Coefficient Quotes

Every woman I had ever met who walked through the world appraised and classified by an extraordinary physicality had also received the keys to an unbearable solitude. It was the coefficient of their beauty, the price they had to pay.

What’s Julie’s number?” Curran glanced at me. “Julie’s fluctuating between thirty-two and thirty-four units. Her shift coefficient is six point five and she’s been at it for sixteen hours.” Dear God, I’d need a damn calculator.

The Mind is the sole coefficient of Time and Space. - Wole Soyinka

Well, the Taco Bell burrito scale of immense magnitude returned an 'r' factor of point eight six. Then when I applied the nose-picking coefficient, I discovered a multivariate numeration of nine dot oh sixteen on the Richter scale.

The most widely accepted measure for calculating income inequality is a century-old formula called the Gini coefficient. It's a gold standard for economists around the globe, along with the World bank, the CIA, and the Paris-based Organization for Economic Cooperation and Development. What it reveals is startling. Today the United States has the most unequal society of all developed nations. America’s level of inequality is comparable to that of Russia, China, Argentina, and the war-torn Democratic Republic of the Congo.

This positive void coefficient remained a fatal defect at the heart of Atom Mirny-1 and overshadowed the operation of every Soviet water-graphite reactor that followed.

You Bastard was thinking: there seems to be some growing dimensional instability here, swinging from zero to nearly forty-five degrees by the look of it. How interesting. I wonder what’s causing it? Let V equal 3. Let Tau equal Chi/4. cudcudcud Let Kappa/y be an Evil-Smelling-Bugger* (* Renowned as the greatest camel mathematician of all time, who invented a math of eight-dimensional space while lying down with his nostrils closed in a violent sandstorm.) differential tensor domain with four imaginary spin co-efficients. . .

Simple problems are hard to solve, Because they need common sense. Simple problems are made complex. Complex things are solved using patterns. Coefficient is introduced along with variable to create a pattern, But coefficient is a constant. To find coefficient, We again use complex patterns to make it look constant.

The Gini Coefficient quantifies how large a percentage of the total income of a society must be redistributed in order to achieve a perfectly equal distribution of wealth.

But eighteen seconds is a long time in neutron physics—and an eternity in a nuclear reactor with a high positive void coefficient.

Those who are unacquainted with the details of scientific investigation have no idea of the amount of labour expended in the determination of those numbers on which important calculations or inferences depend. They have no idea of the patience shown by a Berzelius in determining atomic weights; by a Regnault in determining coefficients of expansion; or by a Joule in determining the mechanical equivalent of heat.

Life, my dear Mamselle, can't be reckoned up correctly without cooking the accounts a bit, and our mistake lies in this: that when we grapple with great things, we never take the human coefficient into consideration. All the confusion comes from that...Don't be upset by the coefficient, Mamselle. It contains all the savor and glamor of life. Otherwise every lout would just drink up life to the dregs, and then put a bullet into his brain...Because then his brain would ask for something beyond life...No matter what happens, keep on living, Mamselle. A living human being is, after all, Nature's most beautiful creation.

Charles had an inbreeding coefficient of 0.254, making him slightly more inbred than a child of two siblings (0.250). He suffered from extensive physical and emotional disabilities, and was a strange (and largely ineffective) king.

Facebook automatically catalogued every tiny action from its users, not just their comments and clicks but the words they typed and did not send, the posts they hovered over while scrolling and did not click, and the people's names they searched and did not befriend. They could use that data, for instance, to figure out who your closest friends were, defining the strength of the relationship with a constantly changing number between 0 and 1 they called a "friend coefficient". The people rated closest to 1 would always be at the top of your news feed.

A good manager drives a project to be good enough, fast enough, cheap enough, and done as much as necessary. A good manager manages the coefficients on these attributes rather than demanding that all those coefficients are 100%. It is this kind of management that Agile strives to enable.

But as no two (theoreticians) agree on this (skin friction) or any other subject, some not agreeing today with what they wrote a year ago, I think we might put down all their results, add them together, and then divide by the number of mathematicians, and thus find the average coefficient of error. (1908)

In these times, when so wide a gulf has opened between the rich and the poor, which, instead of narrowing, as all good men would have it, grows broader daily; it is most important that all ranks and degrees of people should understand whose hands are stretched out to separate these two great divisions of society each of whom, for its strength and happiness, and the future existence of this country, as a great and powerful nation, is dependent on the other.

A government commission into the accident found serious faults with the design, and in 1976 recommended that the void coefficient be lowered, the control rod design be altered, and for ‘fast-acting emergency protection’ to be installed. New designs were drawn up for the rods, but were never installed on any reactors.

statistical indices such as the Gini coefficient give an abstract and sterile view of inequality, which makes it difficult for people to grasp their position in the contemporary hierarchy (always a useful exercise, particularly when one belongs to the upper centiles of the distribution and tends to forget it, as is often the case with economists).

Furious, the beast writhed and wriggled its iterated integrals beneath the King’s polynomial blows, collapsed into an infinite series of indeterminate terms, then got back up by raising itself to the nth power, but the King so belabored it with differentials and partial derivatives that its Fourier coefficients all canceled out (see Riemann’s Lemma), and in the ensuing confusion the constructors completely lost sight of both King and beast. So they took a break, stretched their legs, had a swig from the Leyden jug to bolster their strength, then went back to work and tried it again from the beginning, this time unleashing their entire arsenal of tensor matrices and grand canonical ensembles, attacking the problem with such fervor that the very paper began to smoke. The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F1) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out, fuming factorially, and fell upon the King and hurt him passing sore. But the King, nothing daunted, put on his Markov chain mail and all his impervious parameters, took his increment Δk to infinity and dealt the beast a truly Boolean blow, sent it reeling through an x-axis and several brackets—but the beast, prepared for this, lowered its horns and—wham!!—the pencils flew like mad through transcendental functions and double eigentransformations, and when at last the beast closed in and the King was down and out for the count, the constructors jumped up, danced a jig, laughed and sang as they tore all their papers to shreds, much to the amazement of the spies perched in the chandelier-—perched in vain, for they were uninitiated into the niceties of higher mathematics and consequently had no idea why Trurl and Klapaucius were now shouting, over and over, “Hurrah! Victory!!

Personal knowledge is an intellectual commitment, and as such inherently hazardous. Only affirmations that could be false can be said to convey objective knowledge of this kind. All affirmations published in this book are my own personal commitments; they claim this, and no more than this, for themselves. Throughout this book I have tried to make this situation apparent. I have shown that into every act of knowing there enters a passionate contribution of the person knowing what is being known, and that this coefficient is no mere imperfection but a vital component of his knowledge. And around this central fact I have tried to construct a system of correlative beliefs which I can sincerely hold, and to which I can see no acceptable alternatives. But ultimately, it is my own allegiance that upholds these convictions, and it is on such warrant alone that they can lay claim to the reader’s attention. M. P. Manchester August 1957

The Gini coefficient, devised by the Italian sociologist Corrado Gini in 1912, is a measure of income or wealth disparity in a population. It is usually expressed as a fraction between 0 and 1, and it seems easy to understand, because 0 is the coefficient if everyone owned an equal amount, while 1 would obtain if one person owned everything and everyone else nothing. In our real world of the mid-twenty-first century, countries with a low Gini coefficient, like the social democracies, are generally a bit below 0.3, while highly unequal countries are a bit above 0.6. The US, China, and many other countries have seen their Gini coefficients shoot up in the neoliberal era, from 0.3 or 0.4 up to 0.5 or 0.6, this with barely a squeak from the people losing the most in this increase in inequality, and indeed many of those harmed often vote for politicians who will increase their relative impoverishment. Thus the power of hegemony: we may be poor but at least we’re patriots! At least we’re self-reliant and we can take care of ourselves, and so on, right into an early grave, as the average lifetimes of the poorer citizens in these countries are much shorter than those of the wealthy citizens. And average lifetimes overall are therefore decreasing for the first time since the eighteenth century. Don’t think that the Gini coefficient alone will describe the situation, however; this would be succumbing to monocausotaxophilia, the love of single ideas that explain everything, one of humanity’s most common cognitive errors. The

The most widely accepted measure for calculating income inequality is a century-old formula called the Gini coefficient. It’s a gold standard for economists around the globe, along with the World Bank, the CIA, and the Paris-based Organization for Economic Cooperation and Development. What it reveals is startling. Today the United States has the most unequal society of all developed nations. America’s level of inequality is comparable to that of Russia, China, Argentina, and the war-torn Democratic Republic of the Congo.

In the villages and factories, people from the district committees of the Communist Party travelled around, meeting people. Yet not one of them was capable of giving an answer if they were asked what decontamination was, how children could be protected, or what the coefficients were for radionuclides finding their way into the food chain. Neither could they if asked about alpha, beta and gamma particles, nor about radiobiology, ionizing radiation, let alone isotopes. For them, that was all something from another planet. They gave lectures about the heroism of Soviet people, symbols of military courage, and the wiles of Western intelligence services.

Major changes were made to the RBMK design, including improving the speed at which control rods entered the core during a SCRAM event, lowering the time for a complete insertion from 18 seconds to 12; reducing the positive steam void coefficient of reactivity, and the effect of reactivity if there was a complete void in the core; installation of a Fast Acting Emergency Protection system, complete with an additional 24 control rods; removing the ability to bypass emergency protection systems while the reactor was at power, and, most importantly, a new control rod layout with a longer boron section and no empty/water section ahead of it. The graphite tip remained.264

You can write great books," the great man continued. "Or you can have kids. It's up to you." [...] Writing was a practice. The more you wrote, the better a writer you became, and the more books you produced. Excellence plus productivity, that was the formula for sustained success, and time was the coefficient of both. Children, the great man said, were notorious thieves of time. [...] Writers need to be irresponsible, ultimately, to everything but the writing, free of commitments to everything but the daily word count. Children, by contrast, needed stability, consistency, routine, and above all, commitment. In short, he was saying, children are the opposite of writing.

This was, he told the King, a femfatalatron, an erotifying device stochastic, elastic and orgiastic, and with plenty of feedback; whoever was placed inside the apparatus instantaneously experienced all the charms, lures, wiles, winks and witchery of all the fairer sex in the Universe at once. The femfatalatron operated on a power of forty megamors, with a maximum attainable efficiency—given a constant concupiscence coefficient—of ninety-six percent, while the system's libidinous lubricity, measured of course in kilocupids, produced up to six units for every remote-control caress. This marvelous mechanism, moreover, was equipped with reversible ardor dampers, omnidirectional consummation amplifiers, absorption philters, paphian peripherals, and "first-sight" flip-flop circuits, since Trurl held here to the position of Dr. Yentzicus, creator of the famous oculo-oscular feel theory. There were also all sorts of auxiliary components, like a high-frequency titillizer, an alternating tantalator, plus an entire set of lecherons and debaucheraries; on the outside, in a special glass case, were enormous dials, on which one could carefully follow the course of the whole decaptivation process. Statistical analysis revealed that the femfatalatron gave positive, permanent results in ninety-eight cases of unrequited amatorial superfixation out of a hundred.

In a hypothetical, extremely simple Cloud Ark consisting of only two arklets, only one calculation needed to be performed: namely, the calculation that answered the question “Will Arklet 1 bang into Arklet 2 if both stay on their current courses?” In a three-arklet cloud, it was also necessary to figure out whether Arklet 1 would collide with Arklet 3, and whether 2 and 3 were going to collide. So, that was a total of three calculations. If the cloud expanded to four arklets, six calculations were needed, and so on. In mathematical terms these were known as triangular numbers, a kind of binomial coefficient, but the bottom line was that the number of calculations went up rapidly with the number of arklets in the cloud.

And by the early 1970s our little parable of Sam and Sweetie is exactly what happened to the North American Golden Retriever. One field-trial dog, Holway Barty, and two show dogs, Misty Morn’s Sunset and Cummings’ Gold-Rush Charlie, won dozens of blue ribbons between them. They were not only gorgeous champions; they had wonderful personalities. Consequently, hundreds of people wanted these dogs’ genes to come into their lines, and over many matings during the 1970s the genes of these three dogs were flung far and wide throughout the North American Golden Retriever population, until by 2010 Misty Morn’s Sunset alone had 95,539 registered descendants, his number of unregistered ones unknown. Today hundreds of thousands of North American Golden Retrievers are descended from these three champions and have received both their sweet dispositions and their hidden time bombs. Unfortunately for these Golden Retrievers, and for the people who love them, one of these time bombs happens to be cancer. To be fair, a so-called cancer gene cannot be traced directly to a few famous sires, but using these sires so often increases the chance of recessive genes meeting—for good and for ill. Today, in the United States, 61.4 percent of Golden Retrievers die of cancer, according to a survey conducted by the Golden Retriever Club of America and the Purdue School of Veterinary Medicine. In Great Britain, a Kennel Club survey found almost exactly the same result, if we consider that those British dogs—loosely diagnosed as dying of “old age” and “cardiac conditions” and never having been autopsied—might really be dying of a variety of cancers, including hemangiosarcoma, a cancer of the lining of the blood vessels and the spleen. This sad history of the Golden Retriever’s narrowing gene pool has played out across dozens of other breeds and is one of the reasons that so many of our dogs spend a lot more time in veterinarians’ offices than they should and die sooner than they might. In genetic terms, it comes down to the ever-increasing chance that both copies of any given gene are derived from the same ancestor, a probability expressed by a number called the coefficient of inbreeding. Discovered in 1922 by the American geneticist Sewall Wright, the coefficient of inbreeding ranges from 0 to 100 percent and rises as animals become more inbred.

There’s one major shortcoming inherent to using this method of cooling. Unlike in a typical PWR, the water entering the reactor is the same water that passes through the cooling pumps and then as steam through the turbines, meaning highly irradiated water is present in all areas of the system. A PWR uses a heat exchanger to pass heat from the reactor water to clean, lower pressure water, allowing the turbines to remain free of contamination. This is better for safety, maintenance and disposal. A second problem is that steam is allowed to form in the core, making dangerous steam voids more likely, and further increasing the chances of a positive void coefficient. In ordinary boiling water reactors, which use water as both a coolant and moderator like in a PWR, this would not be such a problem, but it is in a graphite-moderated BWR.

Separation of function is not to be despised, but neither should it be exalted. Separation is not an unbreakable law, but a convenience for overcoming inadequate human abilities, whether in science or engineering. As D'Arcy Thompson, one of the spiritual fathers of the general systems movement, said: As we analyze a thing into its parts or into its properties, we tend to magnify these, to exaggerate their apparent independence, and to hide from ourselves (at least for a time) the essential integrity and individuality of the composite whole. We divided the body into its organs, the skeleton into its bones, as in very much the same fashion we make a subjective analysis of the mind, according to the teaching of psychology, into component factors: but we know very well that judgement and knowledge, courage or gentleness, love or fear, have no separate existence, but are somehow mere manifestations, or imaginary coefficients, of a most complex integral.10 The

Page 50: It is a common misconception that psychological measurements of human abilities are generally more prone to error or inaccuracy than are physical measurements. In most psychological research, and especially in psychometrics, this kind of measurement error is practically negligible. If need be, and with proper care, the error variance can usually be made vanishingly small. In my laboratory, for example, we have been able to measure such variables as memory span, flicker-fusion frequency (a sensory threshold), and reaction time (RT) with reliability coefficients greater than .99 (that is, less than 1 percent of the variance in RT is due to errors of measurement). The reliability coefficients for multi-item tests of more complex mental processes, such as measured by typical IQ tests, are generally about .90 to .95. This is higher than the reliability of people's height and weight measured in a doctor's office! The reliability coefficients of blood pressure measurements, blood cholesterol level, and diagnosis based on chest X-rays are typically around .

The main circulating pumps began to cavitate and fill with steam, reducing the flow of valuable cooling water and allowing steam voids (pockets of steam where there should be water) to form in the core. A positive void coefficient was occurring: the absence of cooling water causing an exponential power increase. In simple terms, more steam = less water = more power = more heat = more steam. Because 4 of the 8 water pumps were running off the decelerating turbine, less and less water was supplied to the reactor as power increased. Throughout the building, ‘knocks’ were heard from the direction of the main reactor hall. Akimov’s control board indicated that the rods hadn’t moved far before freezing, only 2.5 meters from their raised position. Thinking quickly, he released the clutch on their servomotors to allow the heavy rods to fall into the core under their own weight, but they didn’t move: jammed. “I thought my eyes were coming out of my sockets. There was no way to explain it,” recalled Dyatlov, six years later. “It was clear that this was not a normal accident, but something much more terrible. It was a catastrophe.”118

The term ‘inequality’ is a way of framing social problems appropriate to an age of technocratic reformers, who assume from the outset that no real vision of social transformation is even on the table. Debating inequality allows one to tinker with the numbers, argue about Gini coefficients and thresholds of dysfunction, readjust tax regimes or social welfare mechanisms, even shock the public with figures showing just how bad things have become (‘Can you imagine? The richest 1 per cent of the world’s population own 44 per cent of the world’s wealth!’) – but it also allows one to do all this without addressing any of the factors that people actually object to about such ‘unequal’ social arrangements: for instance, that some manage to turn their wealth into power over others; or that other people end up being told their needs are not important, and their lives have no intrinsic worth. The last, we are supposed to believe, is just the inevitable effect of inequality; and inequality, the inevitable result of living in any large, complex, urban, technologically sophisticated society. Presumably it will always be with us. It’s just a matter of degree.

So they rolled up their sleeves and sat down to experiment -- by simulation, that is mathematically and all on paper. And the mathematical models of King Krool and the beast did such fierce battle across the equation-covered table, that the constructors' pencils kept snapping. Furious, the beast writhed and wriggled its iterated integrals beneath the King's polynomial blows, collapsed into an infinite series of indeterminate terms, then got back up by raising itself to the nth power, but the King so belabored it with differentials and partial derivatives that its Fourier coefficients all canceled out (see Riemann's Lemma), and in the ensuing confusion the constructors completely lost sight of both King and beast. So they took a break, stretched their legs, had a swig from the Leyden jug to bolster their strength, then went back to work and tried it again from the beginning, this time unleashing their entire arsenal of tensor matrices and grand canonical ensembles, attacking the problem with such fervor that the very paper began to smoke. The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F_1) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out fuming factorially, and fell upon the King and hurt him passing sore. But the King, nothing daunted, put on his Markov chain mail and all his impervious parameters, took his increment Δk to infinity and dealt the beast a truly Boolean blow, sent it reeling through an x-axis and several brackets—but the beast, prepared for this, lowered its horns and—wham!!—the pencils flew like mad through transcendental functions and double eigentransformations, and when at last the beast closed in and the King was down and out for the count, the constructors jumped up, danced a jig, laughed and sang as they tore all their papers to shreds, much to the amazement of the spies perched in the chandelier—perched in vain, for they were uninitiated into the niceties of higher mathematics and consequently had no idea why Trurl and Klapaucius were now shouting, over and over, "Hurrah! Victory!!

Using graphite as a moderator can be highly dangerous, as it means that the nuclear reaction will continue - or even increase - in the absence of cooling water or the presence of steam pockets (called ‘voids’). This is known as a positive void coefficient and its presence in a reactor is indicative of very poor design. Graphite moderated reactors were used in the USA in the 1950s for research and plutonium production, but the Americans soon realised their safety disadvantages. Almost all western nuclear plants now use either Pressurised Water Reactors (PWRs) or Boiling Water Reactors (BWRs), which both use water as a moderator and coolant. In these designs, the water that is pumped into the reactor as coolant is the same water that is enabling the chain reaction as a moderator. Thus, if the water supply is stopped, fission will cease because the chain reaction cannot be sustained; a much safer design. Few commercial reactor designs still use a graphite moderator. Other than the RBMK and its derivative, the EGP-6, Britain’s Advanced Gas-Cooled Reactor (AGR) design is the only other graphite-moderated reactor in current use. The AGR will soon be joined by a new type of experimental reactor at China’s Shidao Bay Nuclear Power Plant, which is currently under construction. The plant will house two graphite-moderated ‘High Temperature Reactor-Pebble-bed Modules’ reactors, the first of which is undergoing commissioning tests as of mid-2019.

Although Dyatlov, Shift Foreman Akimov, and Senior Reactor Control Engineer Toptunov had violated some operating regulations, they were ignorant of the deadly failing of the RBMK-1000 that meant that insertion of the control rods, instead of shutting down the reactor at the end of the test, could initiate a runaway chain reaction. Every one of the investigators behind the report now agreed that the fatal power surge that destroyed the reactor had begun with the entry of the rods into its core. ‘Thus the Chrnobyl accident comes within the standard pattern of most severe accidents in the world. It begins with an accumulation of small breaches of the regulations. … These produce a set of undesirable properties and occurrences that, when taken separately, do not seem to be particularly dangerous, but finally an initiating event occurs that, in this particular case, was the subjective actions of the personnel that allowed the potentially destructive and dangerous qualities of the reactor to be released.’ IAEA experts revealed at last the true magnitude of the technical cover-up surrounding the causes of the disaster: the long history of previous RBMK accidents, the dangerous design of the reactor, its instability, and the way its operators had been misled about its behavior. In dense scientific detail, it described the inherent problems of the positive void coefficient and the fatal consequences of the control rod ‘tip’ effect. (pp. 347-348)

Much more than skeleton, it is flash, I mean the carrion flesh, which disturb and alarm us – and which alleviates us as well. The Buddhists monks gladly frequented charnel houses: where corner desire more surely and emancipate oneself from it? The horrible being a path of liberation in every period of fervor and inwardness, our remains have enjoyed great favor. In the Middle Ages, a man made a regimen of salvation, he believed energetically: the corpse was in fashion. Faith was vigorous than, invincible; it cherished the livid and the fetid, it knew the profits to be derived from corruption and gruesomeness. Today, an edulcorated religion adheres only to „nice” hallucinations, to Evolution and to Progress. It is not such a religion which might afford us the modern equivalent of the dense macabre. „Let a man who aspires to nirvana act so that nothing is dear to him”, we read in a Buddhist text. It is enough to consider these specters, to meditate on the fate of the flash which adhered to them, in order to understand the urgency of detachment. There is no ascesis in the double rumination on the flesh and on the skeleton, on the dreadful decrepitude of the one and the futile permanence of the other. It is a good exercise to sever ourselves now and then from our face, from our skin, to lay aside this deceptive sheathe, then to discard – if only for a moment – that layer of grease which keeps us from discerning what is fundamental in ourselves. Once exercise is over, we are freer and more alone, almost invulnerable. In other to vanquish attachments and the disadvantages which derive from them, we should have to contemplate the ultimate nudity of a human being, force our eyes to pierce his entrails and all the rest, wallow in the horror of his secretions, in his physiology of an imminent corpse. This vision would not be morbid but methodical, a controlled obsession, particularly salutary in ordeals. The skeleton incites us to serenity; the cadaver to renunciation. In the sermon of futility which both of them preach to us happiness is identified with the destruction of our bounds. To have scanted no detail of such a teaching and even so to come to terms with simulacra! Blessed was the age when solitaries could plumb their depths without seeming obsessed, deranged. Their imbalance was not assigned a negative coefficient, as is the case for us. They would sacrifice ten, twenty years, a whole life, for a foreboding, for a flash of the absolute. The word „depth” has a meaning only in connection with epochs when the monk was considered as the noblest human exemplar. No one will gain – say the fact that he is in the process of disappearing. For centuries, he has done no more than survive himself. To whom would he address himself, in a universe which calls him a „parasite”? In Tibet, the last country where monks still mattered, they have been ruled out. Yet is was a rare consolation to think that thousands of thousands of hermits could be meditating there, today, on the themes of the prajnaparamita. Even if it had only odious aspects, monasticism would still be worth more than any other ideal. Now more then ever, we should build monasteries … for those who believe in everything and for those who believe in nothing. Where to escape? There no longer exist a single place where we can professionally execrate this world.

What this means is that the (Infinity) of points involved in continuity is greater than the (Infinity) of points comprised by any kind of discrete sequence, even an infinitely dense one. (2) Via his Diagonal Proof that c is greater than Aleph0, Cantor has succeeded in characterizing arithmetical continuity entirely in terms of order, sets, denumerability, etc. That is, he has characterized it 100% abstractly, without reference to time, motion, streets, noses, pies, or any other feature of the physical world-which is why Russell credits him with 'definitively solving' the deep problems behind the dichotomy. (3) The D.P. also explains, with respect to Dr. G.'s demonstration back in Section 2e, why there will always be more real numbers than red hankies. And it helps us understand why rational numbers ultimately take up 0 space on the Real Line, since it's obviously the irrational numbers that make the set of all reals nondenumerable. (4) An extension of Cantor's proof helps confirm J. Liouville's 1851 proof that there are an infinite number of transcendental irrationals in any interval on the Real Line. (This is pretty interesting. You'll recall from Section 3a FN 15 that of the two types of irrationals, transcendentals are the ones like pi and e that can't be the roots of integer-coefficient polynomials. Cantor's proof that the reals' (Infinity) outweighs the rationals' (Infinity) can be modified to show that it's actually the transcendental irrationals that are nondenumerable and that the set of all algebraic irrationals has the same cardinality as the rationals, which establishes that it's ultimately the transcendetnal-irrational-reals that account for the R.L.'s continuity.)

Perspective does not appear to me to be a subjective deformation of things but, on the contrary, to be one of their properties, perhaps their essential property. It is precisely because of it that the perceived possesses in itself a hidden and inexhaustible richness, that it is a 'thing'...Far from introducing a coefficient of subjectivity into perception, it provides us with the assurance of communicating with a world which is richer than what we know of it, that is, of communicating with a real world...The perceived is grasped in an indivisible manner as an 'in-itself,' that is, as gifted with an interior which I will never have finished exploring; and as 'for-me,' that is, as given 'in person' through its momentary aspects. Neither this metallic spot which moves while I glance toward it, nor even the geometric and shiny mass which emerges from it when I look at it, nor finally, the ensemble of perspectival images which I have been able to have of it are the ashtray; they do not exhaust the meaning of the 'this' by which I designate it; and, nevertheless, it is the ashtray which appears in all of them...Thus, to do justice to our direct experience of things it would be necessary to maintain at the same time, against empiricism, that they are beyond their sensible manifestations and, against intellectualism, that they are not unities in the order of judgment, that they are embodied in their apparitions. The 'things' in naive experience are evident as perspectival beings ...I grasp in a perspectival appearance, which I know is only one of its possible aspects, the thing itself which transcends it. A transcendence which is nevertheless open to my knowledge--this is the very definition of a thing as it is intended by naive consciousness.

Harley begins to panic. “Coming to Earth?! Our Earth? But I don’t want to die. There is so much I haven’t done yet – like learn Modularity Theorem!” “What is Modularity Theorem?” I ask. “The theorem states that any elliptic curve over Q can be obtained via a rational map with integer coefficients from the classical modular curve (N) for integer N and is a curve with integer coefficients with an explicit definition. If N is the smallest integer for which the parameterization can be sourced,

The spread is often measured by the Gini coefficient, named after Corrado Gini, an Italian economist who worked in the first half of the twentieth century. Gini’s coefficient, or simply the Gini, is a number that lies between 0 (perfect equality—everyone has the same) and 1 (perfect inequality, with one person having everything). It measures how far people are apart on average. (If you really want know the details, it is the average difference in income between all pairs of people divided by twice the average income. If there are two of us, and you have everything, the difference between us is twice the mean, and the Gini is 1. If we both have the same, the difference between us is 0, and so is the Gini.)

Perhaps the most obvious difference between modern social and personality psychology is that the former is based almost exclusively on experiments, whereas the latter is usually based on correlational studies. […] In summary, over the past 50 years social psychology has concentrated on the perceptual and cognitive processes of person perceivers, with scant attention to the persons being perceived. Personality psychology has had the reverse orientation, closely examining self-reports of individuals for indications of their personality traits, but rarely examining how these people actually come off in social interaction. […] individuals trained in either social or personality psychology are often more ignorant of the other field than they should be. Personality psychologists sometimes reveal an imperfect understanding of the concerns and methods of their social psychological brethren, and they in particular fail to comprehend the way in which so much of the self-report data they gather fails to overcome the skepticism of those trained in other methods. For their part, social psychologists are often unfamiliar with basic findings and concepts of personality psychology, misunderstand common statistics such as correlation coefficients and other measures of effect size, and are sometimes breathtakingly ignorant of basic psychometric principles. This is revealed, for example, when social psychologists, assuring themselves that they would not deign to measure any entity so fictitious as a trait, proceed to construct their own self-report scales to measure individual difference constructs called schemas or strategies or construals (never a trait). But they often fail to perform the most elementary analyses to confirm the internal consistency or the convergent and discriminant validity of their new measures, probably because they do not know that they should. […] an astonishing number of research articles currently published in major journals demonstrate a complete innocence of psychometric principles. Social psychologists and cognitive behaviorists who overtly eschew any sympathy with the dreaded concept of ‘‘trait’’ freely report the use of self-report assessment instruments of completely unknown and unexamined reliability, convergent validity, or discriminant validity. It is almost as if they believe that as long as the individual difference construct is called a ‘‘strategy,’’ ‘‘schema,’’ or ‘‘implicit theory,’’ then none of these concepts is relevant. But I suspect the real cause of the omission is that many investigators are unfamiliar with these basic concepts, because through no fault of their own they were never taught them.

Each project ship must maintain its coefficient of frustration,” went the private admonition. “Frustration must come from both human and mechanical sources.” They

Briefly, the Long Count consists of a tabulation of days elapsed since the supposed inception of the calendar, the total being expressed as so many cycles of differing magnitudes. The largest of these cycles is the baktun, containing 144,000 days; next, the katun, 7,200 days; then the tun, with 360 days; the uinal, with 20; and the smallest of all, always at the bottom of the column, the kin of one day. Each of these in the days of Maya ascendancy was shown with its own hieroglyph, while to its left stood the coefficient by which it was to be multiplied.

the level itself wasn't a mere number, but a certain coefficient that, among other things, amplified and diminished damage when the level gap was wide enough.

Pearson’s correlation coefficient is represented by the Greek letter rho (ρ) for the population parameter and r for a sample statistic. This coefficient is a single number that measures both the strength and direction of the linear relationship between two continuous variables. Values can range from -1 to +1.

Pearson’s correlation coefficient is unaffected by scaling issues. Consequently, a statistical assessment is better for determining the precise strength of the relationship.

Pearson’s correlation measures only linear relationships. Consequently, if your data contain a curvilinear relationship, the correlation coefficient will not detect it.

What is a good correlation? How high should it be? These are commonly asked questions. I have seen several schemes that attempt to classify correlations as strong, medium, and weak. However, there is only one correct answer. The correlation coefficient should accurately reflect the strength of the relationship. Take a look at the correlation between the height and weight data, 0.705. It’s not a very strong relationship, but it accurately represents our data. An accurate representation is the best-case scenario for using a statistic to describe an entire dataset.

squared is a primary measure of how well a regression model fits the data. This statistic represents the percentage of variation in one variable that other variables explain. For a pair of variables, R-squared is simply the square of the Pearson’s correlation coefficient. For example, squaring the height-weight correlation coefficient of 0.705 produces an R-squared of 0.497, or 49.7%. In other words, height explains about half the variability of weight in preteen girls.

values and coefficients are they key regression output. Collectively, these statistics indicate whether the variables are statistically significant and describe the relationships between the independent variables and the dependent variable. Low p-values (typically < 0.05) indicate that the independent variable is statistically significant. Regression analysis is a form of inferential statistics. Consequently, the p-values help determine whether the relationships that you observe in your sample also exist in the larger population. The coefficients for the independent variables represent the average change in the dependent variable given a one-unit change in the independent variable (IV) while controlling the other IVs.

The low p-values indicate that both education and IQ are statistically significant. The coefficient for IQ (4.796) indicates that each additional IQ point increases your income by an average of approximately $4.80 while controlling everything else in the model. Furthermore, the education coefficient (24.215) indicates that an additional year of education increases average earnings by $24.22 while holding the other variables constant.

This graph shows all the observations together with a line that represents the fitted relationship. As is traditional, the Y-axis displays the dependent variable, which is weight. The X-axis shows the independent variable, which is height. The line is the fitted line. If you enter the full range of height values that are on the X-axis into the regression equation that the chart displays, you will obtain the line shown on the graph. This line produces a smaller SSE than any other line you can draw through these observations. Visually, we see that that the fitted line has a positive slope that corresponds to the positive correlation we obtained earlier. The line follows the data points, which indicates that the model fits the data. The slope of the line equals the coefficient that I circled. This coefficient indicates how much mean weight tends to increase as we increase height. We can also enter a height value into the equation and obtain a prediction for the mean weight. Each point on the fitted line represents the mean weight for a given height. However, like any mean, there is variability around the mean. Notice how there is a spread of data points around the line. You can assess this variability by picking a spot on the line and observing the range of data points above and below that point. Finally, the vertical distance between each data point and the line is the residual for that observation.

There have even been attempts to calculate income levels and Gini coefficients for Palaeolithic mammoth hunters (they both turn out to be very low).1 It’s almost as if we feel some need to come up with mathematical formulae justifying the expression, already popular in the days of Rousseau, that in such societies ‘everyone was equal, because they were all equally poor.

Perhaps the most important idea is that the rate of reversion to the mean relates to the coefficient of correlation. If the correlation between two variables is 1.0, there is no reversion to the mean. If the correlation is 0, the best guess about what the next outcome will be is simply the average. In other words, when there's no correlation between what you do and what happens, you'll see total reversion to the mean. That's why there's always a small expected loss when you play roulette, whether you've just lost or won chips. Simply having a sense of the correlations for various events can help guide us in making predictions.

Let us look at the correlation between temperature, humidity and wind speed and all other features. Since the data also contains categorical features, we cannot only use the Pearson correlation coefficient, which only works if both features are numerical. Instead, I train a linear model to predict, for example, temperature based on one of the other features as input. Then I measure how much variance the other feature in the linear model explains and take the square root. If the other feature was numerical, then the result is equal to the absolute value of the standard Pearson correlation coefficient. But this model-based approach of “variance-explained” (also called ANOVA, which stands for ANalysis Of VAriance) works even if the other feature is categorical. The “variance-explained” measure lies always between 0 (no association) and 1 (temperature can be perfectly predicted from the other feature). We calculate the explained variance of temperature, humidity and wind speed with all the other features. The higher the explained variance (correlation), the more (potential) problems with PD plots. The following figure visualizes how strongly the weather features are correlated with other features.

Any inhabitant with a negative attachment coefficient (in which case it is referred to as a coefficient of ironic detachment) will be placed on probation pending review of the individual’s suitability for continued inclusion within the U31 diegetic space.

It turns out Evite had a built-in viral coefficient—when people send online invitations, the person receiving that invitation may turn around and send invitations to others, and so on. Unbeknownst to Selina (until she kicked the cord), Evite had been growing of its own volition.

Being able to react in significantly different ways given only slightly different observational outcomes to an event creates a sort of temporal 'updraft' for the subjective experiencer, whereby they gain access to greater possible timelines by virtue of having a high 'butterfly effect' coefficient.

Soon, there were other things that began to trouble Tyler. One type of experiment he and Erika were tasked with doing involved retesting blood samples on the Edisons over and over to measure how much their results varied. The data collected were used to calculate each Edison blood test’s coefficient of variation, or CV. A test is generally considered precise if its CV is less than 10 percent. To Tyler’s dismay, data runs that didn’t achieve low enough CVs were simply discarded and the experiments repeated until the desired number was reached. It was as if you flipped a coin enough times to get ten heads in a row and then declared that the coin always returned heads. Even within the “good” data runs, Tyler and Erika noticed that some values were deemed outliers and deleted. When Erika asked the group’s more senior scientists how they defined an outlier, no one could give her a straight answer. Erika and Tyler might be young and inexperienced, but they both knew that cherry-picking data wasn’t good science

After all, imagine we framed the problem differently, the way it might have been fifty or 100 years ago: as the concentration of capital, or oligopoly, or class power. Compared to any of these, a word like ‘inequality’ sounds like it’s practically designed to encourage half-measures and compromise. It’s possible to imagine overthrowing capitalism or breaking the power of the state, but it’s not clear what eliminating inequality would even mean. (Which kind of inequality? Wealth? Opportunity? Exactly how equal would people have to be in order for us to be able to say we’ve ‘eliminated inequality’?) The term ‘inequality’ is a way of framing social problems appropriate to an age of technocratic reformers, who assume from the outset that no real vision of social transformation is even on the table. Debating inequality allows one to tinker with the numbers, argue about Gini coefficients and thresholds of dysfunction, readjust tax regimes or social welfare mechanisms, even shock the public with figures showing just how bad things have become (‘Can you imagine? The richest 1 per cent of the world’s population own 44 per cent of the world’s wealth!’) – but it also allows one to do all this without addressing any of the factors that people actually object to about such ‘unequal’ social arrangements: for instance, that some manage to turn their wealth into power over others; or that other people end up being told their needs are not important, and their lives have no intrinsic worth. The last, we are supposed to believe, is just the inevitable effect of inequality; and inequality, the inevitable result of living in any large, complex, urban, technologically sophisticated society. Presumably it will always be with us. It’s just a matter of degree. Today, there is a veritable boom of thinking about inequality: since 2011, ‘global inequality

Thus, multiple regression requires two important tasks: (1) specification of independent variables and (2) testing of the error term. An important difference between simple regression and multiple regression is the interpretation of the regression coefficients in multiple regression (b1, b2, b3, …) in the preceding multiple regression model. Although multiple regression produces the same basic statistics discussed in Chapter 14 (see Table 14.1), each of the regression coefficients is interpreted as its effect on the dependent variable, controlled for the effects of all of the other independent variables included in the regression. This phrase is used frequently when explaining multiple regression results. In our example, the regression coefficient b1 shows the effect of x1 on y, controlled for all other variables included in the model. Regression coefficient b2 shows the effect of x2 on y, also controlled for all other variables in the model, including x1. Multiple regression is indeed an important and relatively simple way of taking control variables into account (and much easier than the approach shown in Appendix 10.1). Key Point The regression coefficient is the effect on the dependent variable, controlled for all other independent variables in the model. Note also that the model given here is very different from estimating separate simple regression models for each of the independent variables. The regression coefficients in simple regression do not control for other independent variables, because they are not in the model. The word independent also means that each independent variable should be relatively unaffected by other independent variables in the model. To ensure that independent variables are indeed independent, it is useful to think of the distinctively different types (or categories) of factors that affect a dependent variable. This was the approach taken in the preceding example. There is also a statistical reason for ensuring that independent variables are as independent as possible. When two independent variables are highly correlated with each other (r2 > .60), it sometimes becomes statistically impossible to distinguish the effect of each independent variable on the dependent variable, controlled for the other. The variables are statistically too similar to discern disparate effects. This problem is called multicollinearity and is discussed later in this chapter. This problem is avoided by choosing independent variables that are not highly correlated with each other. A WORKING EXAMPLE Previously (see Chapter 14), the management analyst with the Department of Defense found a statistically significant relationship between teamwork and perceived facility productivity (p <.01). The analyst now wishes to examine whether the impact of teamwork on productivity is robust when controlled for other factors that also affect productivity. This interest is heightened by the low R-square (R2 = 0.074) in Table 14.1, suggesting a weak relationship between teamwork and perceived productivity. A multiple regression model is specified to include the effects of other factors that affect perceived productivity. Thinking about other categories of variables that could affect productivity, the analyst hypothesizes the following: (1) the extent to which employees have adequate technical knowledge to do their jobs, (2) perceptions of having adequate authority to do one’s job well (for example, decision-making flexibility), (3) perceptions that rewards and recognition are distributed fairly (always important for motivation), and (4) the number of sick days. Various items from the employee survey are used to measure these concepts (as discussed in the workbook documentation for the Productivity dataset). After including these factors as additional independent variables, the result shown in Table 15.1 is

regression results. Standardized Coefficients The question arises as to which independent variable has the greatest impact on explaining the dependent variable. The slope of the coefficients (b) does not answer this question because each slope is measured in different units (recall from Chapter 14 that b = ∆y/∆x). Comparing different slope coefficients is tantamount to comparing apples and oranges. However, based on the regression coefficient (or slope), it is possible to calculate the standardized coefficient, β (beta). Beta is defined as the change produced in the dependent variable by a unit of change in the independent variable when both variables are measured in terms of standard deviation units. Beta is unit-less and thus allows for comparison of the impact of different independent variables on explaining the dependent variable. Analysts compare the relative values of beta coefficients; beta has no inherent meaning. It is appropriate to compare betas across independent variables in the same regression, not across different regressions. Based on Table 15.1, we conclude that the impact of having adequate authority on explaining productivity is [(0.288 – 0.202)/0.202 =] 42.6 percent greater than teamwork, and about equal to that of knowledge. The impact of having adequate authority is two-and-a-half times greater than that of perceptions of fair rewards and recognition.4 F-Test Table 15.1 also features an analysis of variance (ANOVA) table. The global F-test examines the overall effect of all independent variables jointly on the dependent variable. The null hypothesis is that the overall effect of all independent variables jointly on the dependent variables is statistically insignificant. The alternate hypothesis is that this overall effect is statistically significant. The null hypothesis implies that none of the regression coefficients is statistically significant; the alternate hypothesis implies that at least one of the regression coefficients is statistically significant. The

regression lines that describe the relationship of the independent variables for each group (called classification functions). The emphasis in discriminant analysis is the ability of the independent variables to correctly predict values of the nominal variable (for example, group membership). Discriminant analysis is one strategy for dealing with dependent variables that are nominal with three or more categories. Multinomial logistic regression and ordinal regression have been developed in recent years to address nominal and ordinal dependent variables in logic regression. Multinomial logistic regression calculates functions that compare the probability of a nominal value occurring relative to a base reference group. The calculation of such probabilities makes this technique an interesting alternative to discriminant analysis. When the nominal dependent variable has three values (say, 1, 2, and 3), one logistic regression predicts the likelihood of 2 versus 1 occurring, and the other logistic regression predicts the likelihood of 3 versus 1 occurring, assuming that “1” is the base reference group.7 When the dependent variable is ordinal, ordinal regression can be used. Like multinomial logistic regression, ordinal regression often is used to predict event probability or group membership. Ordinal regression assumes that the slope coefficients are identical for each value of the dependent variable; when this assumption is not met, multinomial logistic regression should be considered. Both multinomial logistic regression and ordinal regression are relatively recent developments and are not yet widely used. Statistics, like other fields of science, continues to push its frontiers forward and thereby develop new techniques for managers and analysts. Key Point Advanced statistical tools are available. Understanding the proper circumstances under which these tools apply is a prerequisite for using them.

SUMMARY A vast array of additional statistical methods exists. In this concluding chapter, we summarized some of these methods (path analysis, survival analysis, and factor analysis) and briefly mentioned other related techniques. This chapter can help managers and analysts become familiar with these additional techniques and increase their access to research literature in which these techniques are used. Managers and analysts who would like more information about these techniques will likely consult other texts or on-line sources. In many instances, managers will need only simple approaches to calculate the means of their variables, produce a few good graphs that tell the story, make simple forecasts, and test for significant differences among a few groups. Why, then, bother with these more advanced techniques? They are part of the analytical world in which managers operate. Through research and consulting, managers cannot help but come in contact with them. It is hoped that this chapter whets the appetite and provides a useful reference for managers and students alike. KEY TERMS   Endogenous variables Exogenous variables Factor analysis Indirect effects Loading Path analysis Recursive models Survival analysis Notes 1. Two types of feedback loops are illustrated as follows: 2. When feedback loops are present, error terms for the different models will be correlated with exogenous variables, violating an error term assumption for such models. Then, alternative estimation methodologies are necessary, such as two-stage least squares and others discussed later in this chapter. 3. Some models may show double-headed arrows among error terms. These show the correlation between error terms, which is of no importance in estimating the beta coefficients. 4. In SPSS, survival analysis is available through the add-on module in SPSS Advanced Models. 5. The functions used to estimate probabilities are rather complex. They are so-called Weibull distributions, which are defined as h(t) = αλ(λt)a–1, where a and 1 are chosen to best fit the data. 6. Hence, the SSL is greater than the squared loadings reported. For example, because the loadings of variables in groups B and C are not shown for factor 1, the SSL of shown loadings is 3.27 rather than the reported 4.084. If one assumes the other loadings are each .25, then the SSL of the not reported loadings is [12*.252 =] .75, bringing the SSL of factor 1 to [3.27 + .75 =] 4.02, which is very close to the 4.084 value reported in the table. 7. Readers who are interested in multinomial logistic regression can consult on-line sources or the SPSS manual, Regression Models 10.0 or higher. The statistics of discriminant analysis are very dissimilar from those of logistic regression, and readers are advised to consult a separate text on that topic. Discriminant analysis is not often used in public

Who am I? I am one who finds his life a question, whose life is always being put in question, which is what gives life its salt. We seek but do not find, not quite, not if we are honest, which does not discourage the religious heart but drives it on and heightens the passion, for this is one more encounter with the impossible. We may and we must have our opinions on the subject; we must finally reach a judgment and take a stand about life, but my advice is to attach a coefficient of uncertainty to what we say, for even after we have taken a stand, we still do not know who we are.

Hence, the energy for independent thoughts is additive except for a term log[B(n1,n2)], the log of a binomial coefficient. Since binomial coefficients are always bigger than (or equal to) one, it follows that energy is super-additive. Combining thoughts demand more and more mental power as the sizes increase: the MIND is limited in the complexity of thoughts.

Simple Regression   CHAPTER OBJECTIVES After reading this chapter, you should be able to Use simple regression to test the statistical significance of a bivariate relationship involving one dependent and one independent variable Use Pearson’s correlation coefficient as a measure of association between two continuous variables Interpret statistics associated with regression analysis Write up the model of simple regression Assess assumptions of simple regression This chapter completes our discussion of statistical techniques for studying relationships between two variables by focusing on those that are continuous. Several approaches are examined: simple regression; the Pearson’s correlation coefficient; and a nonparametric alterative, Spearman’s rank correlation coefficient. Although all three techniques can be used, we focus particularly on simple regression. Regression allows us to predict outcomes based on knowledge of an independent variable. It is also the foundation for studying relationships among three or more variables, including control variables mentioned in Chapter 2 on research design (and also in Appendix 10.1). Regression can also be used in time series analysis, discussed in Chapter 17. We begin with simple regression. SIMPLE REGRESSION Let’s first look at an example. Say that you are a manager or analyst involved with a regional consortium of 15 local public agencies (in cities and counties) that provide low-income adults with health education about cardiovascular diseases, in an effort to reduce such diseases. The funding for this health education comes from a federal grant that requires annual analysis and performance outcome reporting. In Chapter 4, we used a logic model to specify that a performance outcome is the result of inputs, activities, and outputs. Following the development of such a model, you decide to conduct a survey among participants who attend such training events to collect data about the number of events they attended, their knowledge of cardiovascular disease, and a variety of habits such as smoking that are linked to cardiovascular disease. Some things that you might want to know are whether attending workshops increases

knowledge of cardiovascular disease and whether such knowledge reduces behaviors that put people at risk for cardiovascular disease. Simple regression is used to analyze the relationship between two continuous variables. Continuous variables assume that the distances between ordered categories are determinable.1 In simple regression, one variable is defined as the dependent variable and the other as the independent variable (see Chapter 2 for the definitions). In the current example, the level of knowledge obtained from workshops and other sources might be measured on a continuous scale and treated as an independent variable, and behaviors that put people at risk for cardiovascular disease might also be measured on a continuous scale and treated as a dependent variable. Scatterplot The relationship between two continuous variables can be portrayed in a scatterplot. A scatterplot is merely a plot of the data points for two continuous variables, as shown in Figure 14.1 (without the straight line). By convention, the dependent variable is shown on the vertical (or Y-) axis, and the independent variable on the horizontal (or X-) axis. The relationship between the two variables is estimated as a straight line relationship. The line is defined by the equation y = a + bx, where a is the intercept (or constant), and b is the slope. The slope, b, is defined as Figure 14.1 Scatterplot or (y2 – y1)/(x2 – x1). The line is calculated mathematically such that the sum of distances from each observation to the line is minimized.2 By definition, the slope indicates the change in y as a result of a unit change in x. The straight line, defined by y = a + bx, is also called the regression line, and the slope (b) is called the regression coefficient. A positive regression coefficient indicates a positive relationship between the variables, shown by the upward slope in Figure 14.1. A negative regression coefficient indicates a negative relationship between the variables and is indicated by a downward-sloping line. Test of Significance The test of significance of the regression coefficient is a key test that tells us whether the slope (b) is statistically different from zero. The slope is calculated from a sample, and we wish to know whether it is significant. When the regression line is horizontal (b = 0), no relationship exists between the two variables. Then, changes in the independent variable have no effect on the dependent variable. The following hypotheses are thus stated: H0: b = 0, or the two variables are unrelated. HA: b ≠ 0, or the two variables are (positively or negatively) related. To determine whether the slope equals zero, a t-test is performed. The test statistic is defined as the slope, b, divided by the standard error of the slope, se(b). The standard error of the slope is a measure of the distribution of the observations around the regression slope, which is based on the standard deviation of those observations to the regression line: Thus, a regression line with a small slope is more likely to be statistically significant when observations lie closely around it (that is, the standard error of the observations around the line is also small, resulting in a larger test statistic). By contrast, the same regression line might be statistically insignificant when observations are scattered widely around it. Observations that lie farther from the

regression line will have larger standard deviations and, hence, larger standard errors. The computer calculates the slope, intercept, standard error of the slope, and the level at which the slope is statistically significant. Key Point The significance of the slope tests the relationship. Consider the following example. A management analyst with the Department of Defense wishes to evaluate the impact of teamwork on the productivity of naval shipyard repair facilities. Although all shipyards are required to use teamwork management strategies, these strategies are assumed to vary in practice. Coincidentally, a recently implemented employee survey asked about the perceived use and effectiveness of teamwork. These items have been aggregated into a single index variable that measures teamwork. Employees were also asked questions about perceived performance, as measured by productivity, customer orientation, planning and scheduling, and employee motivation. These items were combined into an index measure of work productivity. Both index measures are continuous variables. The analyst wants to know whether a relationship exists between perceived productivity and teamwork. Table 14.1 shows the computer output obtained from a simple regression. The slope, b, is 0.223; the slope coefficient of teamwork is positive; and the slope is significant at the 1 percent level. Thus, perceptions of teamwork are positively associated with productivity. The t-test statistic, 5.053, is calculated as 0.223/0.044 (rounding errors explain the difference from the printed value of t). Other statistics shown in Table 14.1 are discussed below. The appropriate notation for this relationship is shown below. Either the t-test statistic or the standard error should be shown in parentheses, directly below the regression coefficient; analysts should state which statistic is shown. Here, we show the t-test statistic:3 The level of significance of the regression coefficient is indicated with asterisks, which conforms to the p-value legend that should also be shown. Typically, two asterisks are used to indicate a 1 percent level of significance, one asterisk for a 5 percent level of significance, and no asterisk for coefficients that are insignificant.4 Table 14.1 Simple Regression Output Note: SEE = standard error of the estimate; SE = standard error; Sig. = significance.

Table 14.1 also shows R-square (R2), which is called the coefficient of determination. R-square is of great interest: its value is interpreted as the percentage of variation in the dependent variable that is explained by the independent variable. R-square varies from zero to one, and is called a goodness-of-fit measure.5 In our example, teamwork explains only 7.4 percent of the variation in productivity. Although teamwork is significantly associated with productivity, it is quite likely that other factors also affect it. It is conceivable that other factors might be more strongly associated with productivity and that, when controlled for other factors, teamwork is no longer significant. Typically, values of R2 below 0.20 are considered to indicate weak relationships, those between 0.20 and 0.40 indicate moderate relationships, and those above 0.40 indicate strong relationships. Values of R2 above 0.65 are considered to indicate very strong relationships. R is called the multiple correlation coefficient and is always 0 ≤ R ≤ 1. To summarize up to this point, simple regression provides three critically important pieces of information about bivariate relationships involving two continuous variables: (1) the level of significance at which two variables are associated, if at all (t-statistic), (2) whether the relationship between the two variables is positive or negative (b), and (3) the strength of the relationship (R2). Key Point R-square is a measure of the strength of the relationship. Its value goes from 0 to 1. The primary purpose of regression analysis is hypothesis testing, not prediction. In our example, the regression model is used to test the hypothesis that teamwork is related to productivity. However, if the analyst wants to predict the variable “productivity,” the regression output also shows the SEE, or the standard error of the estimate (see Table 14.1). This is a measure of the spread of y values around the regression line as calculated for the mean value of the independent variable, only, and assuming a large sample. The standard error of the estimate has an interpretation in terms of the normal curve, that is, 68 percent of y values lie within one standard error from the calculated value of y, as calculated for the mean value of x using the preceding regression model. Thus, if the mean index value of the variable “teamwork” is 5.0, then the calculated (or predicted) value of “productivity” is [4.026 + 0.223*5 =] 5.141. Because SEE = 0.825, it follows that 68 percent of productivity values will lie 60.825 from 5.141 when “teamwork” = 5. Predictions of y for other values of x have larger standard errors.6 Assumptions and Notation There are three simple regression assumptions. First, simple regression assumes that the relationship between two variables is linear. The linearity of bivariate relationships is easily determined through visual inspection, as shown in Figure 14.2. In fact, all analysis of relationships involving continuous variables should begin with a scatterplot. When variable

(e). Hence the expressions are equivalent, as is y = ŷ + e. Certain assumptions about e are important, such as that it is normally distributed. When error term assumptions are violated, incorrect conclusions may be made about the statistical significance of relationships. This important issue is discussed in greater detail in Chapter 15 and, for time series data, in Chapter 17. Hence, the above is a pertinent but incomplete list of assumptions. Getting Started Conduct a simple regression, and practice writing up your results. PEARSON’S CORRELATION COEFFICIENT Pearson’s correlation coefficient, r, measures the association (significance, direction, and strength) between two continuous variables; it is a measure of association for two continuous variables. Also called the Pearson’s product-moment correlation coefficient, it does not assume a causal relationship, as does simple regression. The correlation coefficient indicates the extent to which the observations lie closely or loosely clustered around the regression line. The coefficient r ranges from –1 to +1. The sign indicates the direction of the relationship, which, in simple regression, is always the same as the slope coefficient. A “–1” indicates a perfect negative relationship, that is, that all observations lie exactly on a downward-sloping regression line; a “+1” indicates a perfect positive relationship, whereby all observations lie exactly on an upward-sloping regression line. Of course, such values are rarely obtained in practice because observations seldom lie exactly on a line. An r value of zero indicates that observations are so widely scattered that it is impossible to draw any well-fitting line. Figure 14.2 illustrates some values of r. Key Point Pearson’s correlation coefficient, r, ranges from –1 to +1. It is important to avoid confusion between Pearson’s correlation coefficient and the coefficient of determination. For the two-variable, simple regression model, r2 = R2, but whereas 0 ≤ R ≤ 1, r ranges from –1 to +1. Hence, the sign of r tells us whether a relationship is positive or negative, but the sign of R, in regression output tables such as Table 14.1, is always positive and cannot inform us about the direction of the relationship. In simple regression, the regression coefficient, b, informs us about the direction of the relationship. Statistical software programs usually show r rather than r2. Note also that the Pearson’s correlation coefficient can be used only to assess the association between two continuous variables, whereas regression can be extended to deal with more than two variables, as discussed in Chapter 15. Pearson’s correlation coefficient assumes that both variables are normally distributed. When Pearson’s correlation coefficients are calculated, a standard error of r can be determined, which then allows us to test the statistical significance of the bivariate correlation. For bivariate relationships, this is the same level of significance as shown for the slope of the regression coefficient. For the variables given earlier in this chapter, the value of r is .272 and the statistical significance of r is p ≤ .01. Use of the Pearson’s correlation coefficient assumes that the variables are normally distributed and that there are no significant departures from linearity.7 It is important not to confuse the correlation coefficient, r, with the regression coefficient, b. Comparing the measures r and b (the slope) sometimes causes confusion. The key point is that r does not indicate the regression slope but rather the extent to which observations lie close to it. A steep regression line (large b) can have observations scattered loosely or closely around it, as can a shallow (more horizontal) regression line. The purposes of these two statistics are very different.8 SPEARMAN’S RANK CORRELATION

COEFFICIENT The nonparametric alternative, Spearman’s rank correlation coefficient (r, or “rho”), looks at correlation among the ranks of the data rather than among the values. The ranks of data are determined as shown in Table 14.2 (adapted from Table 11.8): Table 14.2 Ranks of Two Variables In Greater Depth … Box 14.1 Crime and Poverty An analyst wants to examine empirically the relationship between crime and income in cities across the United States. The CD that accompanies the workbook Exercising Essential Statistics includes a Community Indicators dataset with assorted indicators of conditions in 98 cities such as Akron, Ohio; Phoenix, Arizona; New Orleans, Louisiana; and Seattle, Washington. The measures include median household income, total population (both from the 2000 U.S. Census), and total violent crimes (FBI, Uniform Crime Reporting, 2004). In the sample, household income ranges from $26,309 (Newark, New Jersey) to $71,765 (San Jose, California), and the median household income is $42,316. Per-capita violent crime ranges from 0.15 percent (Glendale, California) to 2.04 percent (Las Vegas, Nevada), and the median violent crime rate per capita is 0.78 percent. There are four types of violent crimes: murder and nonnegligent manslaughter, forcible rape, robbery, and aggravated assault. A measure of total violent crime per capita is calculated because larger cities are apt to have more crime. The analyst wants to examine whether income is associated with per-capita violent crime. The scatterplot of these two continuous variables shows that a negative relationship appears to be present: The Pearson’s correlation coefficient is –.532 (p < .01), and the Spearman’s correlation coefficient is –.552 (p < .01). The simple regression model shows R2 = .283. The regression model is as follows (t-test statistic in parentheses): The regression line is shown on the scatterplot. Interpreting these results, we see that the R-square value of .283 indicates a moderate relationship between these two variables. Clearly, some cities with modest median household incomes have a high crime rate. However, removing these cities does not greatly alter the findings. Also, an assumption of regression is that the error term is normally distributed, and further examination of the error shows that it is somewhat skewed. The techniques for examining the distribution of the error term are discussed in Chapter 15, but again, addressing this problem does not significantly alter the finding that the two variables are significantly related to each other, and that the relationship is of moderate strength. With this result in hand, further analysis shows, for example, by how much violent crime decreases for each increase in household income. For each increase of $10,000 in average household income, the violent crime rate drops 0.25 percent. For a city experiencing the median 0.78 percent crime rate, this would be a considerable improvement, indeed. Note also that the scatterplot shows considerable variation in the crime rate for cities at or below the median household income, in contrast to those well above it. Policy analysts may well wish to examine conditions that give rise to variation in crime rates among cities with lower incomes. Because Spearman’s rank correlation coefficient examines correlation among the ranks of variables, it can also be used with ordinal-level data.9 For the data in Table 14.2, Spearman’s rank correlation coefficient is .900 (p = .035).10 Spearman’s p-squared coefficient has a “percent variation explained” interpretation, similar

to the measures described earlier. Hence, 90 percent of the variation in one variable can be explained by the other. For the variables given earlier, the Spearman’s rank correlation coefficient is .274 (p < .01), which is comparable to r reported in preceding sections. Box 14.1 illustrates another use of the statistics described in this chapter, in a study of the relationship between crime and poverty. SUMMARY When analysts examine relationships between two continuous variables, they can use simple regression or the Pearson’s correlation coefficient. Both measures show (1) the statistical significance of the relationship, (2) the direction of the relationship (that is, whether it is positive or negative), and (3) the strength of the relationship. Simple regression assumes a causal and linear relationship between the continuous variables. The statistical significance and direction of the slope coefficient is used to assess the statistical significance and direction of the relationship. The coefficient of determination, R2, is used to assess the strength of relationships; R2 is interpreted as the percent variation explained. Regression is a foundation for studying relationships involving three or more variables, such as control variables. The Pearson’s correlation coefficient does not assume causality between two continuous variables. A nonparametric alternative to testing the relationship between two continuous variables is the Spearman’s rank correlation coefficient, which examines correlation among the ranks of the data rather than among the values themselves. As such, this measure can also be used to study relationships in which one or both variables are ordinal. KEY TERMS   Coefficient of determination, R2 Error term Observed value of y Pearson’s correlation coefficient, r Predicted value of the dependent variable y, ŷ Regression coefficient Regression line Scatterplot Simple regression assumptions Spearman’s rank correlation coefficient Standard error of the estimate Test of significance of the regression coefficient Notes   1. See Chapter 3 for a definition of continuous variables. Although the distinction between ordinal and continuous is theoretical (namely, whether or not the distance between categories can be measured), in practice ordinal-level variables with seven or more categories (including Likert variables) are sometimes analyzed using statistics appropriate for interval-level variables. This practice has many critics because it violates an assumption of regression (interval data), but it is often

Multiple Regression   CHAPTER OBJECTIVES After reading this chapter, you should be able to Understand multiple regression as a full model specification technique Interpret standardized and unstandardized regression coefficients of multiple regression Know how to use nominal variables in

Hence, the energy for independent thoughts is additive except for a term log[B(n1,n2)], the log of a binomial coefficient. Since binomial coefficients are always bigger than (or equal to) one, it follows that energy is super-additive. Combining thoughts demand more and more mental power as the sizes increase:

The two key factors that drive viral growth are the viral coefficient and the viral cycle time.

Gini coefficient, scores

The next point was made by Newton, who discussed the question: ‘When it does not go in a straight line then what?’ And he answered it this way: that a force is needed to change the velocity in any manner. For instance, if you are pushing a ball in the direction that it moves it will speed up. If you find that it changes direction, then the force must have been sideways. The force can be measured by the product of two effects. How much does the velocity change in a small interval of time? That’s called the acceleration, and when it is multiplied by the coefficient called the mass of an object, or its inertia coefficient, then that together is the force. One can measure this.

Whether you’re currently a marketing executive or a college grad about to enter the field—the first growth hackers have pioneered a new way. Some of their strategies are incredibly technical and complex. The strategies also change constantly; in fact, occasionally it might work only one time. This book is short because it sticks with the timeless parts. I also won’t weigh you down with heavy concepts like “cohort analysis” and “viral coefficients.”* Instead, we will focus on the mindset—it’s far and away the most important part. I start and end with my own experiences in this book, not because I am anyone special but because I think they illustrate a microcosm of the industry itself. The old way—where product development and

Covariance Coefficient

Manifesto" I know that dying is how we escape the rest of our lives. I think that trees send us a message: do not believe you are lucky. The skins of apples and the peeler will marry; it's simply a question of when. Believe in mourning and carrion birds. Look how their fleshy treasures dissolve in the sun before their very eyes. To love something you must have considered what it means to do without. You must have thought about it—the coefficient of the body is another body—but do not forget that there are people who are willing to staple your palm to your chest. Know there are places it isn't wise to go. Begin again if you must: there are ways to make up for what you have been before, the dust in the corners that collects you. Sympathy is overrated. Rethink how lack becomes everyone's master, drives us into town and spends our money. Quiet: the trees are napping. Water meets itself again. We reach for the days that precede us and the world keeps us from knowing too much. The body loves music, the abandoned road of it; each day a peel lengthens in the shadow of blossoms, fabric weaves itself into light. Pay attention to the patterns. They repeat— terraces erode, groves lie fallow— order is cognate of joy.

The reason special names are given to these quadratic irrationalities is that any quadratic algebraic integer is a linear combination (with ordinary integers as coefficients) of 1 and one of these fundamental quadratic algebraic integers.

You may show what atomic groupings are necessary in order that life may emerge out of matter, sentience out of life, or intellect out of sentience; but you cannot thereby reduce life, let alone sentient life and intellectual life, to terms of matter; you have only succeeded in tabulating the material coefficients of things which are not themselves material. I do not mean that Mr Russell would not be able to put up a case against this argument; I only complain that he simplified his task by pretending to misunderstand what the argument was; by assuming that it was merely physical when as a matter of fact it is metaphysical.

Colorizing gardens was a complex task that involved matching the osmotic coefficients of the different plants with the specific gravities of the dyes—and that was before you got started on pressure density evaporation rates and seasonal hue variation.

Moreover, America’s inequality is worse than other wealthy nations. The Gini coefficient is a common measure of a country’s inequality. It measures inequality from 0 (perfect equality) to 1 (complete inequality). According to the Organization for Economic Co-operation and Development in 2017, “the Gini coefficient in the U.S. stood at 0.434.” This number “was higher than in any other of the G-7 countries, in which the Gini ranged from 0.326 in France to 0.392 in the UK, and inching closer to the level of inequality observed in India (0.495).

The distribution of income in a society is called the 'Gini coefficient,' named after an Italian sociologist named Corrado Gini, who published a paper on the topic in 1912. A society where one person earns all the money and everyone else earns none, effectively has a Gini coefficient of 1.0; and a society where everyone earns the same amount has a coefficient of zero. Neither is desirable. Moderate differences in income motivate people because they have a reasonable chance of bettering their circumstances, and extreme differences discourage people because their efforts look futile. A study of 21 small-scale societies around the world found that hunter-gatherers like the Hadza—who presumably represent the most efficient possible system for survival in a hostile environment—have Gini coefficients as low as .25. In other words, they are far closer to absolute income equality than to absolute monopoly. Because oppression from one's own leaders is as common a threat as oppression from one's enemies, Gini coefficients are one reliable measure of freedom. Hunter-gatherer societies are not democracies—and many hold women in subordinate family roles—but the relationship between those families and their leaders is almost impervious to exploitation. In that sense, they are freer than virtually all modern societies. According to multiple sources, including the Congressional Budget Office, the United States has one of the highest Gini coefficients of the developed world, .42, which puts it at roughly the level of Ancient Rome. (Before taxes, the American Gini coefficient is even higher—almost .6—which is on par with deeply corrupt countries like Haiti, Namibia, and Botswana.) Moreover, the wealth gap between America's richest and poorest families has doubled since 1989. Globally, the situation is even more extreme: several dozen extremely rich people control as much wealth as the bottom half of humanity—3.8 billion people.

It's tempting to imagine that economic injustice destabilizes societies to the point where they collapse and have to reform themselves, but the opposite appears to be true. Countries with large income disparities, such as the United States, are among the most powerful and wealthy countries in the world, perhaps because they can protect themselves with robust economies and huge militaries. They're just not very free. Even societies with income disparities that are truly off the chart—medieval Europe had a Gini coefficient of .79—are relatively stable until a cataclysmic event like the plague triggers a radical redistribution of wealth. During the last decades, progressive reforms have reduced the Gini coefficient—and stabilized the economies—in many Latin American countries. From every standpoint—morally, politically, economically—such reforms are clearly the right things to do. But throughout the great sweep of human history, egalitarian societies with low Gini coefficients rarely dominate world events. From the Han Dynasty of Ancient China to the Roman Empire to the United States, there seems to be a sweet spot of economic injustice that is moderately unfair to most of its citizens but produces extremely powerful societies. Economist Walter Scheidel calculates that 3,500 years ago, such large-scale states controlled only 1 percent of the Earth's habitable landmass but represented at least half the human population. By virtually any metric, that's a successful society. 'For thousands of years, most of humanity lived in the shadow of these behemoths,' Scheidel writes. 'This is the environment that created the 'original one percent,' made up of competing but often closely intertwined elite groups.' The question, then, is how do ordinary people protect their freedom in the face of such highly centralized state control?

The coefficient of variation is a measure of variability that is computed as the ratio between the standard deviation and the mean of a probability distribution. You can think of it as a general measure of the relative breadth of a probability distribution. Since the square of the standard deviation is the variance of a distribution, this means that queues vary linearly with variance, a point worth remembering.

Her briefing had pointed out that the maximum angle that a dune could assume was independent of the local gravity, so this looked like dunes on Earth. The angle depended only on the dirt’s coefficient of friction,

your tile should have a dynamic coefficient of friction (DCOF) of greater than 0.42; you can find this in the manufacturer’s specifications.

The Gini coefficient, devised by the Italian sociologist Corrado Gini in 1912, is a measure of income or wealth disparity in a population. It is usually expressed as a fraction between 0 and 1, and it seems easy to understand, because 0 is the coefficient if everyone owned an equal amount, while 1 would obtain if one person owned everything and everyone else nothing. In our real world of the mid-twenty-first century, countries with a low Gini coefficient, like the social democracies, are generally a bit below 0.3, while highly unequal countries are a bit above 0.6. The US, China, and many other countries have seen their Gini coefficients shoot up in the neoliberal era, from 0.3 or 0.4 up to 0.5 or 0.6, this with barely a squeak from the people losing the most in this increase in inequality, and indeed many of those harmed often vote for politicians who will increase their relative impoverishment.

Parkinson’s Law, proposed the “coefficient of inefficiency”: Once a committee grows to more than eight members, it becomes less efficient with each new member added, becoming useless once it hits twenty.

The Gini coefficient, devised by the Italian sociologist Corrado Gini in 1912, is a measure of income or wealth disparity in a population. It is usually expressed as a fraction between 0 and 1, and it seems easy to understand, because 0 is the coefficient if everyone owned an equal amount, while 1 would obtain if one person owned everything and everyone else nothing.

You cannot buy a viral coefficient. Its growth depends directly on the value of your product or service provides and whether people are sharing that with others. Grand openings are directly related to your initial viral coefficient.

When basic things get more abundant, it’s the poor who benefit the most. This fact is not captured in Gini coefficients. As such, comparing the impact of changes in TPs over time on different groups of people may be much more informative than using Gini coefficients.

The term ‘inequality’ is a way of framing social problems appropriate to an age of technocratic reformers, who assume from the outset that no real vision of social transformation is even on the table. Debating inequality allows one to tinker with the numbers, argue about Gini coefficients and thresholds of dysfunction, readjust tax regimes or social welfare mechanisms, even shock the public with figures showing just how bad things have become (‘Can you imagine? The richest 1 per cent of the world’s population own 44 per cent of the world’s wealth!’) – but it also allows one to do all this without addressing any of the factors that people actually object to about such ‘unequal’ social arrangements: for instance, that some manage to turn their wealth into power over others; or that other people end up being told their needs are not important, and their lives have no intrinsic worth.

A study of emissions from US metropolitan areas between 1999 and 2008 found that, contrary to expectations, CO2 emissions scale proportionally with city size, and that larger cities are not metabolically more efficient than smaller ones.38 The scaling coefficient was only 7 percent lower than 1.0—that is, every 1 percent rise in population led to a 0.93 percent rise of emissions.

Once we get the regression results, we would calculate a t-statistic, which is the ratio of the observed coefficient to the standard error for that coefficient.* This t-statistic is then evaluated against whatever t-distribution is appropriate for the size of the data sample (since this is largely what determines the number of degrees of freedom). When the t-statistic is sufficiently large, meaning that our observed coefficient is far from what the null hypothesis would predict, we can reject the null hypothesis at some level of statistical significance. Again, this is the same basic process of statistical inference that we have been employing throughout the book. The fewer the degrees of freedom (and therefore the “fatter” the tails of the relevant t-distribution), the higher the t-statistic will have to be in order for us to reject the null hypothesis at some given level of significance. In the hypothetical regression example described above, if we had four degrees of freedom, we would need a t-statistic of at least 2.13 to reject the null hypothesis at the .05 level (in a one-tailed test). However, if we have 20,000 degrees of freedom (which essentially allows us to use the normal distribution), we would need only a t-statistic of 1.65 to reject the null hypothesis at the .05 level in the same one-tailed test.