grand mean ${\mu }_{\stackrel{̿}{X}}=\mu$. If two of them are different, the process is classified as OOC. The quantities ${LCL}_X=\overline{\overline{x}}-3\overline{s}/\sqrt{k}$ and ${UCL}_X=\overline{\overline{x}}+3\overline{s}/\sqrt{k}$ are the Control Limits of the CC. When the Ranges R_i=max(x_ij)-min(x_ij) are considered for each sample we have ${LCL}_X=\overline{\overline{x}}-A_2\overline{R}$, $U{CL}_X=\overline{\overline{x}}+A_2\overline{R}$ and ${LCL}_R=D_3\overline{R}$, U${CL}_R=D_4\overline{R}$, where $\overline{R}$ is the “mean range” and the coefficients A₂, D₃, D₄ are tabulated and depend on the sample size k [6–21,25–36].

See the Appendix B: it is important for understanding our ideas.

The interval LCL_X^-------UCL_X is the “Confidence Interval” with “Confidence Level” CL=1-α=0.9973 for the unknown mean ${\mu }_{X\left(t\right)}$ of the Stochastic Process X(t) [25–36].  The interval LCL_R^----------UCL_R is the “Confidence Interval” with “Confidence Level” CL=1-α=0.9973 for the unknown Range of the Stochastic Process X(t) [25–36].

Notice that, ONLY for normally distributed data, the length of the Control Interval (UCL_X-LCL_X, which is the Confidence Interval) equals the length of the Probability Interval, PI (U-L): UCL_X-LCL_X=U-L.

The error highlighted, i.e. the confusion between the Probability Interval and the Control Limits (Confidence Interval!) has no consequences for decisions when the data are Normally distributed, as considered by Shewhart. On the contrary, it has BIG consequences for decisions WHEN the data are Non-Normally distributed [4,5,24] .

We think that the paper “Quality of Methods for Quality is important”, [1] appreciated and mentioned by J. Juran at the plenary session of the EOQC (European Organization for Quality Control) Conference (1989), should be considered and meditated.

2.3. Statistics and RIT

We are going to present the fundamental concepts about RIT (Reliability Integral Theory) that we use for computing the Control Limits of CCs. Since in the example chosen, 4 data will be “compounded” we use a “4 units Stand-by system”, depicted by 5 states (Figure 4): 0 is the state with all units not-failed; 1 is the state with the first unit failed; 2 is the state with the second unit failed; and so on, until the system enters the state 5 where all the 4 units are failed (down state, in yellow): any transition provides a datum to be used for the computations. RIT can be found in the author’s books…

Figure 4. A “4 units Stand-by system” and its states.

Figure 4. A “4 units Stand-by system” and its states.

RIT can be used for parameters estimation and Confidence Intervals (CI), (Galetto 1981, 1982, 1995, 2010, 2015, 2016), in particular for Control Charts (Deming, 1986, 1997, Shewhart 1931, 1936, Galetto 2004, 2006, 2015). In fact, any Statistical or Reliability Test can be depicted by an “Associated Stand-by System” [25–36] whose transitions are ruled by the kernels b_k,j(s); we write the fundamental system of integral equations for the reliability tests, whose duration t is related to interval 0^-----t; the collected data t_j can be viewed as the times of the various failures (of the units comprising the System) [t₀=0 is the start of the test, t is the end of the test and g is the number of the data (4 in the Figure 4)]

Firstly, we assume that the kernel $b_{j,j+1}\left(s-t_j\right)$ is the pdf of the exponential distribution $f$($s-t_j$|$\mu ,{\sigma }^2$)$=\lambda e^{-\lambda (s-t_j)}$, where $\lambda$ is the failure rate of each unit and $\lambda =1/\theta$: $\theta$ is the MTTF of each unit. We state that $R_j\left(t-t_j\right)$ is the probability that the stand-by system does not enter the state g (5 in Figure 4), at time t, when it starts in the state j (0, 1, …, 4) at time t_j, ${\overline{W}}_j\left(t-t_j\right)$ is the probability that the system does not leave the state j, $b_{j,j+1}\left(s-t_j\right)ds$ is the probability that the system makes the transition j→j+1, in the interval s^-----s+ds.

The system reliability $R_0\left(t\right)$  is the solution of the mathematical system of the Integral Equations (8)

$$\begin{array}{c}{R}_j\left(t-t_j\right)={\overline{W}}_j\left(t-t_j\right)+{\int }_{t_j}^t{b}_{j,j+1}\left(t-t_j\right)R_{j+1}\left(t-s\right)ds \mathrm{}\\ \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{}\mathrm{}\mathrm{j}=0,\mathrm{}1,\dots ,,\mathrm{}\mathrm{}\mathrm{g}-1,\mathrm{}\mathrm{}\mathrm{}{R}_g\left(t|t_g\right)={\overline{W}}_g\left(t|t_g\right)\end{array}$$

(8)

With $\lambda e^{-\lambda (s-t_j)}$ we obtain the solution (see Figure 5, putting the Mean Time To Failure MTTF=θ=123 days, $\lambda =1/\theta$)

$$R_0\left(t\right)=e^{-\lambda t}\left[1+\lambda t+{\left(\lambda t\right)}^2/2!+{\left(\lambda t\right)}^3/3!+{\left(\lambda t\right)}^4/4!\right]$$

(8a)

The reliability system (8) can be written in matrix form,

$$R\left(t-r\right)=\overline{W}\left(t-r\right)+{\int }_r^{t}B\left(s-r\right)R\left(s\right)ds$$

(9)

At the end of the reliability test, at time t, we know the data (the times of the transitions t_j) and the “observed” empirical sample D={x₁, x₂, …, x_g}, where x_j=t_j – t_j-1 is the length between the transitions; the transition instants are t_j = t_j-1  + x_j giving the “observed” transition sample D*={t₁, t₂, …, t_g-1, t_g, t=end of the test} (times of the transitions t_j).

We consider now that we want to estimate the unknown MTTF=θ=1/λ of each item comprising the “associated” stand-by system [24,25,26,27,28,29,30]: each datum is a measurement from the exponential pdf; we compute the determinant $\mathrm{d}\mathrm{e}\mathrm{t}B\left(s\right|r; \theta , D^{*})={(1/\theta )}^g\mathrm{e}\mathrm{x}\mathrm{p}[-T\left(t\right)]$ of the integral system (9), where $T\left(t\right)$ is the “Total Time on Test” $T\left(t\right)=\sum _1^g{x}_i$ [${\mathit{t}}_0$ in the Figure 5]: the “Associated Stand-by System” [25,26,27,28,29,30,31,32,33] in the Statistics books provides the pdf of the sum of the RV X_i of the “observed” empirical sample D={x₁, x₂, …, x_g}. At the end time t of the test, the integral equations, constrained by the constraintD*, provide the equation

$$(\partial lndetB(s|r; \theta , D^{*}))/\partial \theta =\theta /g-T(t)=0$$

(10)

It is important to notice that, in the case of exponential distribution [11,12,13,14,15,16,25,26,27,28,29,30,31,32,33,34,35,36], it is exactly the same result as the one provided by the MLM Maximum Likelihood Method.

If the kernel $b_{j,j+1}\left(s-t_j\right)$ is the pdf $f$($s-t_j$|$\mu ,{\sigma }^2$)$=e^{-{\left(s-t_j-\mu \right)}^2/{(2\sigma }^2)}/\left(\sqrt{2\pi }\sigma \right)$ the data are normally distributed, $X~N\left({\mu }_X,{\sigma }_X^2\right)=e^{-{\left(x-{\mu }_X\right)}^2/\left(2{\sigma }_X^2\right)}/\left(\sqrt{2\pi }{\sigma }_X\right)$, with sample size n, then we get the usual estimator $\overline{X}=\sum X_i/n$ such that $E\left(\overline{X}\right)={\mu }_X$.

of a “4 units Stand-by system” with MTTF=θ=123 days; ${\mathbf{t}}_0$ is the total time on test of the 4 units. To compute the CI (with CL=0.8), find the abscissas of the intersections at ${\mathrm{R}}_0\left({\mathsf{\lambda }}_{\mathrm{L}}{\mathbf{t}}_0\right)=0.9$ and ${\mathrm{R}}_0\left({\mathsf{\lambda }}_{\mathrm{U}}{\mathbf{t}}_0\right)=0.1$ ….

The same happens with any other distribution provided that we write the kernel $b_{i,i+1}\left(s\right)$.

The reliability function $R_0\left(t|\theta \right)$, [formula (8)], with the parameter $\theta$, of the “Associated Stand-by System” provides the Operating Characteristic Curve (OC Curve, reliability of the system) [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36] and allows to find the Confidence Limits (${\theta }_L$ Lower and ${\theta }_U$Upper) of the “unknown” mean $\theta$, to be estimated, for any type of distribution (Exponential, Weibull, Rayleigh, Normal, Gamma, …); by solving, with unknown $\theta$, the two equations $R_0$(${\mathit{t}}_0$|$\theta$)$=1-\alpha /2 and R_0\left({\mathit{t}}_0\right|\theta )=\alpha /2$; we get the two values (${\theta }_L$, ${\theta }_U$) such that

$$R_0\left({\mathit{t}}_{\mathit{o}}|{\theta }_L\right)=\alpha /2 and R_0\left({\mathit{t}}_{\mathit{o}}|{\theta }_U\right)=1-\alpha /2$$

(11)

where ${\mathit{t}}_{\mathit{o}}$ is the (computed) “total of the length of the transitions x_i=t_j - t_j-1 data of the empirical sampleD” and CL=$1-\alpha$ is the Confidence Level. CI=${\theta }_L$^--------${\theta }_U$ is the Confidence Interval: ${\theta }_L=1/{\lambda }_U$ and ${\theta }_U=1/{\lambda }_L$.

For example, with Figure 5, we can derive ${\theta }_L=62.5 days=1/{\lambda }_U$ and ${\theta }_U=200 days=1/{\lambda }_L$, with CL=0.8. It is quite interesting that the book [14] Meeker et al., “Statistical Intervals: A Guide for Practitioners and Researchers”, John Wiley & Sons (2017) use the same ideas of FG (shown in the formula 11) for computing the CI; the only difference is that the author FG defined the procedure in 1982 [26], 35 years before Meeker et al.

2.4. ControlCharts for TBE Data. Some Ideas for Phase I Analysis

Let’s consider now TBE (Time Between Event) data, exponentially or Weibull distributed. Quite a lot of authors (in the “Garden … [24]”) compute wrongly the Control Limits of these CCs.

The formulae, shown in the section “ControlCharts for Process Management”, are based on the Normal distribution (thanks to the CLT; see the excerpts 3, 3a and 3b); unfortunately, they are used also for NON_normal data (e.g. see formulae (1)): for that, sometimes, the NON_normal data are transformed “with suitable transformations” in order to “produce Normal data” and to apply those formulae (above) [e.g. Montgomery in his book].

Sometimes we have few data and then we use the so called “Individual ControlCharts” I-CC. The I-CCs are very much used for exponentially (or Weibull) distributed data: they are also named “rare events ControlCharts for TBE (Time Between Events) data”, I-CC_TBE.

The Jarret data (1979) are used also in the paper (found online, 2024, March 1) [4] (Kumar, Chakraborti et al. with various presence in the “Garden … [24]”) who decided to consider the paper [5] (Zhang et al. also present in the “Garden …]”): they use the first 30 observed time intervals as phase 1 and start the monitoring at m = 31. You find the original data in the Table 1 in the paper [3]; moreover, for the benefit of the readers we provide them in the section 3 “Results”.

It is a very good example for understanding better the problem and the consequences of the difference between PI (Probability Intervals) and the Control Limits, using RIT.

Let’s see what the authors say: Kumar, Chakraborti et al. , Journal of Quality Technology, 2016, present the case by writing (highlight due to FG):

5. An Illustrative Example

In this Section, we use the data from Table 6 of Zhang et al. (2006) (see also Jarrett 1979) in order to illustrate the application of the proposed charting schemes. The data consist of the time intervals in days between explosions in coal mines (i.e., the events) from 15 March 1981 to 22 March 1962 (190 observations in total) in Great Britain. As in Zhang et al. (2006), we consider the first m=30 observations to be from the in-control process, from which we estimate that ${\widehat{\mathit{\lambda }}}_0=0.0081$ (or equivalently, the mean TBE is approximately 123 days). In the sequel, we assume that this is the true in-control value ${\mathit{\lambda }}_0$. Since our numerical analysis showed that r=4 is the best choice we apply the t₄-chart … Thus, the remaining 160 observations are first converted by accumulating a set of four consecutive failure times and the corresponding observations are shown in Table… These are the times until the fourth failure, and are used for monitoring the process in order to detect a change in the mean TBE; an increase (which means process improvement) or a decrease (process deterioration).

Excerpt 5. From Kumar, Chakraborti et al., “Journal of Quality Technology”, 2016.

In the paper of Zhang et al. (2006) we read:

The second example presented here uses real data taken from Jarrett (1979). The data set consists of time intervals in days between explosions in coal mines from March 15, 1851 to March 22, 1962. The data are reproduced in Table 6. We first establish the ARL-unbiased exponential chart with the first 30 observations using the proposed approach. The phase I analysis is presented in Table …. Then the established control limits are used to monitor the subsequent data, i.e., from number 31 to number 190. The resultant ARL-unbiased exponential chart for all the data is plotted in Fig…., where the dotted lines represent the estimated control limits which stop updating at point number 30 and the continuing straight lines represent the established control limits. It is revealed, surprisingly, that the mean of the time intervals between explosions remained constant for a very long period of time (about 40 years), and the accident rate only started to decrease sometime after the 125th explosion. There is an alarm at data point number 80 (value = 0), which is considered here a false alarm.

Excerpt 6. From Zhang et al. (2006), “IIE Transactions”.

Notice that both the papers [4,5] are (and were) present in the “Garden … [24]”.

Zhang et al., 2006, compute the Control Limits from the first 30 data and find LCL=0.268 and UCL=1013.9 (you can see them in their Table 7, that is “our” excerpt 11).

All the data [30+40 t₄] are very interesting for our analysis; we recap the two important points, given by the authors (Kumar et al.):

… first m=30 observations to be from the in-control process, from which we estimate … the mean TBE approximately, 123 days; we name it θ₀.

… we apply the t₄-chart… Thus, … converted by accumulating a set of four consecutive failure times … the times until the fourth failure, used for monitoring the process to detect a change in the mean TBE.

The 3 authors (Kumar, Chakraborti et al.) state: “… the control limits … t4-chart are seen to be equal to LCL=63.95, UCL=1669.28 with CL (Centre Line)=451.79.”

Notice that the authors Zhang et al. and Kumar, Chakraborti et al. find different Control Limits to be used for monitoring the same process: a very interesting situation; the reason is that they use “different” statistics in Phase II.

The FG findings for the Phase I, using the first 30 data, compute different Control Limits with RIT: RIT solves the I-CC_TBE with exponentially distributed data as those of Table 1, considered by Zhang et al. and Kumar et al.

In the previous section, we computed the CI=${\theta }_L$^--------${\theta }_U$ of the parameter $\theta$, using the (subsample) “transition times durations”: ${\mathit{t}}_{\mathit{O}}$=“total of the transition times durations (length of the transitions x_i=t_j - t_j-1 data) in the empirical sample (subsample with n=4 only, as an example)” and Confidence Level CL=$1-\alpha$.

When we deal with a I-CC_TBE we compute the LCL and UCL of the mean θ through the empirical mean ${\overline{\mathit{t}}}_{\mathit{O}}={\mathit{t}}_{\mathit{O}}/\mathrm{n}$ of each transition, for the n=30 data (Phase I of Zhang et al. and Kumar et al.); we solve the two following equations (12) for the two unknown values LCL and UCL, for $R\left({\overline{\mathit{t}}}_{\mathit{O}} |\theta \right)$ of each item in the sample, similar to (11)

$$R\left({\overline{\mathit{t}}}_{\mathit{O}}|LCL\right)=\alpha /2, R\left({\overline{\mathit{t}}}_{\mathit{O}}|UCL\right)=1-\alpha /2$$

(12)

where now ${\overline{\mathit{t}}}_{\mathit{O}}={\mathit{t}}_{\mathit{O}}$/n is the “mean, to be attributed, to the single lengths of the single transitions x_i=t_j-t_j-1 data in the empirical sampleD with the Confidence Level CL=$1-\alpha$: $LCL=1/{\lambda }_U$ and $UCL=1/{\lambda }_L$.

In the next sections we can see the Scientific Results found by a Scientific Theory (we anticipate them: the Control Limits are LCL=18.0 days and UCL=88039.3 days).

3. Results

In these sections we provide the scientific analysis of the Jarret data [3] and compare our result with those of Chakraborty [4,5]: the findings are completely different and the decisions, consequently, should be different, with different costs of wrong decisions.

3.1. ControlCharts for TBE Data. Phase I Analysis

The Jarret data are in the Table 1.

Excerpt 7. The CC of the 190 data from “Improved Shewhart-TypeCharts for Monitoring Times Between Events”, Journal of Quality Technology, 2016, (Kumar, Chakraborti et al): the first 30 are used to find the Control Limits for the other 40 t₄ (time between “4 failures”: 4*40=160).

Excerpt 8. The CC of the 190 data [named by the authors “Figure 7. ARL-unbiased exponential chart for the coal mining data”] from “Design of exponential control charts using a sequential sampling scheme”, IIE Transactions, (Zhang et al., 2006) [the first 30 data are used to find the Control Limits].

Notice that the authors Zhang et al. and Kumar, Chakraborti et al. find different Control Limits to be used for monitoring the same process: a very interesting situation; the reason is that they use “different” statistics in Phase II. The results are in the excerpts 7 and 8.

For exponentially distributed data (12) becomes (13) [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33], k=1, with CL=$1-\alpha$

$$e^{-[{\overline{\mathit{t}}}_{\mathit{O}}/\mathit{L}CL]}=1-\alpha /2 \mathrm{and} e^{-[{\overline{\mathit{t}}}_{\mathit{O}}/\mathit{U}CL]}=\alpha /2$$

(13)

The endpoints of the CI=$LCL$^--------$UCL$ are the Control Limits of the I-CC_TBE.

This is the right method to extract the “true” complete information contained in the sample (see the Figure 9).

The Figure 9 is justified by the Theory [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33] and is related to the formulae [(12), (13) for k=1], for the I-CC_TBE charts.

Remember the book Meeker et al., “Statistical Intervals: A Guide for Practitioners and Researchers”, John Wiley & Sons (2017): the authors use the same ideas of FG; the only difference is that FG invented 30 years before, at least.

Compare the formulae [(13), for k=1], theoretically derived with a sound Theory [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33], with the ones in the Excerpt [in the Appendix C (a small sample from the “Garden … [24]”)] and notice that the two Minitab authors (Santiago&Smith) use the “empirical mean ${\overline{\mathit{t}}}_{\mathit{O}}$” in place of the ${\theta }_0$ in the Figure 1: it is the same trick of replacing $\stackrel{̿}{x}$ to the mean μ which is valid for the Normal distributed data only; e.g., see the formulae (1)!

Analysing the first 30 data of the two articles (Zhang et al. and Kumar, Chakraborti et al.) we get a total ${\mathit{t}}_{\mathit{O}}$=3568 days and a mean ${\overline{\mathit{t}}}_{\mathit{O}}={\mathit{t}}_{\mathit{O}}$/30=118.9 days; notice that it is rather different from the value 123 computed by (Kumar, Chakraborti et al.). Fixing α=0.0027, with RIT we find the CI$=${72.6$={\theta }_L$^--------${\theta }_U=220.4$} for the parameter $\theta$, and the Control Limits LCL=18.0 days and UCL=88039.3 days.

Compare these with the LCL_Zhang=0.268 and UCL_Zhang=1013.9 (Zhang et al., 2006) and the “LCL_Kumar=63.95, UCL_Kumar=1669.28 with Centre_Line_Kumar=451.79” (Kumar, Chakraborti et al., for the t₄-chart).

Quite big differences… with profound consequences on the decision on the states IC or OOC of the process; the Figure 1, and the “scientific” formula (13) justify our findings.

Now we try to explain why those authors (Zhang et al. and Kumar, Chakraborti et al.) got their results.

In Zhang et al. (“Design of exponential control charts using a sequential sampling scheme”, IIE Transactions); at page 1107, we find the values L_u and U_u versus LCL_Zhang and UCL_Zhang

We do not know the cause of the “little” difference.

The interesting point is that with these Control Limits the Process “appears” IC (In Control), for the first m=30 observations, Phase I; see the excerpt 11.

So, one is induced to think that the mean ${\overline{\mathit{t}}}_{\mathit{O}}={\mathit{t}}_{\mathit{O}}$/30=118.9 days can be used to find the λ₀=1/118.9 for using the Control Limits in the Phase II (the next 160 data) (see the excerpt 12, with the words “plugging into …”).

Notice that the formulae in excerpt 9 are very similar to those in Appendix C (related to [24])”.

This fact generates an IC which is not real. See the Figure 9.

The analysis of the first 30 data show that three possible distributions can be considered, using Minitab 21: Weibull, Gamma, and Exponential; since the Weibull is acceptable, we could use it. See the Table 2.

α=0.0027, α*=0.00372, γ_α*=1.2927.

Excerpt 9. Formulae for the Control Limits for the first 30 data, IIE Transactions, (Zhang et al., 2006).

1000.

Anyway, for comparison with Zhang et al., 2006, we use the exponential distribution, which is not the “best” to be considered: as you can see the Process is OOC, with 7 points below the LCL.

Therefore, these data should be discarded for the computation of λ₀.

Hence, the Control Limits (Zhang et al., 2006), based on the estimate “assumed as true” λ₀=0.0081

cannot be used for the next 160 data.

Is the statement (assumption!) “… first m=30 observations to be from the in-control process…” sound?

NO!

By formulae (13) we find the Figure 9 that proves that the process is OOC.

Using the formulae in the Excerpt 9, those authors do not extract the maximum information from the data in the Process Control.

Before ending this section, let’s see what MINITAB, which use the ideas of Santiago & Smith, provides us in Phase I (Figure 7a)

Notice that JMP (using the ideas of Santiago & Smith), provides us in Phase I the same type of information (Figure 7b)

For the software Minitab, the process is IC, the same as Zhang et al. and Kumar et al.; same result could have been found by JMP (Appendix B) and SAS, and all the authors in the “Garden [24]”…

3.2. ControlCharts for TBE Data. Phase II Analysis

We saw in the previous section what usually it is done during the Phase I of the application of CCs: estimation of the mean and standard deviation; later, their values are assumed as “true known” parameters of the data distribution, in view of the Phase II.

In particular, for TBE individual data the exponential distribution is assumed with a known parameter λ₀ or θ₀.

We consider now what it is done during the Phase II of the application of CCs for TBE data individual exponentially distributed.

We go on with the paper “Improved Shewhart-TypeCharts for ….”, Journal of Quality Technology, 2016, (Kumar, Chakraborti et al. with various presence in the “Garden …”) who analysed the Jarret data. In their paper we read:

Thus we focus on Shewhart-type TBE charts and try to improve their performance. The process is said to be in-control (IC) when λ=λ₀, where λ₀ is the given (known) or specified value of the failure rate. It should be noted here that when λ₀ is not known from previous knowledge, it has to be estimated from a preliminary IC sample. In the sequel, we assume that λ₀ is known or that it has been estimated from a (sufficiently) large Phase I sample. Omissis….. Therefore, following Kumar and Chakraborti (2014), the UCL and LCL given in (2) can be more conveniently rewritten

Excerpt 10. From “Improved … Monitoring Times Between Events”, J. Quality Technology, ‘16.

They combining 4 data to generate a t₄ chart giving the formulae in the excerpt 10 (with r in place of 4; notice the authors mentioned… ).

Notice the formulae: the mentioned authors provide their LCL and UCL which are actually the Probability Limits L and U of the Probability Interval (PI) and NOT the Control Limits of the Control Chart, as it is easily seen By using the Theory of CIs (Figure 1 and Figure 5).

All the Jarret data [30+40 t₄] are very interesting for our analysis; we recap the two important points, given by the authors (Chakraborti et al.):

… first m=30 observations to be from the in-control process, from which we estimate … the mean TBE approximately, 123 days; we name it θ₀.

… we apply the t₄-chart… Thus, … converted by accumulating a set of four consecutive failure times … the times until the fourth failure, used for monitoring the process to detect a change in the mean TBE.

The 3 authors (Chakraborti et al.) state: “… the control limits … t4-chart are seen to be equal to LCL=63.95, UCL=1669.28 with CL (Centre Line)=451.79”, named by them “ARL-unbiased {1/1, 1/1}”.

The 3 authors (Chakraborti et al.) state also: “… the control limits … t4-chart are seen to be equal to LCL=217.13, UCL=852.92 with CL (Centre Line)=451.79”, named by them “ARL-unbiased {M:3/4, M:3/4}”.

Dropping out the OOC data (from the first 30 observations), in Phase I, with RIT, we find that now the process is IC: the distribution fitting the remaining data is the Weibull with parameters η=140.6 days and β=1.39; since the CI of the shape parameter is 0.98^-----2.15, with CL=90%, we can assume β=1 (exponential with θ=127.9); therefore, we have the “true” LCL=18.6, quite different from the LCLs of the authors (Chakraborti et al.).

Considering the 40 t₄ data, the distribution fitting the data is the Weibull with parameters η=990.2 days and β=1.18; since the 1∈CI of the shape parameter, with CL=90%, we can assume β=1 (exponential with θ=924.5); therefore, we have the “true” LCL=72.9 and UCL=1987, quite different from the Control Limits of the authors (Kumar, Chakraborti et al.): hence there is a profound consequence on the analysis of the t4-chart; see the Figure 12.

with

Excerpt 11. From “Improved … Monitoring Times Between Events”, J. Quality Technology, ‘16.

Considering, on the contrary, the value θ=127.9 from the first [<30] INDIVIDUAL observations of the IC process, transformed into the one for the t₄ chart, we have the “true” LCL=40.38 and UCL=1100.37; so, we have 4 LCLs and 4 UCLs (see Figure 12).

The 3 authors (Kumar, Chakraborti et al.) show the formulae for the Control Limits (excerpt 11):

Now we face a problem: in which way could the 3 authors (Kumar, Chakraborti et al.) compute their Control Limits from the individual first m=30 observations?

FG (in spite of excerpt 11) did not find the way: is θ₀=1/λ₀ the value estimated from the first m=30 observations? He suspects that they used a trick (not shown): they use the fist 30 data to find θ₀=123 days (from the total t₀ of days in the Phase I) and then, they consider the total as though it were 4t₀, i.e. 30 t₄, to find LCL and UCL for the t₄-chart … We did it, but we could not find them…

Doing that, Chakraborti et al. missed the fact that the process, in the individual first m=30 observations was OOC and they should not use the ″λ₀ specified value of the failure rate, that has to be estimated from a preliminary IC sample.″

Since the data are assumed (by the 3 authors) Exponentially distributed it follows from the Theory that “… by accumulating a set of four consecutive failure times (times until the fourth failure)…” we should find the t₄ data (determinations of the RV T₄) Erlang distributed. Since k=4 we could expect that the CLT could apply and find that the 40 t₄ data (determinations of the 40 RV T₄) follow “approximately” the Normal distribution.

Considering the 40 t₄ data and searching for the distribution, we are disillusioned: the distributions Normal, Lognormal, Exponential, Gamma, Smallest Extreme, Largest Extreme, Logistic, Loglogistic do not fit the data; only the Weibull, and the Box-Cox and Johnson transformations seem adequate… See the Gamma (Erlang) fitting in the Figure 8.

Therefore, the authors formulae for the Control Limits are inadequate in three ways, because

the gamma (Erlang) distribution does not apply, with CL=95%

then, the formulae in the excerpt 11 cannot be applied.

the formulae, in their paper, $\mathit{P}\left[{\mathit{T}}_{\mathit{r}}>\mathit{U}\mathit{C}\mathit{L}\right]={\mathit{\alpha }}_0/2$ and $\mathit{P}\left[{\mathit{T}}_{\mathit{r}}<\mathit{L}\mathit{C}\mathit{L}\right]={\mathit{\alpha }}_0/2$ are generated by the confusion (of the authors) between LCL and L and UCL and U, as you can see in the Figure 9, based on the non-applicable Gamma distribution; you see the vertical line intercepting the two probability lines in the points L and U such that $\mathit{P}\left[{\mathit{T}}_{\mathit{r}}>\mathit{U}\right]={\mathit{\alpha }}_0/2$ and $\mathit{P}\left[{\mathit{T}}_{\mathit{r}}<\mathit{L}\right]={\mathit{\alpha }}_0/2$ versus the horizontal line, at ${\overline{\mathit{t}}}_0$, intercepting the two lines at LCL and UCL.

It is clear that the two intervals, L^-----U and LCL^-----UCL, are different and have different definitions, meaning and length, through the Theory [6,7,8,11,12,13,14,15,16,25,26,27,28,29,30,31,32,33,34,35,36]. Notice, in the Figure 9, the logarithmic scale for both axes (to have readable intervals).

It should be noted that we drew two horizontal lines, one at ${\overline{\mathit{t}}}_0$ (the experimental mean) and the other at ${\mathit{\theta }}_0$ (the assumed known mean) to show the BIG difference between the interception points: we showed the TRUE LCL and UCL; the reader can guess that the WRONG Control Limits (not shown in the figure) have quite different values from the TRUE Control Limits.

and at ${\mathsf{\theta }}_0$ ….

The reader is humbly asked to be very attentive in the analysis of the Figure 9: FG thanks him!

Using the original 160 data, divided in 40 samples of size 4, we could compare the estimates with those found by the 40 t₄ (in the paper “Improved … for Monitoring TBE”, Journal of Quality Technology, 2016”); you see them in the Table 3.

Notice that the Figure 10a shows the same behaviour as the Phase I figure; it would be interesting to understand IF, with that, the authors would have been able to find an ARL=370: the process is Out Of Control both considering the first individual 30 data and the last individual 160 data ….

How many OOC are in the Jarret data?

Analysing the 40 t4 data with Minitab (which uses the Santiago&Smith formulae, Appendix C) we get Figure 11a, and with JMP we get Figure 11b;

See the authors Acknowledgement (in the paper)….

Notice that the OOC points in the Figure 10 disappear when we plot the 40 t4, as you can see in the Figure 12, where we draw 4 LCLs and 4 UCLs, from Kumar, Chakraborti et al. analysis and FG analysis (with RIT). We have the Figure 12.

Notice theFigures 11a and 11b. Compare it with Figure 12. What can we deduce from this analysis?

That the Method is fundamental to draw sound decisions.

Which is the best Method? Only the one which is Scientific, based on a sound Theory. Which one?

From the Theory [6,7,8,11,12,13,14,15,16,25,26,27,28,29,30,31,32,33,34,35,36] we know that we must assess the “true (to be used in the Phase II)” Control Limits from the data of an IC Process in the Phase I: therefore, only LCL_G1=40.38 and UCL_G1=1100.37 are the Scientific Control Limits; you can compare with the others in Figure 12.

From Figure 12 we see that the first 20 t₄ have a mean lower than the last 20 t₄: the mean time between events increased with calendar time. We can assess that by computing the two mean values: their ratio is 3.18 and we can see if it is significant and we find that it is so with CL=0.9973 (0.0027 probability of being wrong).

See alsoFigures 13a and 13b.

FromFigures 11a, 11b, 12, 13a and 13b we saw that the mean time between explosions was changing with time: it became larger (improved); a method that should show better this behaviour is the EWMA Control Chart. We do not analyse this point; also, for this chart there is the problem of the confusion between the intervals L^-------U and LCL^-------UCL.

4. Discussion

We decided to use the Jarrett (1979) data in Table 1, from the paper [3] and the analysis by Kumar, Chakraborti, Rakitzis, (2017) in the Journal of Quality Technology [4] and that by Zhang, Xie, Goh (2006) in the IIE Transactions [5], (papers that you can find also in the “Garden of flowers” [24] and in the Appendix C).

We got different results from those authors: the cause is that they use the Probability Limits of the PI (Probability Interval) as they were the Control Limits (so named by them) of the ControlCharts.

The proof of the confusion between the intervals L^-------U (Probability Interval) and LCL^-------UCL (Confidence Interval) in the domain of ControlCharts (for Process Management) highlight the importance and novelty of these ideas in the Statistical Theory and in the applications.

For the “location” parameter in the CCs, from the Theory, we know that two mean ${\mu }_{\overline{X}\left(t_q\right)}$ (parameter), q=1,2, …, n, and any other mean ${\mu }_{\overline{X}\left(t_r\right)}$(parameter), r=1,2, …, n, are different, with risk α, if their estimates are not both included in their common Confidence Interval as the CI of the grand mean ${\mu }_{\stackrel{̿}{X}}=\mu$ (parameter) is.

Let’s consider the formula (4) and apply it to a “Normal model” (due to CLT, and assuming known variance), sequentially we can write the “real” fixed interval L^----U comprising the RV $\stackrel{̿}{X}$ (vertical interval) and the Random Interval comprising the unknown mean $\mu$ (horizontal interval) (Figure 14)

$$P\left[L=\mu -{\displaystyle \frac{\sigma z_{1-\frac{\alpha }{2}}}{\sqrt{k}}}\le \stackrel{̿}{X}\le \mu +{\displaystyle \frac{\sigma z_{1-\frac{\alpha }{2}}}{\sqrt{k}}}=U\right]=P\left[\stackrel{̿}{X}-{\displaystyle \frac{\sigma z_{1-\frac{\alpha }{2}}}{\sqrt{k}}}\le \mu \le \stackrel{̿}{X}+{\displaystyle \frac{\sigma z_{1-\frac{\alpha }{2}}}{\sqrt{k}}}\right]$$

(14)

When the RV $\stackrel{̿}{X}$ assume its determination (numerical value) $\stackrel{̿}{x}$ (grand mean) the Random Interval becomes the Confidence Interval for the parameter μ, with CL=1-α: risk α that the horizontal line does not comprise the “mean” μ.

This is particularly important for the Individual ControlCharts for Exponential, Weibull and Gamma distributed data: this is what Deming calls “Profound Knowledge (understanding variation)” [9,10]. In this case, the Figure 14 looks like the Figure 1, where you see the Confidence Interval, the realisation of the horizontal Random Interval.

The case we considered shows clearly that the analyses, in the Process Management, taken so far have been wrong and the decisions have been misleading, when the collected data follow a Non-Normal distribution [24].

Since a lot of papers (related to Exponential, Weibull and Gamma distributions), with the same problem as that of “The garden of flowers” [24], are published in reputed Journals we think that the title “History is written by the winners. Reflections on ControlCharts for Process Control” is suitable for this paper: the authors of the wrong papers [24] are the winners.

.

Our study is limited to the Individual ControlCharts with Exponentially, Weibull and Gamma distributed data.

Further studies should consider other distributions which cannot be transformed into the three above distributions: Exponential, Weibull and Gamma.

5. Conclusions

With our figures (and the Appendix C, that is a short extract from the “Garden … [24]”) we humbly ask the readers to look at the references [1-57] and find how much the author has been fond of Quality and Scientificness in the Quality (Statistics, Mathematics, Thermodynamics, …) Fields.

The errors, in the “Garden … [24]”, are caused by the lack of knowledge of sound statistical concepts about the properties of the parameters of the parent distribution generating the data, and the related Confidence Intervals. For the I-CC_TBE the computed Control Limits (which are actually the Confidence Intervals), in the literature are wrong due to lack of knowledge of the difference between Probability Intervals (PI) and Confidence Intervals (CI); see the Figure 17 (remembering also the Figure 14 and Figure 1). Therefore, the consequent decisions about Process IC and OOC are wrong.

We saw that RIT is able to solve vario[1–57us problems in the estimation (and Confidence Interval evaluation) of the parameters of distributions for ControlCharts. The basics of RIT have been given.

We could have shown many other cases (from papers not mentioned here, that you can find in [22,23,24]) where errors were present due to the lack of knowledge of RIT and sound statistical ideas.

Following the scientific ideas of Galileo Galilei, the author many times tried to compel several scholars to be scientific (Galetto 1981-2025). Only Juran appreciated the author’s ideas when he mentioned the paper “Quality of methods for quality is important” at the plenary session of EOQC Conference, Vienna. [1]

For the control charts, it came out that RIT proved that the TCharts, for rare events and TBE (Time Between Events), used in the software Minitab, SixPack, JMP or SAS are wrong. So doing the author increased the h-index of the mentioned authors who published wrong papers.

RIT allows the scholars (managers, students, professors) to find sound methods also for the ideas shown by Wheeler in Quality Digest documents.

We informed the authors and the Journals who published wrong papers by writing various letters to the Editors…: no “Corrective Action”, a basic activity for Quality has been carried out by them so far. The same happened for Minitab Management. We attended a JMP forum in the JMP User Community and informed them that their “ControlCharts for Rare Events” were wrong: they preferred to stop the discussion, instead to acknowledge the JMP faults [56,57].

So, dis-quality continues to be diffused people and people continue taking wrong decisions…

Deficiencies in products and methods generate huge cost of Dis-quality (poor quality) as highlighted by Deming and Juran. Any book and paper are products (providing methods): their wrong ideas and methods generate huge cost for the Companies using them. The methods given here provide the way to avoid such costs, especially when RIT gives the right way to deal with Preventive Maintenance (risks and costs), Spare Parts Management (cost of unavailability of systems and production losses), Inventory Management, cost of wrong analyses and decisions.

Figure 15. Probability Intervals L^-----U versus Confidence Intervals LCL^-----UCL in ControlCharts.

Figure 15. Probability Intervals L^-----U versus Confidence Intervals LCL^-----UCL in ControlCharts.

We think that we provided the readers with the belief that Quality of Methods for Quality is important.

The reader should remember the Deming’s statements and the ideas in [6-57].

Unfortunately, many authors do not know Scientifically the role (concept) of Confidence Intervals (Appendix B) for Hypothesis Testing.

Therefore, they do not extract the maximum information form the data in the Process Control.

Control Charts are a means to test the hypothesis about the process states, H₀={Process In Control} versus H₁={Process Out Of Control}, with stated risk α=0.0027.

We have a big problem about Knowledge: sound Education is needed.

We think that the Figure 16 conveys the fundamental ideas about the need of Theory for devising sound Methods, to be used in real applications in order to avoid the Dis-quality Vicious Circle.

Humbly, given our commitment to Quality and our long-life love for it [1-57], we would venture to quote Voltaire: [6–57

“It is dangerous to be right in matters on which the established men are wrong.” because “Many are destined to reason wrongly; others, not to reason at all; and others, to persecute those who do reason.” So, “The more often a stupidity is repeated, the more it gets the appearance of wisdom.” and “It is difficult to free fools from the chains they revere.”

Figure 16. Knowledge versus Ignorance, in Tools and Methods.

Figure 16. Knowledge versus Ignorance, in Tools and Methods.

Let’s hope that Logic and Truth prevail and allow our message to be understood (Figure 15 and Figure 16).

The objective of collecting and analysing data is to take the right action. The computations are merely a means to characterize the process behaviour. However, it is important to use the right Control Limits take the right action about the process states, i.e., In Control versus Out Of Control.

On July-August 2024 we again verified (through Six????? new downloaded papers) that the Pandemic Disease about the (wrong) Control Limits, that are actually the Probability Limits of the PI is still present (notice the Journals):

Zameer Abbas et al., (30 June 2024): “Efficient and distribution-free charts for monitoring the process location for individual observations”, Journal of Statistical Computation and Simulation,

Marcus B. Perry (June 2024) [University of Alabama 674 Citations] “Joint monitoring of location and scale for modern univariate processes”, Journal of Quality Technology.

E. Afuecheta et al., (2023) “A compound exponential distribution with application to control charts”, Journal of Computational and Applied Mathematics [the authors use data of Santiago&Smith (Appendix C) and erroneously find that the UTI process IC].

N. Kumar (2019), “Conditional analysis of Phase II exponential chart for monitoring times to an event”, Quality Technology & Quantitative Management

N. Kumar (2021), “Statistical design of phase II exponential chart with estimated parameters under the unconditional and conditional perspectives using exact distribution of median run length”, Quality Technology & Quantitative Management

S. Chakraborti et al. (2021), “Phase II exponential charts for monitoring time between events data: performance analysis using exact conditional average time to signal distribution”, Journal of Statistical Computation and Simulation

Other papers with the same problem have been downloaded from June 2024 to December 2024…

There will be any chance that the Pandemic Disease ends? See the Excerpt 12: notice the (ignorant) words “plugging into …”. The only way out is Knowledge… (Figure 16): Deming’s [7,8] Profound Knowledge, Metanoia, Theory.

Excerpt 12. From “Conditional analysis of Phase II exponential chart… an event”, Q. Tech. & Quantitative Mgt, ’19.

LCL=0.001

Notice the LCL, the Mean and the UCL of both charts.

Compute the mean of all the data and you find a different value: therefore, the mean in the charts is not the mean of the process!

If one analyses the data with Minitab, he finds the Figure A3.

You see that now the UTI process is IC: notice the LCL, Mean and UCL.

A natural question arises: which of the three figures is correct?

Actually, they all are wrong, as you can see from the Figure A4:

The author offered JMP to become a better statistical software provider by solving the flaw according to JMP advertising:

Our Purpose

Our purpose is to empower scientists and engineers via our statistical discovery software. That’s pretty straightforward, and it’s never wavered. Sometimes we work directly with the data explorers themselves, sometimes with their companies, and other times with colleges and universities so that the next generation of scientists and engineers will count on JMP by the time they enter the workforce.

Although our purpose is clearly connected to our software, it doesn’t end there. As an employer, JMP purposefully creates and maintains a culture of camaraderie that allows the personalities of our diverse and inclusive employee base to shine through. JMP is an equal opportunity employer.

And as inhabitants of this planet, we intentionally measure and work to mitigate our impact on the world. Explore our Data for Green program to see how we’re working to empower other organizations to make a difference too.

The catchphrase “corporate social responsibility” could be used for a lot of what we do for our employees, communities, education and the Earth. But starting with JMP founder John Sall, we just try to do the right thing.

No reaction … and therefore NO Corrective Action.