+. 01-OCT-2001 +( probabilities Pr(A).and.Pr(B) = Pr(A)*Pr(B) Pr(A).or.Pr(B) = Pr(A)+Pr(B)-Pr(A)*Pr(B) ~ Pr(A)+Pr(B) +( error propagation in calculations What is the error propagation in the RATE calculation ? Example : RATE = (COST+/-5%) divided by (HOURS+/-20%) The normal rule applied is the sum of the percentage errors RATE=COST/HOURS +/-25% Actually the true maximum error is just over 26%, but the usual rule is just to add percentages when multiplying or dividing. But when dividing, to be fully accurate, you can divide the sum by (1 - (square of error in denominator)) above: (.05 + .2)/(1 - .2*.2) = .25/.96 = .26 or 26% (When multiplying it is fuly accurate to just add the percentage errors) For additive/subtractive the errors always add. Not the percentage errors though, just the absolute erros. So, for example: 80 [+/-7] + 20 [+/-2] is 100 [+/-9] +( BETA Distribution From: "Clarence Maday" Date: Mon, 11 Mar 2002 08:56:53 -0500 Subject: Re: [tocexperts] New file uploaded to tocexperts Generalized Beta distribution: a = min b = max m = mode r = (b-m)/(m-a) From Arena, the distribution is a + (b-a)Beta(p,q) where p = (4+3r+r^2)/(1+r^2) q = (1+3r+4r^2)/(1+r^2) Is this what you are looking for? +( Benford's Law This week I'd like to talk about an astonishing mathematical phenomenon called "Benford's Law". It's not directly related to ASIT or inventive thinking but is very interesting and inspiring, so I felt obliged to report on this little known mathematical phenomenon. Consider any list of numbers that was obtained from the financial records of a corporate, or from geographic, scientific and demographic data.It comes as a great surprise that, if the numbers under investigation are not entirely random but somehow socially or naturally related, the distribution of the first digit is not uniform but the following 1 will be the first digit about 30% of cases, 2 will come up in about 18% of cases, 3 in 12%, 4 in 9%, 5 in 8%, etc. For the more mathematically inclined - the first digit, D, appears with the frequency proportional to log (1 + 1/D).This is known as Benford's Law. The astonishing fact is that this law is correct for ANY list of meaningful numbers that are socially or naturally related. It also astonishing that none disputes it or offers a competing law related to digits. The law was discovered by the American astronomer Simon Newcomb in 1881 who noticed that the first few pages of his logarithm tables books were more worn than the last few and from this he surmised that he was consulting the first pages-which gave the logs of numbers with low digits-more often In 1938, Frank Benford arrived at the same formula after a comprehensive investigation of listings of data covering a variety of natural phenomena. In 1961 Roger Pinkham discovered an interesting property of the Benford's probabilities.It turned out that these probabilities (i.e. 30%, 18%, 12%, in 9%, 8%, etc) are scale invariant. In other words, if a set of numbers followed Benford's law closely, and if all the numbers in the set were multiplied by a nonzero constant (such as 22.04 or 0.323), then the new set of numbers would also follow Benford's law closely. Only the probabilities of Benford's law had this amazing property. This scale invariance explains why Benford's law works on financial data throughout the world, even though the data are expressed in different currencies. Benford's law has surprising applications in financial fraud detection. Because human choices are not random, invented numbers are unlikely to follow Benford's law. The interesting thing is that the more the deceivers try to make their acts look random the easier it is for CPA's using Beford's law to expose them. Dr. Theodore P. Hill asks his mathematics students at the Georgia Institute of Technology to go home and either flip a coin 200 times and record the results, or merely pretend to flip a coin and fake 200 results. The following day he runs his eye over the homework data, and to the students' amazement, he easily fingers nearly all those who faked their tosses. A person trying to fake 200 flips of a coin would never list 6 or more series of the same side although in true randomness these series have a quite high probability of occurrence. As I said this article seems to have nothing to do with ASIT although I think that the Benford's law is similar to ASIT in the way it applies to such a wide variety different domains. +( STATISTICS : general Statistik fuer Betriebsleiter arithm. Mittelwert : = (a1+a2+a3+...+an)/n = 1/n * SUMi(ai) gewichtet = (a1g1+a2g2+...+angn)/(g1+g2+g3...gn) = 1/n SUMi(gi*ai) geometrisches Mittel : = n.te Wurzel von (a1*a2*a3...*an) = n_SQRT (a1*a2..an) harmonisches Mittel : = n / [ 1/a1 + 1/a2 + ... + 1/an ] Median oder Zentralwert oder mitllerer Wert : der Zentralwert ist einer nach der Groesse der Einzelwerte geordneten Zahlenreihe von beiden Enden der Zahlenreihe gleich weit entfernt. Bei gleicher Gliederzahl faellt der Zentralwert zwischen die beiden Mittelglieder und wird als deren Durchschnitt ausgedrueckt. vv 3, 9, 12, 19, 20, 21, 30 ^^ Modus oder dichtester oder haeufigster oder typischer Wert oder Scheitelwert : ist der Wert, der in einer Reihe am h„ufigsten vorkommt. 3, 7, 5, 7, 6, 7, 8, 7, 10, 2 ^ ^ ^ ^ POISSON DISTRIBUTION ... describes a discrete random variable that can take on the values 0, 1, 2, 3, ... The mean is "my" and the standard deviation is SQRT(my) - this implies that the Poisson distribution is a "one parameter" distribution. The probability mass function (=pmf) is : p(k) = exp(-my) * my^k / k! k = 0, 1, 2, 3, ... and the cumulative distribution function (cdf) G(x) = SUMk(p(x)) k = 0, 1 .... x The poisson distribution is applicable : - the number of phone calls to a switch board or a customer service person - for counting processes that are composed of a nummber of independent counted processes (number of customers, who place orders and together represent the total order count for a business) - customers in a bank or restaurant, demand experienced by a retailer, hits on a web page - number of machine failures EXPONENTIAL DISTRIBUTION The time between the k'th and (k+1)'th arrivals is a continuous random variable with probability mass function (lambda = arrival rate) : g(t) = lamda * exp (-lambda*t) lambda >= 0 cumulative G(t) = 1 - exp (-lambda*t) lambda >= 0 The mean is 1/lambda and the standard deviation is also 1/lambda which makes the exponential distribution a one-parameter distribution also. The special feature of the exponential distribution is that it is "memoryless" (Factory Physics page 96). Poisson applies to counting processes and exponential applies to times between countings. NORMAL DISTRIBUTION p(y) = const/å * exp{-[(y-eta)ý/2åý]} with eta = mean å = standard deviation åý = variance or measure of spread population sample ___________________________________________________________________________ a hypothetical set a set of n observations; of N observations; n is very small N is very large mean eta = ä y/N avg = ä y/n variance V=åý= ä(y-eta)ý/N sý = ä(y-avg)ý/(n-1) std dev å = ûV s = ûsý 66% interval = mean +/- å 95% interval = mean +/- 2å 99.9% interval = mean +/- 3å CENTRAL LIMIT THEOREM says, that the distribution of a linear function of errors will tend to normality as the number of components becomes large and is almost irrespective of the individual distributions of the components. SHEWHART CHART plot of observations in a grid of +/- 2å and +/- 3å this is based on the assumption that the sample mean X_bar is a valid representative of the population mean "eta" Wheeler user more exact statistics to generate lower and upper process limits CUSUM CHART plot of cumulative deviations of (observations-mean) in order to observe if there is a shift of the mean Y = SUM (X-X_bar) ERROR in MEASUREMENTS Variance V ~ k1ýå1ý + k2ýåý + k3ýåý ... where ki are constants = (Y(+å) - Y(-å)) / (y(+å) - y(-å)) and åiý are the variances of each measurement +( Poisson distribution How much is enough? The Poisson Formula can tell you. The Poisson formula is a statistical treasure: It's fairly simple, it doesn't require a lot of data-gathering, and it doesn't even require that you know whether the data is normally distributed. You can use it anytime the question is "How much is enough?" by finding the probability that a number won't be enough. You then decide the chance you want to take of running out. You'll need either a calculator that can do factorials, powers, and variances or a computer spreadsheet program. You'll also need to take the time to get some data, either on the whole population or a representative sample. Finally, you'll need to trust that statistics can have a practical use. W = e^ (-æ) * æ^ r /r (Non-mathematician's version: Probability of "r" occurences is equal to "e" raised to the minus power of the mean times the mean raised to the power of" r" divided by "r".) where "W" means the probability of "r" ("r" = any integer) and "æ" = sample mean divided by the variance and "e" is a constant (approximately 2.718.) Poisson was a French mathematician and physicist. (His name translates to "Fish" in English, so we can call it the "Fishy Formula.") I like to restate the formula as: W = æ^r / ( e^ æ * r) because e (-æ;) is the same as 1/e^ æ; . "r" could be the number of a certain item you sell each week, the number of phone lines in use at any one time, or almost anything that has an integer value. (An integer is a whole number - 5 is an integer, 5.23 isn't.) Count "r" several times to get a sample. Ideally, your sample should be "randomized" but sometimes you can only do it for a period of time, say for several weeks in a row. Just remember, your sample is only representative of when, how and where you take it. Calculate the mean (average) and variance. (Use your calculator or spreadsheet program.) Divide the mean by the variance to get "μ" - pronounced "mu". Let's say that you sample "r" nine times and get 7, 3, 5, 6, 5, 4, 6, 5, 5. The mean calculates to 5.1111, the variance to 1.2100, so μ (mu) is 4.2245. The probability (W) of seeing a 7 comes out to 6.97%, an 8 to 3.68%, a 9 to 1.73%, and a 10 to 0.73%. Notice the probabilities are getting smaller; at some point, they get so small you can ignore them. But, what you really want are the cumulative probabilities: What is the probability of any "r" equal to or greater than a certain value? For that, calculate the individual probabilities and add them up. You can graph them like this: +( accurate or precise From: "Jim Bowles" Date: Tue, 3 Jul 2001 22:54:31 +0100 Image two marksmen with rifles. Assumption: It is vital that they hit their target, say a sniper. The first one shoots and the round is fired off in rapid succession. The bullets hit a tree, just by the snipers head. All four shots form a neat cluster in the bark of the tree. And the sniper is gone. The second one shoots at his chosen target. The first hits the snipers arm. The second his skull. The third his guts, The forth his heart. The message is this Precision is pretty, Accuracy is deadly. A Swiss watch with a good mechanical movement is a precision watch. i.e it has a known and repeatable error. Electronic watches are accurate. Providing the power is there they are always right. From: "Tony Rizzo" Date: Wed, 4 Jul 2001 01:24:45 -0400 Sure thing! Below, the "o" is the target. Each "x" is the point where a shot hit. Target ================> o The following is inaccurate and imprecise: x x x x o x x x The following is accurate and imprecise: x x x x o x x x x x x The following is precise but not accurate: xx x x x o x x The following is accurate and precise. x xxoxx xx From: "Potter, Brian (James B.)" Date: Tue, 3 Jul 2001 12:20:26 -0400 The usual distinction in statistics (and by extension therefrom to science, engineering, and process control) is ... Accurate: the sample average falls near a particular value (typically, a target value, a value predicted by theory, or a value one expects based on previous experience) Precise: the sample variance (or standard deviation) is small Now, a toy example in a manufacturing process: A process cuts long metal bars into shorter ones. The target length is 10 units. Neither Sample of seven bars with lengths Accurate 10, 11, 12, 13, 19, 20, and 20 nor The average of 15 is rather far from Precise: 10 AND the standard deviation just over 4 indicates high variation Accurate: Sample of seven bars with lengths 5, 6, 7, 8, 14, 15, and 15 The average is exactly 10, very accurate but not particularly precise (the standard deviation is slightly over 4) Precise: Sample of eleven bars with lengths 14.925, 15.000, 15.075, 15.000, 14.950, 15.000, 15.050, 15.000, 14.975, 15.000, and 15.025 The standard deviation (from the mean of 15) is less than 0.04, rather precise but not very accurate (far from 10) Accurate Sample of eleven bars with lengths AND 9.925, 10.000, 10.075, 10.000, 9.950, Precise: 10.000, 10.050, 10.000, 9.975, 10.000, and 10.025 The standard deviation (from the mean of 10) is less than 0.04, rather precise AND quite accurate (near 10) In this technical usage, "consistent" and "reproducable" (depending upon context) make good English synonyms for "precise." "Accurate," agian in this technical sense, should speak for itself. --- Date: Tue, 21 Aug 2001 05:49:50 -0700 (PDT) From: Ed Wong It is part of "Reductioninst" thought that the whole is the sum of the parts. And it is my understanding that a large amount of the progress of Western Science over the last 400 years has been due to the cultural change towards a belief in reductionism. Strangely - the more we know about the seperate componenets, the less we seem to know about the sum of these parts - especially now that in some cases we actually have the power to calculate that sum. So - perhaps there are other ways of "knowing" a good enough answer that doesnt rely on pure reductionist thought? By pure - I mean that the focus has been on getting a more "accurate" value for each of the parts so that the sum will become more accurate. A portion of the desire to get more accurate is the attempt to get more "precise". Precise is defined as more sig figures Accurate is defined as closer to the "truth" Thus the focus of many "Project Management" efforts is to get more and more "defined" (accurate and /or precise) times for each of the tasks so that sum can be better calculated. But - there are other ways to get a pretty good answer pretty quickly. Fermi was famous for doing these types of problems. His general reasoning is that overall the errors will tend to cancel, and thus pretty accurate (though not precise) answers can be obtained VERY quickly. For MANY decisons, this is all that is needed. One classic example is the calculation of the "tonage" of the first A bomb. Fermi was sprinkling small bits of paper in the observation bunker and watching the "drift" as the bomb blast came through the observation slit. He was correct to 2 sig figures and had an answer within seconds. Not enought to do "science" on, plenty for determining the types of targets it could (or could not) destroy. My current attaction of ToC is that some pretty good answers can be obtained VERY quickly. There are some examples where ToC did not yeild the "optimum" answer. But it did come very close and were the optimum answer required much more work to determine. Like ToC advises - there is a need to determine WHERE to focus your attention, --- From Tony Rizzo Subject Re [tocexperts] Re Software requirements for S-CCPM Precision can be achived by overestimating duration, as you point out. But precision is not required. Accuracy is required. This means that the estimates should not be biased. Accuracy is the difference between the means of two distributions. One is the distribution of task duration estimates, and the other is the distribution of actual task duration. Precision is an indication of the spread of a distribution. Great precision means that the data points are very close to each other, as you clearly know. A bias exists if the two means are very different from each other. Estimates of project duration require unbiased estimates of task duration. That is, they require estimates of task duration that are drawn from an unbiased distribution of task duration estimates. The best sources of such data are the resources who expect to perform the tasks and who see no need to cover their aspect ratios from the lightning bolts of management. Management, in turn, can become the source of overwhelming amounts of bias in the estimates of duration, by requiring that task estimates be treated as commitments. Management can also become a huge source of bias in the distribution of actual task duration, by failing enable resources to prioritize their task queues, which in turn causes multitasking. Resources, in turn, can become a tremendous source of bias in the distribution of project duration estimates, by failing to plan projects. +( conditional probabilities (bedingte Wahrscheinlichkeit) ein Taxifahrer macht Fahrerflucht und wird dabei beobachtet. Der Zeuge spricht von einem blauen Taxi. Der Zeuge hat eine Trefferwahrscheinlichkeit seiner Beobachtungen von 80%. In der Stadt gibt es nur 85% grüne und 15% blaue Taxis. Wie hoch ist die Wahrscheinlichkeit, daß das Taxi tatsächlich blau war ? 1) 85% grüne * 20% Trefferwahrscheinlichkeit = 17 grüne Taxis werden fälschlicherweise als BLAU gesehen 2) 15% blaue * 80% = 12 blaue Taxis werden richtig erkannt p = 12 / (12+17) = 41% Gesamtwahrscheinlichkeit, daß das Taxi tatsächlich blau war ! +( variation From: "Christopher Mularoni" Subject: [cmsig] Re: Productivity - was The real driver. Date: Tue, 24 Jul 2001 19:34:54 -0400 Sorry can't stay out of this one. All real systems have variability. Systems with variability must be buffered by some combination of Inventory, Capacity, or Time. The best combination depends on the goals and purpose of system. Examples (see Factory Physics) : Fire Departments -Can't buffer with inventory (Can't stock services) -Can't buffer with time (Violates objectives) -Must buffer with capacity Organ Transplants -Can't buffer with WIP (perishable) -Can't buffer with capacity (ethically anyway) -Must buffer with time Ballpoint Pens -Can't buffer with time(who will backorder a cheap pen) -Can't buffer with capacity (too expensive and slow) -Must buffer with inventory --- From: "Philip Bakker" Subject: [cmsig] Re: Live in Peace - conflict resolution Date: Wed, 19 Sep 2001 10:48:25 +0200 At the same time we have discussed several times on this list about uncontrolled variation, common causes and special causes. In that context I want to quote the book of Domenico Lepore and Oded Cohen about the relation between TOC and Deming's System of Profound Knowledge (p. 48-49): "A variation which is controlled makes the process predictable. But uncontrolled variation isn't consistent over time. It is due to special causes, causes which are external to the process. We must absolutely understand what kind of variation is affecting the process we are examining. If the variation is uncontrolled, then the manager's ability to predecit will be seriously undermined, and so too will his capacity to manage. ..... Failing to identify the source of variation, special or common, leads to taking inappropriate actions on the system that may worsen the situation. Deming called that tampering with the system. As Brian Joiner points out in his book, Fourth Generation Management, in order to improve an out-of-control process we must: 1) Gather data immediately so that the special causes which generate the instability of the system are quickly identified. 2) Activate an immediate solution to limit the damage 3) Look for what has determined the occurence of the special cause. 4) Implement a long-term solution. ..... But what usually happens in our organizations? Normally, we don't think about processes. We just act in order to satisfy specifications." Of course this book has nothing to do with the strike on the US. But we have to look at the process. It's also my impression that, although the problem is already huge, it still needs to be made bigger. So many things are needed: we have to protect ourselves, limit the damage, avoid reacting on symptoms, look for deeper causes and implement a long-term solution for all involved. If it's just about plain terrorism as an incident with a very unique and rare special cause, we don't need to bother about a cloud. Just take care of it. Otherwise it's good to look beyond the act. If we build a cloud, it needs to cover the cause(s) of most of our UDE's and also UDE's of the Arabs, Bin Laden or who ever we decide is part of this enormous problem. A couple of problems of all involved parties should be formulated by persons who have good knowledge about them. Then we can build a real generic cloud and gain understanding of the occurence of the special cause. Only then we can start to evaporate the cloud. It's a little strange to talk about processes, variation and tampering when so many emotions and pain are involved. But I hope it can help to straighten out much of our confusion in a time when wise actions are so really needed. By the people of the US and by those in all other countries in the world. --- From: "Bruce Stern" Subject: [cmsig] RE: Challenging statistics basic assumptions Date: Tue, 28 Aug 2001 22:27:49 -0400 Ron makes a very good point here. The use of control charts according to Shewart and Deming was that points beyond the control limits were ECONOMICALLY worth investigating as having an assignable cause (also known as special cause variation). The stuff within the control limits is attributable to no assignable cause (also known as common cause variation). If you want to change the common cause variation you must change the system, since this is the expected result of that system. You can "fix" the special cause cases since you can economically track down what is different about them. This over time will get rid of these pesky outliers or at least reduce their number. However, it won't fix the common cause variation, which is a point Deming hammers away at noting that management is responsible for the system and so it is up to management, not the workers, to fix it. Okay, back to our regularly scheduled program. I think that a reasonable view of the expected variation in the tasks is the place to start. If we wind up with somewhat repetitive tasks over time we can capture the predicted versus actual and refine this estimate. In no case is it likely to be perfect, but I don't believe it should be. Eli S's point about these being non-normal distributions is a very fair one. Given enough tasks, some of that should dampen out. Given enough history and the level of predictability may improve. The bottom line is that doing the project with some reasonable use of CCPM is more likely to result in an on-time completion than without it. ----- Original Message ----- From: "Ronald A. Gustafson" Sent: Saturday, August 18, 2001 7:16 PM I'm in the test and evaluation business. I share your concern about violating the underlying assumptions when applying statistics. We will typically gather data under a wide set of conditions, clump it all together, assume it's normally distributed, and crank numbers. Years ago (20+) I played with a technique that Strategic Air Command used to identify "outliers" for scoring missile accuracy. Personally I don't believe in "outliers" (every score is indicative of system performance), but I always like to investigate different ideas. First I would compute the 3xCEP miss distance (as I recall that includes 99.987% percent of all the scores for a bivariate, normal distribution -- someone else can check if my memory is correct). Any score that exceeded that miss distance was discarded as an "outlier." This procedure is repeated on each "reduced" data set until no more "outliers" are found. I tried this for scores from tests of two completely different systems. In both cases the process would have discarded about 30% of the data as being "outliers." Obviously, a lot was going on in the "tails" of the distributions. Was it important operationally - who knows? It all depends on what's needed for a specific operation. Regardless, rather than just accepting this as some sort of weird distribution, I want to know what's driving performance to the tails (cause-effect). It might be indicative of some inherent problem within the system that is readily corrected. I expect the same is true for task performance in Critical Chain networks. Over the years, I have been in so many debates about statistical significance. What I've found is that all of those debates are useless until you know what is operationally significant. Also, in operations, "risk" is always managed. Only in textbook examples or in simulations can we afford to let things play out to a disastrous ending without taking action. Buffers never tell us what corrective action to take, they only give us time to plan and take that action. We must have some understanding of the cause-effect nature of the situation in order to decide on an action. That understanding is far more important than increasing the "precision" in estimating buffer size. Provide "reasonable" buffers, get to work, and then sort things out as they happen. The "real world" has never been quite what I expected it to be. The best plans are only a starting point. --- On March 4 2001 Brian Potter made a very interesting contribution to the CMSIG. > In his mail 'Pondering Awareness of Variation' he solicited for commentary > on the thought that perhaps awareness of variation comes in degrees. A quote > from that mail > > [Begin quote] > 1. Total unawareness Attributing effects of variation to > a "failure to execute" (probably, the "failure" of > subordinates). > > 2. Glimmer of awareness A _Murphy's Law_ level of > understanding that "things always go wrong," but there > is nothing one can do about it (other than allow "extra" > time, resources, and [of course] money). > > 3. Basic awareness Knowing something about statistical > variation, randomness, reproducibility, and the like > but with a feeling that it is "somebody else's problem" > (probably a subordinate's) > > 4. Awareness It's real; it's my problem, and SPC is a > good control methodology. > > 5. Mature awareness Variation is a reality, and a > continuous improvement management philosophy creates a > dynamic system that can deal with variation. How do I > keep variation from damaging the ultimate results? > Deming, TQMP, 6-Sigma, et al have reached this point. > > 6. Sophistication Variation is a natural phenomenon of > physical systems. How can I design systems that absorb > variation without suffering damaging effects (in > particular, without passing damaging results on to the > customer, but also absorbing variation without feeling > damaging internal impacts)? This is where I see ToC > being, now. > > 7. Exploitation Variation is change with a natural > pattern. How can I design systems with the variation > patterns as PLANNED control parameters? How can I > "surf" on the variation and use it to my advantage? > [End quote] +( mode - median - mean Date: Sat, 22 Sep 2001 10:53:41 -0400 From: Brian Potter Subject: [cmsig] Request for clarity - aggregate variation and AVERAGE] Sorry, guys, this one has been incubating in my "drafts folder" while I let my day job distract me from it (and shine a light on it). Hope the result will be worth the wait. The question about averages (and some day job thinking about how to sell critic al chain to to PD organization at work) led me into variation as well. The _mode_, _median_, and _arithmetic mean_ are three common "measures of _central tendency_" (the result one EXPECTS when picking a specific instance of a randomly distributed value [e.g., the anti-corrosion coating thickness on a coil of coated steel sheet on its way to becoming automobile body panels]). mode: the most frequently occurring sampled value median: as you defined, below arithmetic mean: as you defined below For symmetric distributions (e. g., the _Normal [Gaussian] Distribution_, the _t Distribution_, and the uniform distribution for three common examples) the three will converge to the same value (the population mean) as the number of points inspected grows. For asymmetric distributions (like task or project completion times), the three will have distinct values. In the project timing case (and other situations with a much longer tail on the right end [toward larger values]), the _mode_ will be smallest, the _median_ will be in-between, and the _arithmetic mean_ will be largest. Tony's statement that the average of the sums equals the sum of the averages applies to the arithmetic mean. Arithmetic means are very nice estimators of expected value. They are "unbiased," easy to calculate, easy to understand, and lay the foundation for powerful parametric statistics when one knows "enough" (there's Eli's point, again) about the distribution(s) involved. Much critical chain literature talks about the 50% likelihood point in the task duration distribution or in the project duration distribution. The "50% likelihood point" is the median which can significantly smaller than the mean when the distribution is highly skewed with a long tail towards larger values (as in the project management case). Thus a sum of medians could be MUCH SMALLER than a sum of means for the same population (e.g., a sequence of task timings along a critical chain). Thus, to have a project come in under (or even on) the "sum of the medians" timing would be quite extraordinary. Since the distribution is so skewed, even a single task coming in with above median timing would probably erase the gains from having every other task finish no later than the median time. Thus, finishing in under "sum of the medians" timing probably happens about once every 2 ^ N projects (where "N" is the number of tasks on the critical chain and "^" is the "raised to the power of" operator). For ten tasks in a critical chain (and no complicating integration activities), the early finish is about a one in a thousand shot (1023 taking over sum of the medians time and exactly one coming in at or under sum of the medians time). So, what estimates do we get when we ask folks to estimate "expected task time" or "50% completion chance" task time? With highly skewed distributions, I suspect that "human intuition" leads to something nearer the mode or the median than the mean. If so, some buffer consumption will be the norm. Since the EXPECTED project duration is the sum of the task mean times (which is larger than the sum of the task median times), we expect projects to take longer than the sum of the median task times (and mo re longer than the sum of the task time modes). Thanks to the Central Limit Theorem, project durations should be distributed normally. Thus, (because the normal distribution is symmetric) the mode and median of the project duration EQUAL the mean of the project duration. Perhaps, one consequence of CC/PM is normally distributed project completion times. Interesting (and possibly useful), if true. How might we test this hypothesis? Also, thanks to the Central Limit Theorem, we expect the variance of the project time to become smaller relative to the project time as the number of tasks in the project's critical chain increases. This expectation flies in the face of practical experience with large projects managed by other approaches! Statistical thinking about larger projects suggests that they SHOULD be MORE predictable than smaller projects! That's REALLY interesting. Bigger projects should be more manageable than smaller one s ... How can we test that? Predicted effect of a project management model that does not waste safety time: Total project durations are approximately normally distributed with a mean duration near the sum of the means of the task durations and a variance near the sum of the task duration variances divided by the square root of the number of tasks on the critical chain. -------- Original Message -------- Date: Sat, 18 Aug 2001 21:11:01 -0400 From: "Tony Rizzo" Ah, yes! When I say average, I mean the mean. For example, given the data points, a1, a2, ..., an. In my dictionary, average means (a1 + a2 + ... + an)/n. Yes, the project buffer begins at the mean duration of the critical chain, which equals the sum of the means of the critical chain tasks. However, as you observe, variation can be positive as well as negative. Therefore, as you note, the project buffer explicitly covers only the half of the variation that lies to the right of our estimate of the mean of the critical chain. Chapters 2 and 3 of our critical chain tutorial contain some rather revealing pictures of this. I prefer to speak in terms of averages (means) because that's the only mathematically valid way of doing mathematical models that I know of. Strictly speaking, a chain of events exhibits a distribution of durations. The expected value of that unknown distribution is what one would use for modeling purposes. But, since we don't know the distribution, we estimate the expected value with the mean of whatever data we have, if we have data. If we don't have data, then we estimate it with the educated guess of the people doing the various tasks. ----- Original Message ----- From: "J Caspari" Sent: Saturday, August 18, 2001 3:55 PM > Tony wrote: > > << I just love this sort of discussion. And John, I don't think that you're > mean. In fact, I rather like you. >> > > Thank you. Looking back over the substantive discussions on this list it > seems that these discussions result in our finding out something that we > didn't know before. I'm learning a lot from this one. > > I hope that you are not going to write me one of those sonnet things. :-) > > Would you provide some clarity on a couple of things? > > << The project buffer is our estimate of the aggregate variation in the set > of critical chain tasks. >> > > I'm not sure what the term 'aggregate variation' means. My knowledge of > statistics stops short somewhere near the end of *How to Lie with > Statistics* (Huff, W. W. Norton & Co., 1954) and a search of standard > dictionaries and the WWWeb did not provide any useful hits, Poisson or > otherwise. > > It seems to me that 'aggregate variation' should refer to 'all of the > variation.' In a project environment where we are interested in the > distributions of the duration of projects (comnpletion dates) this would > mean that the range of the variation would be from the earliest completion > to the latest completion. However, if we assume that a project buffer is > established so that it starts where the probability is 50%-50% that the > project will be completed (or even somewhat to the west of that point--but > let's leave that for another discussion), then the variation before that > point would have been ignored. > > << It is every bit as much a feature of the critical chain as are the > estimates of AVERAGE duration of the tasks in that chain. The latter > comprise our estimate of the expected duration of the critical chain. >> > > This AVERAGE, then marks the start of the project buffer. But one thing > that I have learned so far in this multinational discussion is that we may > not be using the term, AVERAGE, and its instances in the same way. > > My understanding of an 'average' comes from Huff's book (p.28) where he > comments << When you are told that something is an average you still don't > know very much about it unless you can find out which of the common kinds of > average it is--mean, median, or mode. >> > > Hence, we might speak of a project buffer that starts at the median estimate > (half above, half below) and which has a mean (arithmetic average, add 'em > up and divide by n) which is well out in to the project buffer. > > Are you using AVERAGE in a particular way? +( SHEWHART XmR and CUSUM chart see Donald Wheeler : Understanding Variation Shewharts Rule 1 for the presentation of data : Data should always be presented in such a way that preserves the evidence in the data for all the predictions that might be made from this data. Shewharts Rule 2 for the presentation of data : Whenever an average, range, or histogram is used to summarize data, the summary should not mislead the user into taking any action that the user would not take if the data were presented in a time series. The first principle of understanding data : No data have meaning apart from their context. The second principle of understanding data : While every data set contains noise, some data sets may contain signals. Therefore before you can detect a signal within any given data set, you must filter out the noise. There are two errors possible when interpreting data : - interpreting noise as if it were data - failing to detect a signal when it is present When people are pressured to meet a target value (=goal) there are three ways they can proceed (Wheeler p. 20, source : Brian Joiner) - they can work to improve the system - they can distort the system - they can distort the databases (Goldratt : tell me how they measure you and I tell you how you behave) Shewharts meaning of the word "control" (Wheeler p. 141) : A phenomenon (process) will be said to be controlled (predictable) when, through the use of past experience, we can predict (describe), at least within limits, how the phenomenon (process) will vary (behave) in the future. Any process shows - routine variation (noise or common cause variation) and - execptional variation (signals) Exceptional variation occurs when either (Wheeler) : - observed data lies outside of the mean +/- 3*sigma - 3 out of 4 consecutive observed data record lie closer to the upper/lower limit than to the mean - 8 or more consequtive observed data records are above or below the mean Shewhart created the "Four Western Electric Zone Rules" (see Zultner in cutter.pdf) : - one point more than three standard deviations away from the center line - two out of three successive points more than two std deviations away from the center line on the same side - four out of five successive points more than one std deviations away from the center line on the same side - eight consecutive points on the same side of the center line another one : - six consecutive points ascending or descending SHEWHART XmR PROCESS CONTROL CHART plot of observations in a grid of +/- 2å and +/- 3å Wheeler uses the expressions - process behaviour chart - voice of the process variables in the formula : X individual data record X_bar (arithmetic) average of all data X_tilde median of all data mR the absolute value of the difference between 2 consecutive data records mR_bar MEAN or (arithmetic) average of all mR mR_tilde median of all mR LNPL lower natural process limit : X_bar - 3*sigma Wheeler : LPCL ~ X_bar - 2,66 * mR_bar for average moving range ~ X_bar - 3,14 * mR_tilde for median moving range UNPL upper natural process limit : X_bar + 3*sigma Wheeler : UPCL ~ X_bar + 2,66 * mR_bar for average moving range ~ X_bar + 3,14 * mR_tilde for median moving range URL upper range limit : mR_bar + ??? Wheeler : URL ~ mR_bar * 3.27 for average moving ramge ~ mR_tilde * 3.87 for median moving range If the plot of x or mR leaves the ranges of X_bar +/- L/U_NPL or mR + mR_bar than the process gives us a signal that something has changed (for the better or the worse). --- Shewhart in his 1931 book explains that - the control chart is irrelevant of the nature of the orig. distribution, therefore a normal distribution is "good enough" - +/- 3 sigma is "good enough" for the study of data (economical) - X_bar is a good enough representative of the mean of the population --- Deming in "Out of Crisis" uses the following formulas : sample standard deviation s = SQRT{SUM[(X-X_bar)ý]/n-1} and if the population mean "eta" is known : s = SQRT{SUM[(X-eta)ý]/N} L/U_PL = X_bar +/- 3* SQRT[X_bar*(1-X_bar)/n] --- Date: Thu, 02 May 2002 18:19:04 -0400 From: Subject: [cmsig] RE: System stability -- how to tell HP Staber wrote: > BTW : is there an explanation for the obscure values in > Wheelers formula's for calculating lower and upper > process limits : REZ> The constants in the common formula for individual (x and mR) charts result from applying the general formula for control charts to the case where n=2. Instead of giving you the general formula plus the tables you need to apply it, the (one and only) number you need from the tables when n=2 is supplied to you in the common formula for individual charts. [Incidentally, the reason tables are used in the general formula, even today, is that the comprehensive formula involves a double integral that is nasty even today to calculate on the fly. So even in the 21st century, SPC texts still give you the same "easy to use" tables developed in the days of slide rules...] ... > mR_bar = average or median mean of all mR REZ> P.S. mR_bar is the MEAN (arithmetic average) NOT the median, which is denonated as mR_tilde, and uses a slightly different formula (which Wheeler gives in the back of "Understanding Variation"). Please be careful which measure of central tendency you are using! > Today I simply use the statistical capabilities of EXCEL > and compute LNPL = AVG - 3* SIGMA (I am not sure if > SIGMA is the right function name in the english version > of EXCEL) REZ> No, no, NO. You give the formula, and then you do not use it, using Excel functions instead! Calculate the formula EXACTLY as written, with NO short-cuts! [The Excel function for standard deviation (and the one in your calculator) is the one used in Descriptive Statistics -- this definition assumes there is no change in the variation of the data you apply it to. This is exactly what we wish to test for. So you must use the Analytic Statistics definition of standard deviation, which is given in the formula. Wheeler has a nice section on the inflated limits that result from this sort of misunderstanding of how to do SPC. Learn from a master...] --- CUSUM CHART plot of cumulative deviations of (observations-mean) in order to observe if there is a shift of the mean --- also in Box, Hunter, Hunter : Statistics For Experimenters p 556 ff --- in Shewhart : Economic Control of Quality of Manufactured Product p 275 ff, 300 and 360 : LNPL = X_bar - 2*t*SIGMA/SQRT(n) and UNPL = X_bar + 2*t*SIGMA/SQRT(n) and the variation in standard deviation is : dSIGMA = +/- 2*t*SIGMA/SQRT(2n) where n number of observations t parameter to controll accuracy of prediction 99.73% t=3 95.45% t=2 68.27% t=1 50.00% t=0.6745 +( STATISTICS : UNCERTAINTY CALCULATIONS http://physics.nist.gov/cuu/Uncertainty/index.html Classification of uncertainty components The uncertainty of the measurement result y arises from the uncertainties u(xi) (or ui for brevity) of the input estimates xi that enter equation (2). Thus, in the example of equation (3), the uncertainty of the estimated value of the power P arises from the uncertainties of the estimated values of the potential difference V, resistance R0, temperature coefficient of resistance b, and temperature t. In general, components of uncertainty may be categorized according to the method used to evaluate them. Type A evaluation method of evaluation of uncertainty by the statistical analysis of series of observations, Type B evaluation method of evaluation of uncertainty by means other than the statistical analysis of series of observations. Representation of uncertainty components Standard Uncertainty Each component of uncertainty, however evaluated, is represented by an estimated standard deviation, termed standard uncertainty with suggested symbol ui, and equal to the positive square root of the estimated variance ui2. Standard uncertainty: Type A An uncertainty component obtained by a Type A evaluation is represented by a statistically estimated standard deviation si, equal to the positive square root of the statistically estimated variance si2, and the associated number of degrees of freedom vi. For such a component the standard uncertainty is ui = si . Standard uncertainty: Type B In a similar manner, an uncertainty component obtained by a Type B evaluation is represented by a quantity uj, which may be considered an approximation to the corresponding standard deviation; it is equal to the positive square root of uj2, which may be considered an approximation to the corresponding variance and which is obtained from an assumed probability distribution based on all the available information. Since the quantity uj2 is treated like a variance and uj like a standard deviation, for such a component the standard uncertainty is simply uj. Evaluating uncertainty components: Type A A Type A evaluation of standard uncertainty may be based on any valid statistical method for treating data. Examples are calculating the standard deviation of the mean of a series of independent observations; using the method of least squares to fit a curve to data in order to estimate the parameters of the curve and their standard deviations; and carrying out an analysis of variance (ANOVA) in order to identify and quantify random effects in certain kinds of measurements. Mean and standard deviation As an example of a Type A evaluation, consider an input quantity Xi whose value is estimated from n independent observations Xi ,k of Xi obtained under the same conditions of measurement. In this case the input estimate xi is usually the sample mean (4) and the standard uncertainty u(xi) to be associated with xi is the estimated standard deviation of the mean (5) Evaluating uncertainty components: Type B A Type B evaluation of standard uncertainty is usually based on scientific judgment using all of the relevant information available, which may include: * previous measurement data, * experience with, or general knowledge of, the behavior and property of relevant materials and instruments, * manufacturer's specifications, * data provided in calibration and other reports, and * uncertainties assigned to reference data taken from handbooks. Below are some examples of Type B evaluations in different situations, depending on the available information and the assumptions of the experimenter. Broadly speaking, the uncertainty is either obtained from an outside source, or obtained from an assumed distribution. Uncertainty obtained from an outside source Multiple of a standard deviation Procedure: Convert an uncertainty quoted in a handbook, manufacturer's specification, calibration certificate, etc., that is a stated multiple of an estimated standard deviation to a standard uncertainty by dividing the quoted uncertainty by the multiplier. Confidence interval Procedure: Convert an uncertainty quoted in a handbook, manufacturer's specification, calibration certificate, etc., that defines a "confidence interval" having a stated level of confidence, such as 95 % or 99 %, to a standard uncertainty by treating the quoted uncertainty as if a normal probability distribution had been used to calculate it (unless otherwise indicated) and dividing it by the appropriate factor for such a distribution. These factors are 1.960 and 2.576 for the two levels of confidence given. Uncertainty obtained from an assumed distribution Normal distribution: "1 out of 2" Procedure: Model the input quantity in question by a normal probability distribution and estimate lower and upper limits a- and a+ such that the best estimated value of the input quantity is (a+ + a-)/2 (i.e., the center of the limits) and there is 1 chance out of 2 (i.e., a 50 % probability) that the value of the quantity lies in the interval a- to a+. Then uj is approximately 1.48 a, where a = (a+ - a-)/2 is the half-width of the interval. Normal distribution: "2 out of 3" Procedure: Model the input quantity in question by a normal probability distribution and estimate lower and upper limits a- and a+ such that the best estimated value of the input quantity is (a+ + a-)/2 (i.e., the center of the limits) and there are 2 chances out of 3 (i.e., a 67 % probability) that the value of the quantity lies in the interval a- to a+. Then uj is approximately a, where a = (a+ - a-)/2 is the half-width of the interval. Normal distribution: "99.73 %" Procedure: If the quantity in question is modeled by a normal probability distribution, there are no finite limits that will contain 100 % of its possible values. However, plus and minus 3 standard deviations about the mean of a normal distribution corresponds to 99.73 % limits. Thus, if the limits a- and a+ of a normally distributed quantity with mean (a+ + a-)/2 are considered to contain "almost all" of the possible values of the quantity, that is, approximately 99.73 % of them, then uj is approximately a/3, where a = (a+ - a-)/2 is the half-width of the interval. Uniform (rectangular) distribution Procedure: Estimate lower and upper limits a- and a+ for the value of the input quantity in question such that the probability that the value lies in the interval a- and a+ is, for all practical purposes, 100 %. Provided that there is no contradictory information, treat the quantity as if it is equally probable for its value to lie anywhere within the interval a- to a+; that is, model it by a uniform (i.e., rectangular) probability distribution. The best estimate of the value of the quantity is then (a+ + a-)/2 with uj = a divided by the square root of 3, where a = (a+ - a-)/2 is the half-width of the interval. Triangular distribution The rectangular distribution is a reasonable default model in the absence of any other information. But if it is known that values of the quantity in question near the center of the limits are more likely than values close to the limits, a normal distribution or, for simplicity, a triangular distribution, may be a better model. Procedure: Estimate lower and upper limits a- and a+ for the value of the input quantity in question such that the probability that the value lies in the interval a- to a+ is, for all practical purposes, 100 %. Provided that there is no contradictory information, model the quantity by a triangular probability distribution. The best estimate of the value of the quantity is then (a+ + a-)/2 with uj = a divided by the square root of 6, where a = (a+ - a-)/2 is the half-width of the interval. Schematic illustration of probability distributions The following figure schematically illustrates the three distributions described above: normal, rectangular, and triangular. In the figures, µt is the expectation or mean of the distribution, and the shaded areas represent ± one standard uncertainty u about the mean. For a normal distribution, ± u encompases about 68 % of the distribution; for a uniform distribution, ± u encompasses about 58 % of the distribution; and for a triangular distribution, ± u encompasses about 65 % of the distribution. Combining uncertainty components Calculation of combined standard uncertainty The combined standard uncertainty of the measurement result y, designated by uc(y) and taken to represent the estimated standard deviation of the result, is the positive square root of the estimated variance uc2(y) obtained from (6) Equation (6) is based on a first-order Taylor series approximation of the measurement equation Y = f(X1, X2, . . . , XN) given in equation (1) and is conveniently referred to as the law of propagation of uncertainty. The partial derivatives of f with respect to the Xi (often referred to as sensitivity coefficients) are equal to the partial derivatives of f with respect to the Xi evaluated at Xi = xi; u(xi) is the standard uncertainty associated with the input estimate xi; and u(xi, xj) is the estimated covariance associated with xi and xj. Simplified forms Equation (6) often reduces to a simple form in cases of practical interest. For example, if the input estimates xi of the input quantities Xi can be assumed to be uncorrelated, then the second term vanishes. Further, if the input estimates are uncorrelated and the measurement equation is one of the following two forms, then equation (6) becomes simpler still. Measurement equation: A sum of quantities Xi multiplied by constants ai. Y = a1X1+ a2X2+ . . . aNXN Measurement result: y = a1x1 + a2x2 + . . . aNxN Combined standard uncertainty: uc2(y) = a12u2(x1) + a22u2(x2) + . . . aN2u2(xN) Measurement equation: A product of quantities Xi, raised to powers a, b, ... p, multiplied by a constant A. Y = AX1a X2b. . . XNp Measurement result: y = Ax1a x2b. . . xNp Combined standard uncertainty: uc,r2(y) = a2ur2(x1) + b2ur2(x2) + . . . p2ur2(xN) Here ur(xi) is the relative standard uncertainty of xi and is defined by ur(xi) = u(xi)/|xi|, where |xi| is the absolute value of xi and xi is not equal to zero; and uc,r(y) is the relative combined standard uncertainty of y and is defined by uc,r(y) = uc(y)/|y|, where |y| is the absolute value of y and y is not equal to zero. Meaning of uncertainty If the probability distribution characterized by the measurement result y and its combined standard uncertainty uc(y) is approximately normal (Gaussian), and uc(y) is a reliable estimate of the standard deviation of y, then the interval y ­ uc(y) to y + uc(y) is expected to encompass approximately 68 % of the distribution of values that could reasonably be attributed to the value of the quantity Y of which y is an estimate. This implies that it is believed with an approximate level of confidence of 68 % that Y is greater than or equal to y ­ uc(y), and is less than or equal to y + uc(y), which is commonly written as Y= y ± uc(y). Expanded uncertainty and coverage factor Expanded uncertainty Although the combined standard uncertainty uc is used to express the uncertainty of many measurement results, for some commercial, industrial, and regulatory applications (e.g., when health and safety are concerned), what is often required is a measure of uncertainty that defines an interval about the measurement result y within which the value of the measurand Y can be confidently asserted to lie. The measure of uncertainty intended to meet this requirement is termed expanded uncertainty, suggested symbol U, and is obtained by multiplying uc(y) by a coverage factor, suggested symbol k. Thus U = kuc(y) and it is confidently believed that Y is greater than or equal to y - U, and is less than or equal to y + U, which is commonly written as Y = y ± U. Coverage factor In general, the value of the coverage factor k is chosen on the basis of the desired level of confidence to be associated with the interval defined by U = kuc. Typically, k is in the range 2 to 3. When the normal distribution applies and uc is a reliable estimate of the standard deviation of y, U = 2 uc (i.e., k = 2) defines an interval having a level of confidence of approximately 95 %, and U = 3 uc (i.e., k = 3) defines an interval having a level of confidence greater than 99 %. Relative expanded uncertainty In analogy with relative standard uncertainty ur and relative combined standard uncertainty uc,r defined above in connection with simplified forms of equation (6), the relative expanded uncertainty of a measurement result y is Ur = U/|y|, y not equal to zero. Examples of uncertainty statements The following are examples of uncertainty statements as would be used in publication or correspondence. In each case, the quantity whose value is being reported is assumed to be a nominal 100 g standard of mass ms. Example 1 ms = 100.021 47 g with a combined standard uncertainty (i.e., estimated standard deviation) of uc = 0.35 mg. Since it can be assumed that the possible estimated values of the standard are approximately normally distributed with approximate standard deviation uc, the unknown value of the standard is believed to lie in the interval ms ± uc with a level of confidence of approximately 68 %. Example 2 ms = (100.021 47 ± 0.000 70) g, where the number following the symbol ± is the numerical value of an expanded uncertainty U = k uc, with U determined from a combined standard uncertainty (i.e., estimated standard deviation) uc = 0.35 mg and a coverage factor k = 2. Since it can be assumed that the possible estimated values of the standard are approximately normally distributed with approximate standard deviation uc, the unknown value of the standard is believed to lie in the interval defined by U with a level of confidence of approximately 95 %. +( failure Date: Wed, 05 Sep 2001 10:07:59 +0200 From: Eli Schragenheim About the definition of failure. We start the implementation with certain expectations. When we don't reach the lower end of our expectations - this is a failure in our eyes. For someone else, with different expectations, it might be a success. Avraham speaks about the measurement bar. Usually we place the bar at the high end of our realistic expectations, because we like to motivate the organization to excel. The risk is when the actual performance fails to reach the agreed bar, but still is well above the low end of the expectations, people might refer to it as a failure while the change was definitely positive. I claim that whenever we notice a gap between prior expectations and actual performance, we have great opportunity to learn something new (to us). Expectations are build upon our current understanding of the cause and effect. When the actual performance does not match our expectations we better check our basic assumptions. When we do that and pinpoint the "wrong paradigm" then we need to update the paradigm, a specific thinking process that involves generalizing from the single case. Certainly when we have several or many cases the generalization is easier to validate. We can learn from the experience of others, provided the cause and effect leads all the way from the inaccurate paradigm to the gap between prior expectations and actual performance. Using a structured analysis, based on some of the TP tools, we can provide the appropriate learning. But, even learning from experience needs to be focused, just like managerial efforts. For me 'learning from experience' should be an integral part of the TOC techniques. It certainly belongs to the overall philosophy. Avraham Mordoch wrote: > > We have to complete the definition of failure of a TOC implementation. I > think that every implementation must start we agreed measure regarding the > success of the implementation. Now, if one puts, as the agreed measure, a > low bar which is easy to jump over, it may be considered by him as a success > story but, at the same time, I will consider it as failure. To start with, > my bar was higher. > > The fact that Tony admits that he had failures and Dave says the have not, > should not drive us to jump to the conclusion that Dave is an excellent TOC > implementation leader and Tony is not. It just tells us that Tony's bars are > higher than Dave's. As a result, very probable, that Tony's implementations > are better. Better meaning better results in the bottom line. > > How many TOC implementations can really show significant and sustained > impact on the organization bottom line? If in an implementation the change > agent has to chase management once in a while, should we consider it as > successful implementation? I doubt. +( special cause variation Exceptional variation occurs when either (Wheeler) : - observed data lies outside of the mean +/- 3*sigma - 3 out of 4 consecutive observed data record lie closer to the upper/lower limit than to the mean - 8 or more consequtive observed data records are above or below the mean Shewhart created the "Four Western Electric Zone Rules" (see Zultner in cutter.pdf) : - on point more than three standard deviations away from the center line - two out of three successive points more than two std deviations away from the center line on the same side - four out of five successive points more than one std deviations away from the center line on the same side - eight consecutive points on the same side of the center line +( Statistik Programme fuer Linux Sender: andy@alice.rhinosaur.lan Newsgroups: at.linux Subject: Re: Suche: Statistik-Programm fr Linux Date: 17 Oct 2002 07:43:53 +0200 > Ich suche ein kostenloses Statistikprogramm (keine reine > Tabellenkalkulation) unter Linux zur Auswertung von Frageb”gen. > > Vielleicht von der Art her soetwas „hnliches wie das sehr bekannte > komerzielle SPSS-Programm. > Probier mal R: http://www.r-project.org/ +( Pareto curriculum vitae http://www.kgh-online.de/infoschul/projekt/haseluenne/oekonome/pareto.htm Den National”konomen galt Vilfredo Pareto als Soziologe und den Soziologen als National”konom. Dabei liegt das Wesentliche von Paretos Lebenswerk in der Verbindung von ™konomie und Soziologie zu einer alles umfassenden Sozialwissenschaft. "Sein scharfer Verstand reichte weit ber die Grenzen der angewandten Wissenschaften hinaus in das Reich reiner und allgemeinster Begriffe: Nur wenige haben jemals mit der gleichen Intensit„t begriffen, daá letzten Endes alle exakten Wissenschaften oder Teilwissenschaften eine Einheit bilden", schrieb Joseph A. Schumpeter ber Pareto. Zwanzig Jahre arbeitete Vilfredo Pareto (1848 bis 1923) fast ausschlieálich an dem Trattato di Sociologia Generale, das 1916 erschien. Das Buch ist sein Lebenswerk - obwohl ausgerechnet jenes Modell nicht enthalten ist, mit dem heute die ™konomen den Namen Pareto vor allem verbinden: das in der Wirtschaftswissenschaft so genannte Pareto-Optimum. Mit diesem Ansatz - auch als Theorie der Wahlakte bezeichnet - schloss er die Lcken in der Grenznutzenlehre (ZEIT Nr. 29/99). Demnach ist das Pareto-Optimum jener Zustand, bei dem kein Mitglied einer Gruppe oder Gesellschaft besser gestellt werden kann, ohne dass zumindest ein anderes schlechter gestellt werden msste. Fr Pareto war die Soziologie stets dogmatisch und mit einer Religion vergleichbar. Er dagegen wollte alles anders machen. "Wer glaubt, die absolute Wahrheit zu besitzen, kann nicht einr„umen, daá es auch noch andere Wahrheiten in der Welt gibt", schrieb Pareto im Trattato. Aus dieser Einsicht heraus unterwarf er seine soziologischen und ”konomischen Theorien einer einheitlichen, der "logisch-erfahrungsm„áigen" Methode. Er war besessen davon, die Gesellschaftsvorg„nge unbeeinflusst zu schildern. Bewertungen wie ungerecht oder gerecht, moralisch oder unmoralisch wollte er nur zulassen, wenn zuvor klargestellt wurde, welchen Dingen diese Begriffe entsprechen sollen. Doch diesen Anspruch erfllte er in seinem Buch keineswegs immer selbst. In Paretos Theorien zum Kampf zwischen den Klassen zeigt sich sein besonderes Verh„ltnis zu Karl Marx. Schon in einem frheren Werk zu den sozialistischen Theorien befasste er sich ausfhrlich mit ihm, wobei er dem Soziologen Marx eine weit gr”áere Bedeutung zumisst als dem ™konomen. Auch fr Pareto ist der Klassenkampf ein Faktum - allerdings wird er vor allem zwischen den Eliten im Streit um die Macht ausgetragen. Fr Pareto wird jedes Volk von einer Elite, einer "ausgew„hlten Klasse", regiert, die zu allen Zeiten und in allen Verh„ltnissen politische Macht ausbt. Die Ausbung der Macht sei das Wesentliche in einer Gesellschaft. Aber "die Eliten sind nicht von Dauer ... sie verschwinden unbestreitbar nach einer gewissen Zeit". Die herrschende Klasse wird immer wieder von Empork”mmlingen aus den Familien der unteren Klassen abgel”st. Pareto nennt das die "Zirkulation der Eliten". Die Geschichte "ist ein Friedhof der Eliten", schreibt er. Die Zirkulation gibt es im Politischen und Ideologischen ebenso wie im Wirtschaftlichen. Durch sie befinden sich die Eliten in einem Zustand andauernder und langsamer Transformation, "die wie ein Strom dahingleitet, der heute anders ist, als er gestern war. Ab und an beobachtet man pl”tzliche und heftige St”rungen, wie wenn ein Strom ber die Ufer tritt, und danach beginnt die neue herrschende Klasse sich ihrerseits zu wandeln: der Strom, in sein Bett zurckgekehrt, flieát von neuem gleichm„áig dahin." Revolutionen gibt es nur bei zu langsamer Erneuerung der Eliten. Im politischen Zyklus gelangt zun„chst eine Gruppe - bei Pareto sind es die "L”wen", die wegen ihrer Ideale stark sind - an die Macht. Doch weil sie sich in Gebrauch und Sicherung der Macht immer mehr von ihren ursprnglichen Idealen entfernt, gewinnt die zweite Elitegruppe, die der "Fchse", immer mehr Einfluss, bis sie selbst die Macht bernimmt. Hat eine herrschende Klasse sich lange Zeit mit Gewalt an der Macht behauptet, kann sie noch eine Zeit lang fortbestehen, indem sie ihren Gegnern den Frieden abkauft. So erkaufte sich laut Pareto das R”mische Imperium den Frieden durch Geld und Ehrungen oder die englische Aristrokatie, indem sie ihre Macht durch eine parlamentarische Monarchie verl„ngerte. In der Wirtschaft ist die Zirkulation „hnlich. Pareto wandte sich dagegen, die Bezieher von Zinseinnahmen und die Unternehmer in einen Topf zu werfen. Er teilte die kapitalistische Klasse in zwei Kategorien ein: die "R", die Rentiers, die Bezieher fester Einkommen und Sparer der Nation einerseits, und die "S", die Spekulanten. Die Spekulanten sind unternehmerisch mit groáem Risiko t„tig. Ihre Gruppe, stark und mutig, l”st die unorganisierten und in Paretos Augen feigen Rentiers ab. Die Rentiers und nicht die Arbeiter bilden die eigentliche unterdrckte Schicht. Denn unter Gewalt, Krieg, Zwangsanleihen, Falschgeld und Steuergesetzen wrden vor allem die Sparer einer Gesellschaft leiden. Pareto hielt die Spekulanten fr den haupts„chlichen Motor des ”konomischen und sozialen Wandels. Dagegen sah er in der Kategorie der Rentner ein m„chtiges Element der Stabilit„t, das oft die Gefahren aufhebt, die durch die Dynamik der Spekulanten entstehen. Die Sparer erfllten eine Aufgabe von gr”áter Wichtigkeit. "Sie „hneln den Bienen, die den Honig in den Bienenst”cken sammeln", vergleicht Pareto. Die Verm”gensanh„ufungen einer Gesellschaft werden indessen mit einiger Regelm„áigkeit durch Kriege oder Angriffe auf das Privateigentum zerst”rt. Entsprechend l„sst sich die Entwicklung des Reichtums einer Gesellschaft als wellenf”rmige Kurve um einen mittleren Trend herum darstellen. Bestechend sind Paretos historische Vergleiche. Oft erstrecken sich seine Analysen ber mehrere Seiten; er untersuchte zum Beispiel das R”mische Reich, die Franz”sische Revolution, das Franziskanertum, den Krieg zwischen Athen und Sparta oder auch die Politik Bismarcks. In der angels„chsischen Welt, aber auch in Frankreich und Italien stieá Paretos Werk auf ein groáes Echo. In Deutschland dagegen wurden Paretos wissenschaftliche Leistungen bis heute eher stiefmtterlich behandelt. Mehr als 80 Jahre nach dem Erscheinen des Trattato gibt es noch immer keine vollst„ndige deutsche šbersetzung des Werkes. Literatur: Vilfredo Pareto: Trait‚ de Sociologie G‚n‚rale Zeller Verlag GmbH & Co.; 1761 S., 260,- DM --- http://cepa.newschool.edu/het/profiles/pareto.htm The Italian economist Vilfredo Pareto was one of the leaders of the Lausanne School <../schools/lausanne.htm> and an illustrious member of the "second generation" of the Neoclassical revolution <../essays/margrev/margrev.htm>. Although only mildly influential during his lifetime, his "tastes-and-obstacles" approach to general equilibrium theory <../essays/paretian/paretocont.htm> were resurrected during the great "Paretian Revival <../schools/paretian.htm>" of the 1930s and have guided much of economics since. Vilfredo Pareto was born in the year of people's revolutions at its epicenter -- Paris, 1848 -- to an Italian aristocratic family. His father, a Ligurian marchese (marquis) and civil engineer, had fled to Paris in 1835 in self-imposed exile, following the example of Mazzini and other Italian nationalists. Vilfredo was the third child (and first son) of his marriage to a Frenchwoman. The Pareto family returned to Piedmont circa 1858. Following his father's footsteps, Vilfredo Pareto studied classics and then engineering at the Polytechnic Institute of Turin. It was here that he acquired his proficiency in mathematics and his basic ideas about mechanical equilibrium that were to characterize his later contributions to economics. After graduating at the top of his class in 1870, Pareto took his first job as a director of the Rome Railway Company. In 1874, Pareto become the managing director of an iron and steel concern, the Societ… Ferriere d'Italia in Florence. Pareto's stay in Florence was marked by political activity, much of it fuelled by his own frustrations with government regulators. After the Cavourist liberal government was replaced with a more interventionist government in Italy in 1876, Pareto was quick to identify the vested political interests that lay behind economic regulation, protectionism and nationalization that proceeded. A democratic republican and free-trader by instinct, Pareto deplored aristocratic and government corporatism. He saw the new Italian parliamentary system as a sham, a "pluto-democracy", a fig leaf for the naked power of the nobility and the wealthy. He sided with the radical democratic movements and the liberals <../schools/manchester.htm> whom, he believed, would replace privilege with meritocracy, restore real democracy, pursue free trade and true competition and promote social welfare. Pareto ran unsuccessfully for office on an opposition platform in the district of Pistoia in 1882. In 1889, after the death of his parents, Pareto changed his lifestyle. He inherited the marchese title, but he never used it. Instead, he quit his job, married a penniless Russian girl from Venice, Alessandrina Bakunin, and moved to a villa in Fiesole. From his retreat, he began writing numerous polemical articles against the government and gave public lectures at a working man's institute. He was quickly targeted as a troublemaker by the authorities. Trailed by police, intimidated by hired thugs, his lectures were often closed down and his applications for teaching jobs blocked. (incidentally, being well-trained with the sword, a crack shot with a pistol and equipped with an aristocratic sense of honor, Pareto never let himself be physically intimidated). His activities brought him to the attention of Maffeo Pantaleoni , then Italy's leading Neoclassical economist. A friendship sparked between the two men, and Pantaleoni introduced Pareto to economic theory, particularly the Walrasian <../schools/lausanne.htm> strand. Pareto, a quick learner with exceptionally good mathematical aptitude, took to it immediately and published several theoretical articles in the Giornale degli economisti. In the meantime, L‚on Walras was looking for someone to take over his chair in political economy at the University of Lausanne <../schools/lausanne.htm> in Switzerland. Pantaleoni recommended Pareto to him -- "He is an engineer like you; he is an economist not like you, but wishing to become like you, if you help him." Walras and Pareto disagreed on many economic policy issues such as free trade and the role of the State. They also had opposing temperaments -- Walras was a timid, bourgeois idealist while Pareto remained his caustic, disputatious, aristocratic self. In spite of this, Walras decided that Pareto ought to succeed him. Pareto was appointed in 1893, and his position at Lausanne made permanent in 1894. Although courteous and respectful to each other in public, Walras and Pareto did not get along very well. Doubtlessly, there were many people in Italy who were glad to see Pareto safely hidden away in Switzerland. But from his new academic perch, Pareto's nerve only increased. His attacks on the Italian government continued in his monthly column to the Giornale degli economisti and in foreign journals. He assisted and even housed many socialists and radicals that had been chased out of Italy (particularly after the 1898 May riots). When the Dreyfus affair broke in France, Pareto put his poison to work against the anti-Semitic authorities. Pareto also set himself to work, producing a three-volume edition of his lecture notes, Cours d'‚conomie politique (1896, 1897). This was more than merely an restatement of the doctrines of the Lausanne School <../schools/lausanne.htm>. Interspersed with his presentations of pure economic theory were numerous asides on methodology and applied economics and extensive sociological observations. His recent reading of Karl Marx and Social Darwinists like Herbert Spencer leaves its imprimatur. Mathematics was neatly relegated to footnotes and corners. In the Cours, his main economic contributions was his exposition of "Pareto's Law" of income distribution. He argued that in all countries and times, the distribution of income and wealth follows a regular logarithmic pattern that can be captured by the formula: log N = log A + m log x where where N is the number of income earners who receive incomes higher than x, and A and m are constants. Over the years, Pareto's Law has proved remarkably resilient in empirical studies. Pareto was also troubled with the concept of "utility". In its common usage, utility meant the well-being of the individual or society, but Pareto realized that when people make economic decisions, they are guided by what they think is desirable for them, whether or not that corresponds to their well-being. Thus, he introduced the term "ophelimity" to replace the worn-out "utility". Preferences was what Pareto wanted to get at. Another contribution of the Cours was Pareto's criticism of the marginal productivity theory of distribution <../essays/margrev/distrib.htm>, pointing out that it would fail in situations where there is imperfect competition or limited substitutability between factors. He'd repeat his criticisms in future writings. Also of importance was Pareto's observation that since the equilibrium is merely a solution to a set of simultaneous equations, then it is at least theoretically possible that a socialist or collectivist economy could "calculate" this solution and so attain exactly the same outcome as in a system guided by free markets. This proposition was picked up and extended by Enrico Barone and became the first shot of the famous socialist calculation debate <../essays/paretian/social.htm>, . In a famous 1900 Rivista article, Pareto suddenly changed direction. Heretofore a radical democrat, Pareto now decided to declare himself an anti-democrat. The disturbances of the 1890s in Italy and France led Pareto to realize that, far from restoring true democracy, meritocracy and promoting social welfare, the radical movements were really just seeking to replace one ‚lite with another ‚lite, the privileges and structures of power remaining intact. The struggle was not for a good society, but a squabble among ‚lites over whom exactly was to going to govern. And the ideals and theories they claimed to fight for? Just propaganda, Pareto declared, the way upwardly-mobile folks incite the helpless, hopeless mob to take to the streets on their behalf. For Pareto, humanitarianism, liberalism, socialism, communism, fascism, whatever, were all the same in the end. All ideologies were just smokescreens foisted by "leaders" who really only aspired to enjoy the privileges and powers of the governing ‚lite. Pareto decided to have none of it -- and went on a crusade to expose the sham of political ideology and doctrine. He condemned socialists <../schools/utopia.htm> of all stripes roundly in a 1902 book, but took particular aim at logically demolishing the "new gospel" of Marxian economics <../schools/marxian.htm>. As revealed in the Cours and in his own introduction to an abridged 1893 edition of Karl Marx 's Capital, Pareto applauded Marxian theories of class struggle and even thought historical materialism was on the right track (albeit not deep and general enough, in his view). But he deplored Marx's Wizard-of-Oz-like conclusion. For Pareto, class struggle is eternal; the promised "classless" society that would emerge under communism was merely ideological fodder for socialist leaders to lay on their flock. Of course, as a good Neoclassical, Pareto could not fathom the labor theory of value either. In 1906, Pareto published his Manual of Political Economy, his magnum opus on pure economics and moved him out of the shadow of Walras . Unlike the Cours, the Manual concentrates on presenting pure economics in an explicitly mathematical form (especially after it was heavily revised for the 1909 French edition). The Walrasian equations are still there, but the focus is on formulating equilibrium in terms of solutions to individual problems of "objectives and contraints <../essays/paretian/paretequil.htm>". To illustrate this, the indifference curve of Edgeworth (1881) was employed extensively -- both in his theory of the consumer and, another great novelty, in his theory of the producer <../essays/product/decision.htm>. It is in the Manual that we find the first representation of what has since become known (and misnamed) as the "Edgeworth-Bowley" box <../essays/margrev/walrex.htm>. Like Irving Fisher (1892), Pareto stumbled on the idea that cardinal utility could be dispensed with. Preferences were the primitive datum, and utility a mere representation of preference-ordering. With this, Pareto not only inaugurated modern microeconomics, but he also demolished the "unholy alliance" of economics and utilitarianism <../schools/utilitar.htm>. In its stead, he introduced the notion of Pareto-optimality <../essays/paretian/paretoptimal.htm>, the idea that a society is enjoying maximum ophelimity when no one can be made better off without making someone else worse off. (for more details, see our discussion of the Paretian general equilibrium system <../essays/paretian/paretocont.htm>). His sociological <../schools/optimist.htm> observations also begin to indicate the future course of his ideas. In 1900, Pareto had entered into a brief controversy in the Giornale degli economisti with Benedetto Croce. Croce had criticized economists' positivistic approach, particularly the assumption of "rational economic man". Pareto defended economists, but, at the same time, realized that the conventional defense was not even convincing enough to himself. Why did the predictions of economics fail to correspond to reality? Why were its policy recommendations, to him logically irrefutable, not adopted? The explanation, he concluded, echoing Georges Sorel , was simply that much of human activity was driven not by logical action, but rather by non-logical action. On this, of course, economics has nothing to say -- which is why, ultimately, economics will always fail empirically. Pareto realized that he had to move beyond economics to look for his answer. Pareto retired from his chair at Lausanne in 1907, gradually passing on his teaching responsibilities to Pasquale Boninsegni <../schools/lausanne.htm>. He moved to Villa Angora in C‚ligny, near Lake Geneva. There he was nursing a heart disease, surrounded by a dozen cats, his enormous personal library, a cellar full of superb wines and a large cabinet of exquisite liquers. His wife ran off in 1901, but, as an Italian citizen, he could not legally divorce her. A Frenchwoman, Jane R‚gis moved in shortly afterwards, and they remained devoted companions for the rest of his life. He only married her in 1923, after he became a citizen of the city-state of Fiume and thus overcame the legal obstacles to divorce. Pareto used his time at C‚ligny to write his Trattato di sociologia generale, which was finally published, after wartime delays, in 1916. This was his great sociological <../schools/optimist.htm> masterpiece. He explains how human action can be neatly reduced to residue and derivation. People act on the basis of non-logical sentiments (residues) and invent justifications for them afterwards (derivations). The derivation is thus just the content and form of the ideology itself. But the residues are the real underlying problem, the particular cause of the squabbles that leads to the "circulation of ‚lites". The underlying residue, he thought, was the only proper object of sociological enquiry. Residues are non-logical sentiments, rooted in the basic aspirations and drives of people. He identifies six classes of residues, all of which are present but unevenly distributed across people -- so the population is always a heterogeneous, differentiated mass of different psychic-types. The most important residues are Class I the "instinct for combining" (innovation) and Class II, the "persistence of aggregates" (conservation). Class I types rule by guile, and are calculating, materialistic and innovating. Class II types rule by force and are more bureaucratic, idealistic and conservative. Pareto's theory of society claimed that there was a tendency to return to an equilibrium where a balanced amount of Class I and Class II people are present in the governing ‚lites. People are always entering and leaving the ‚lite thereby tending to restore the natural balance. On occasion, when it gets too lopsided, an ‚lite will be replaced en masse by another If there are too many Class I people in a governing ‚lites, this means that violent, conservative Class II's are in the lower echelons, itching and capable of taking power when the Class I's finally make a mess of things by too much cunning and corruption (he regarded Napoleon III's France and the Italian "pluto-democratic" system as an example). If the governing ‚lite is composed mostly of Class II types, then it will fall into a bureaucratic, inefficient and reactionary mess, easy prey for calculating upwardly-mobile Class I's (e.g. Tsarist Russia). Pareto colored his sociological theory with numerous classical and contemporary illustrations of his theory. He published two more books (1920, 1921) expanding on the theme. His quasi-mystical arguments about the non-logical motivations attracted many Italian Fascists (Mussolini himself claimed to have attended his lectures at Lausanne). Pareto, however, was largely disdainful of the Fascist movement -- he never had patience for ideologies or ideologues -- but he found them quite amusing. When Mussolini's small band of Class II Fascists marched on Rome in 1922 and brought the whole Class I-dominated Italian government tumbling down, Pareto mumbled triumphantly in his sick-bed, "I told you so!". He was not unhappy at the turn of events. The Fascists showered Pareto with honors from afar, making him a Senator of the Kingdom of Italy, inviting him to join the Italian delegation to the Geneva Disarmament Conference, asking him to contribute to the Fascist party periodicals, etc. He declined most of the honors, but spoke favorably of certain early reforms undertaken by the Fascists. However, he also warned them to avoid despotism, censorship and economic corporatism. When the Fascists clamped down on freedom of expression in Italian universities, Pareto managed to rouse himself to write a protest. Pareto died a mere ten months into Mussolini's reign -- before the uglier aspects of Fascism became obvious. The Fascists continued to use his name unreservedly to give intellectual veneer to their movement. Writing in 1938 on the legacy of Pareto, the economist (and Fascist) Luigi Amoroso <../schools/lausanne.htm> would have the gumption to write (and Econometrica the editorial lapse to publish) the following: "Just as the weaknesses of the flesh delayed, but could not prevent, the triumph of Saint Augustine, so a rationalistic vocation retarded but did not impede the flowering of the mysticism of Pareto. For that reason, Fascism, having become victorious, extolled him in life, and glorifies his memory, like that of a confessor of its faith." (Luigi Amoroso <../schools/lausanne.htm>, "Vilfredo Pareto", Econometrica, 1938: p.21) Despite his association with Fascism, Pareto's sociological work has been taken seriously, going through recurring phases of popularity and critical scrutiny. Freudian psychology has given much weight to some of his notions. It is not so much its main thrust, but its roughness, simplicity and incompleteness that are the main sources of complaint. Pareto's economics have had a much greater impact. Pareto managed to construct a proper school around himself at Lausanne <../schools/lausanne.htm>, including G.B. Antonelli <../schools/lausanne.htm>, Boninsegni <../schools/lausanne.htm>, Amoroso <../schools/lausanne.htm> and others as disciples. Outside this small group, his work also influenced W.E. Johnson <../schools/engmath.htm>, Eugen Slutsky and Arthur Bowley . But Pareto's big break came posthumously in the 1930s and 1940s, a period which we have decided to call the "Paretian Revival <../schools/paretian.htm>". His "tastes-and-obstacles" approach to demand were resurrected by John Hicks and R.G.D. Allen (1934) and extended and popularized by John Hicks (1939), Maurice Allais (1943) and Paul Samuelson (1947). Pareto's work on welfare were resurrected by Harold Hotelling , Oskar Lange and the "New Welfare Economics" <../essays/paretian/paretosocial.htm> movement. Finally, Pareto's ruminations on the potential efficiency of a collectivist society were aired in the Socialist Calculation Debate <../essays/paretian/social.htm> that arose between the Paretians <../schools/paretian.htm> and the Austrians <../schools/austrian.htm>. Major Works of Vilfredo Pareto Principii Fondamentali della Teorie dell' Elasticit…, 1869. "Della logica delle nuove scuole economiche", speech to Accademia dei Gerogofili, 1877. "L'Italie ‚conomique <../texts/pareto/paretoitalie.pdf>", 1891, Revue des deux mondes "Les nouvelles th‚ories ‚conomiques", 1892, Le monde ‚conomique "Considerazioni sui principi fondamentali dell'economia politica pura", 1893, Giornale degli Economisti. "Introduction" to K. Marx , Capital, 1893. Le‡on d'‚conomie pure … l'Universit‚ de Lausanne , 1893 (unpublished) "The Parliamentary Regime in Italy", 1893, American Poli Sci Quarterly La libert‚ ‚conomique et les ‚v‚nements d'Italie. "La courbe des revenus ", 1896, Le Monde economique (French/Italian) Cours d'‚conomie politique profess‚ … l'universit‚ de Lausanne, 3 volumes, 1896-7. "The New Theories of Economics ", 1897, JPE. "Comment se pose le problŠme de l'‚conomie pure?", Notes to Association Stella,1898 (publ. 1965) "Un' Applicazione di teorie sociologiche", 1900, Rivista Italiana di Sociologia (transl. in English as The Rise and Fall of the Elites) "On the Economic Phenomenon", 1900, GdE (repr. 1953, IEP) "Le nuove toerie economiche (con in appendice le equazioni dell' equilibrio dinamico)", 1901, GdE "De l'‚conomique, discours d'installation de M.V. Pareto … professeur ordinaire", Lausanne, 1901 (publ. 1965) Les systŠmes socialistes, 1902. L'‚conomie pure, resum‚ du cours donn‚ a l'Ecole des Hautes Etudes Sociales de Paris, 1902 "Review of Aupetit", 1902, Revue d'econ politique "Anwendungen der Mathematik auf National”konomie", 1903, Encyklop”die der Mathematischen Wissenschaften "Il Crepuscolo della Libert…", 1905, Rivista d'Italia. Manual of Political Economy , 1906 (Italian; French transl., 1909, English transl, 1971). "L'‚conomie et la sociologie au point de vue scientifique", 1907, Rivista di Scienza. "Economie math‚matique", 1911, in Gauthier-Villars, Encyclopedie des sciences mathematiques. Le mythe vertuiste et la litt‚rature immorale. 1911 "Introduction" to G. Osorio, Th‚orie mathematique de l'‚change, 1913. Trattato di Sociologia Generale, 1916. (transl. in English as Mind and Society. Extracts (1) , (2) , (3) ; extract in Spanish ) "Discorso per il Giubileo", 1917, La Riforma Sociale - Jubillee speech manuscript "Formi di fenomeni economici e previsioni", 1917, Riv di Sci Banc Fatti e Teorie, 1920 Trasformazione della Democrazia, 1921. Mon Journal, 1958 Scritti sociologici di Vilfredo Pareto, 1966. Oeuvres complŠtes de Vilfredo Pareto , ed. G. Busino Resources on Vilfredo Pareto HET Pages: The Paretian System <../essays/paretian/PARETOcont.htm>: Equilibrium <../essays/paretian/paretequil.htm>, Efficiency <../essays/paretian/paretoptimal.htm>, Social Welfare <../essays/paretian/Paretosocial.htm>, the Production Decision <../essays/product/decision.htm>, Marginal Productivity Theory of Distribution <../essays/margrev/distrib.htm> "Review of Bortkiewicz's Anwendungen and Pareto's Anwendungen <../texts/edgeworth/edgew3bortk.pdf>", by Francis Ysidro Edgeworth , 1903, EJ "Review of Pareto's Manuale di Economia Politica <../texts/wicksteed/wickspareto06.htm>", by Philip H. Wicksteed , 1906, EJ "Recent Contributions to Mathematical Economics, I & II <../texts/edgeworth/edgew2theta.pdf>", by Francis Ysidro Edgeworth ,1915, EJ "Review of Pareto's Cours d'‚conomie politique, Vol. 1 <../texts/pareto/paretosorel1.pdf>" by Georges Sorel <../schools/utopia.htm>, 1896, Le devenir social, Ann. 2, N. 5, May. "Review of Pareto's Cours d'‚conomie politique, Vol. 2 <../texts/pareto/paretosorel2.pdf>" by Georges Sorel <../schools/utopia.htm>, 1897, Le devenir social, Ann. 3, N. 5, May. "Review of Pareto's Manuale di Economia Politica <../texts/pareto/paretohalbwachs.pdf>" by M. Halbwachs, 1906, L'ann‚e sociologique, Ann. 10 "Review of Jevons's Theory, Pareto's Manuel and Marshall's Principles <../texts/pareto/paretosimiand.pdf>" by Fran‡ois Simiand, 1909, L'ann‚e sociologique, Ann. 11 "Review of Pareto's Manuel d'‚conomie politique" by E. d'Eichthal, 1909, Revue critique d'histoire et de litt‚rature "Review of Pareto's Trait‚ de sociologie g‚n‚rale <../texts/pareto/paretobougle.pdf>" by C. Bougl‚, 1919, Revue historique "Review of Pareto's Trait‚ de sociologie g‚n‚rale <../texts/pareto/paretofbd.pdf>" by F. Bd., 1919, Revue critique d'histoire et de litt‚rature Cercle d'‚tudes Paretiennes at Lausanne. Centre d'‚tudes interdisciplinaires Walras-Pareto at Lausanne Le Fonds Pareto at CWP Lausanne Il Fondo Pareto of the Banca Popolare de Sondria Annoucement of Oeuvres complŠtes de Vilfredo Pareto , ed. G. Busino Biographie de Pareto at CWP Lausanne Biography of Pareto - in English. "Vilfredo Pareto: Concise Overview of His Life, Works, and Philosophy " by Fr. James Thornton "Pareto in Toscana " by Alberto Zanni, 1999, SdPE. "More on Slutsky's Equation as Pareto's Solution ", by C.E. Weber, 1999, HOPE "La raccolta dei documenti di Vilfredo Pareto ", by P. C. Ferrara, 1997 Vilfredo Pareto page at Dead Sociologists Index Vilfredo Pareto page at Marxists.org "Vilfredo Pareto: Il Gioco del Potere? " by Franco Gianola Pareto dressed as a Bedouin Arab on the way to a masked ball. (source ) Pareto Page at McMaster Pareto Page at Akamac Pareto Page at Laura Forgette Selected Primary and Secondary Works of Vilfredo Pareto at Bristol, UK Pareto Page at R. Dixon, Melbourne Walras, Pareto: L'Ecole de Lausanne - in French Essay on "Pareto Analysis" Other Images of Pareto - (1) , (2) Books on Pareto Pareto page at Britannica.com Philosophy of Pareto Pareto Page (in French) Pareto Consulting ?! (don't ask) +( Pareto principle and TOC To Subject [tocexperts] Comparison System Ah, I have the answer to my own question. The question was, "More importantly, the tests must compare TOC to some other existing theory, and check which is a better predictor of future performance. What might that be?" Turns out Tony and I discussed this a couple of years ago. The system model we can compare to is the Pareto model, the current system model you find in all management books, and a mainstay of Six Sigma. Let Y be Throughput towards the Goal. The Pareto model defines Y as the system output, and Xi as the independent variables. It claims Y = f ( X1, X2, ...Xn). TOC says Y = f (Xj) only, where Xj is the constraint. The 'cost world' that Eli defines is a special case of the Pareto model, where Y = X1+X2...+Xn So, now we have not only a comparison system, but a quantitative definition of TOC. It ought to be testable. Regards, Larry Leach