I would agree that most of the Six Sigma world has left its quality roots behind, chasing cost cutting and forgetting Deming, SPC, the Taguchi Loss Function and its implications (which concepts are what the whole idea is, after all, based on). I believe that one reason for the emphasis on enumerative study methods is because they are, unfortunately, what are taught in most statistics courses at the high school and university level. Hopefully, some of the research in the list of papers I provided above will shake the enumerative statistics world enough to effect some change.

There are very few textbooks outside the sort of "deep quality" realm that even discuss analytic or enumerative studies. To advance the notion of analytic studies--and to see it survive us--what is probably needed is some sort of analytic statistics society with enough academic credibility to publish its own journal. 

As far as the 1.5-sigma shift goes, it doesn't exist...despite it's presence in international standards as a "convention." In the real world, assuming some reasonable attempt at statistical process control, a process whose data display a sustained but undetected 1.5 sigma shift CANNOT exist. Platitudes such as "Shift Happens" are cute but do not reflect the real world. 

I would refer the reader to Wheeler and Chambers' Understanding Statistical Process Control for an excellent example of the use of control charts to solve the problem you have posed. Through the use of rational subgrouping, the manager of that process was able to look at hour-to-hour, cycle-to-cycle and cavity-to-cavity variation. Cavity-to-cavity variation turned out to be the greatest driver of variation. The manager used the charts (and the process knowledge of his molding machine operators) to reduce between-cavity variation, and then monitor it using a dynamic and very sensitive method. 

I would say that you don't need hypothesis tests for comparing two processes ro process streams.  In my 35 yers of solving hundreads if not thousands of complex (physics based) problems, certainly executing thousands of process characterizations (emperical models that characterize teh design space of a set of inputs to critical outputs), capabilty studys, Measurement Systems analyses, V&V test plans I have never calculated a p value or performed any tests of statistical significance.  If the study is designed appropriately the visual evidence available in a properly design graph will display all of the evidence you need.  On teh other hand, I've seen hundreds of statistical tests that resulted in a p value less .05 and when I looked at the study design and graphed the resutl it was clear that the 'statistical significance had no practical importance or was simply incorrect becuase the process itself was non homogenous.  Many statisticians - not just Deming or Wheeler - have demonstrated this.  

A great paper to introduce yourself to the enumerative vs analytic study question is Deming's seminal paper: “On Probability as a Basis for Action”, American  Statistician,  November 1975, Vol. 29, No. 4, pp. 146-152, available for free at:  https://www.deming.org/media/pdf/145.pdf

A great paper to introduce yourself to critical questioning of the usefulness of the 'hypothesis test' is "The Null Ritual - What you always wanted to know about significance testing but were afraid to ask" by Gerd Gigerenzer, Stefan Krauss, and Oliver Vitouch.  Published in: D. Kaplan (Ed.). (2004). The Sage handbook of quantitative methodology for the social sciences (pp. 391–408). Th ousand Oaks, CA: Sage. © 2004 Sage Publications.  Available for free at http://library.mpib-berlin.mpg.de/ft/gg/gg_null_2004.pdf

 Some other papers you might find interesting:

“The Insignificance of Statistical Significance Testing”, Johnson, Douglas H., Journal of Wildlife Management,  Vol. 63, Issue 3, pp. 763-772, 1999 http://www.ecologia.ufrgs.br/~adrimelo/lm/apostilas/critic_to_p-value.pdf

“The Case Against Statistical Significance Testing”, Carver, Ronald P., Harvard Educational Review, Vol 48, Issue 3, pp 378-399, 1978 http://healthyinfluence.com/wordpress/wp-content/uploads/2015/04/Carver-SSD-1978.pdf

“What Statistical Significance Testing Is and What It Is Not”, Shaver, James P., Journal of Experimental Education, No.61, pp. 293-316, 1993 

Thank you for your kind words Kay.

In the earliest reference for which I am aware, Deming proposed the differences between the two types of studies in his seminal work on curve-fitting, Some Theory of Sampling. You can still find Dover editions of that book. 

Deming stated that the difference between the two types of studies is the intent. Essentially, enumerative studies deal with a population. You are sampling (collecting statistics) from the population to estimate its parameters, with the intention of taking action on some aspect of the population. The data are static: At least in principle you know every member of the population and can attribute any characteristics of those members to the moment in time when you drew the sample. It is like examining a snapshot with a lot of holes in it, using what you can glean from the information you have to fill in those holes and estimate what the whole picture would look like.

Analytic studies are intended for action on a cause system. They are not static, but dynamic. In an analytic study there is no population of interest. We are trying to sample from the past or present, and studying those data to try to extrapolate into the future. It's a little like watching a movie and looking at what's causing the current actions on the screen, and using that information to try to figure out what will happen next. 

Here's where it gets tricky (example courtesy of David Kerridge): In Out of the Crisis, Deming used an example to illustrate operational definitions. It was a destructive testing example to test whether a blanket was 50% wool. In the operational definition, he outlines a test procedure where the analyst punches 10 holes one inch in diameter, centered by random numbers, then tests the content of that sample. If the sample proves to be greater than 50% wool (plus or minus 2% if I remember right), then the blanket can be considered to be "50% wool."

So, that test would be an enumerative study...the "population" is all the fabric in the blanket, the sample comprises the circles punched from the wool, and we are extrapolating from those circles to the remainder of the blanket. It would be appropriate, in this case, to compute at least a confidence interval around the average wool content from the sample. 

However suppose this test were conducted at some regular time interval...say, once or twice per day. If we use the numbers to run an averages and ranges chart (or an individuals and ranges chart) to get the average and control limits over time (to look for special causes or trends), we have used our enumerative study results to get data for an analytic study. If we were doing that, then we would just get the average wool content from each sample, and not run any sort of confidence interval or t-test. 

Hopefully this helps, Mohamed. If you've read the rest of this discussion, you can see that there is (lamentably) little in the literature any more about this topic.