Siméon-Denis Poisson - Collected Quotes

"For each of the elements into which we have divided the amount of fluid matter, its shape will be altered during the time dt, and also its volume will change if the fluid is compressible; but, since its mass must remain unaltered, it follows that, if we seek to determine its volume and its density at the end of time t + dt, their product will necessarily be the same as after time t. (Siméon-Denis Poisson, "Traité de Méecanique" vol. II, 1811)

"In many different fields, empirical phenomena appear to obey a certain general law, which can be called the Law of Large Numbers. This law states that the ratios of numbers derived from the observation of a very large number of similar events remain practically constant, provided that these events are governed partly by constant factors and partly by variable factors whose variations are irregular and do not cause a systematic change in a definite direction." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"In ordinary life, the words chance and probability are almost synonymous and most often used indifferently. However, if necessary to distinguish their meaning, we attach here the word chance to events taken independently from our knowledge, and retain its previous definition [!] for the word probability. Thus, by its nature an event has a greater or lesser chance, known or unknown, whereas its probability is relative to our knowledge about it." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"[...] in the game of heads or tails, the arrival of heads results from the constitution of the tossed coin. It can be regarded as physically impossible that the chances of both outcomes are the same; however, if that constitution is unknown to us, and we did not yet try out the coin, the probability of heads is for us absolutely the same as that of tails. Actually, we have no reason to believe in one of these events rather than in the other one. This will not be the same after many tosses of the coin: the chance of each side does not change during the trials, but for someone who knows their results, the probability of the future occurrence of heads and tails varies in accord with the number of times they happened." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"The measure of the probability of an event is the ratio of the number of cases favourable for it to the total number of favourable andcontrary cases, all of them equally possible or having the same chance. That proposition signifies that when this ratio is the same for two events, we have the same reason to believe in the occurrence of either of them. Otherwise, we have more reason to believe in the arrival of that event for which it is larger." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"The probability of an event is our reason to believe that it will occur or occurred. [...] Probability depends on our knowledge about an event; for the same event it can differ for different persons. Thus, if a person only knows that an urn contains white and black balls, whereas another person alsoknows that there are more white balls than black ones, the latter has more grounds to believe in the extraction of a white ball. In other words, for him, that event has a higher probability than for the former." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"[...] the rules for establishing the probability of an observed event given the probability of its cause, which are the basis of the theory under consideration, require taking into account all the presumptions prior to the observation, if only they are thought to exist, or if proven that they are not absent." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"The calculus of probability is equally applicable to things of all kinds, moral and physical and, if only in each case observations provide the necessary numerical data, it does not at all depend on their nature." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"The constancy of ratios between the number of times that an event had occurred and the very large number of trials, which establishes itself and is manifested in spite of the variations of the chance of that event during these trials, tempts us to attribute this remarkable regularity to some ceaselessly acting occult cause. However, the theory of probability determines that the constancy of those ratios is a natural state of things belonging to physical and moral categories and maintains all by itself without any aid by some alien cause. On the contrary, it can only be hindered or disturbed by an intervention of a similar [alien] cause." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"[...] the law of large numbers governs phenomena produced by known forces acting together with accidental causes whose effect lacks any regularity." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"The law of large numbers is noted in events which are attributed to pure chance since we do not know their causes or because they are too complicated. Thus, games, in which the circumstances determining the occurrence of a certain card or certain number of points on a die infinitely vary, can not be subjected to any calculus. If the series of trials is continued for a long time, the different outcomes nevertheless appear in constant ratios. Then, if calculations according to the rules of a game are possible, the respective probabilities of eventual outcomes conform to the known Jakob Bernoulli theorem. However, in most problems of contingency a prior determination of chances of the various events is impossible and, on the contrary, they are calculated from the observed result." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible' (equally like to happen)." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"The phenomena of any kind are subject to a general law, which one can call the Law of Large Numbers. It consists in the fact, that, if one observes very large numbers of phenomena of the same kind depending on constant or irregularly changeable causes, however not progressively changeable, but one moment in the one sense, the other moment in the other sense; one finds ratios of these numbers which are almost constant." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"The probability of an event is the reason we have to believe that it has taken place, or that it will take place." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"Things of all kinds are subject to a universal law which may be called the law of large numbers. It consists in the fact that, if one observes very considerable numbers of events of the same nature, dependent on constant causes and causes which vary irregularly, sometimes in one direction, sometimes in the other, it is to say without their variation being progressive in any definite direction, one shall find, between these numbers, relations which are almost constant." (Siméon-Denis Poisson, "Poisson’s Law of Large Numbers", 1837)

"Without the aid of the calculus of probability you run a great risk of being mistaken about the necessity of that conclusion. However, the calculus leaves nothing vague here and in addition provides necessary rules for determining the chance of the change of the causes indicated by comparing the observed facts at different times." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"That which can affect our senses in any manner whatever, is termed matter." (Siméon-Denis Poisson, "A Treatise of Mechanics", 1842)

"Life is good for only two things, discovering mathematics and teaching mathematics." (Simeon-Denis Poisson) [in Mathematical Magazine, Volume 64, Number 1, February 1991]

"That which can affect our senses in any manner whatever, is termed matter." (Siméon-Denis Poisson) 

"The engineer should receive a complete mathematical education, but for what should it serve him? To see the different aspects of things and to see them quickly; he has no time to hunt mice." (Siméon-Denis Poisson)

References: Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", "Règles générales des probabilités", 1837 [source] 

On Numbers: Large Numbers II

"A good description of the data summarizes the systematic variation and leaves residuals that look structureless. That is, the residuals exhibit no patterns and have no exceptionally large values, or outliers. Any structure present in the residuals indicates an inadequate fit. Looking at the residuals laid out in an overlay helps to spot patterns and outliers and to associate them with their source in the data." (Christopher H Schrnid, "Value Splitting: Taking the Data Apart", 1991)

"Skewness is a measure of symmetry. For example, it's zero for the bell-shaped normal curve, which is perfectly symmetric about its mean. Kurtosis is a measure of the peakedness, or fat-tailedness, of a distribution. Thus, it measures the likelihood of extreme values." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Data that are skewed toward large values occur commonly. Any set of positive measurements is a candidate. Nature just works like that. In fact, if data consisting of positive numbers range over several powers of ten, it is almost a guarantee that they will be skewed. Skewness creates many problems. There are visualization problems. A large fraction of the data are squashed into small regions of graphs, and visual assessment of the data degrades. There are characterization problems. Skewed distributions tend to be more complicated than symmetric ones; for example, there is no unique notion of location and the median and mean measure different aspects of the distribution. There are problems in carrying out probabilistic methods. The distribution of skewed data is not well approximated by the normal, so the many probabilistic methods based on an assumption of a normal distribution cannot be applied." (William S Cleveland, "Visualizing Data", 1993)

"The logarithm is one of many transformations that we can apply to univariate measurements. The square root is another. Transformation is a critical tool for visualization or for any other mode of data analysis because it can substantially simplify the structure of a set of data. For example, transformation can remove skewness toward large values, and it can remove monotone increasing spread. And often, it is the logarithm that achieves this removal." (William S Cleveland, "Visualizing Data", 1993)

"Factoring big numbers is a strange kind of mathematics that closely resembles the experimental sciences, where nature has the last and definitive word. […] as with the experimental sciences, both rigorous and heuristic analyses can be valuable in understanding the subject and moving it forward. And, as with the experimental sciences, there is sometimes a tension between pure and applied practitioners." (Carl B Pomerance, "A Tale of Two Sieves", The Notices of the American Mathematical Society 43, 1996)

"Clearly, the mean is greatly influenced by extreme values, but it can be appropriate for many situations where extreme values do not arise. To avoid misuse, it is essential to know which summary measure best reflects the data and to use it carefully. Understanding the situation is necessary for making the right choice. Know the subject!" (Herbert F Spirer et al, "Misused Statistics" 2nd Ed, 1998)

"Big numbers warn us that the problem is a common one, compelling our attention, concern, and action. The media like to report statistics because numbers seem to be 'hard facts' - little nuggets of indisputable truth. [...] One common innumerate error involves not distinguishing among large numbers. [...] Because many people have trouble appreciating the differences among big numbers, they tend to uncritically accept social statistics (which often, of course, feature big numbers)." (Joel Best, "Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists", 2001)

"Use a logarithmic scale when it is important to understand percent change or multiplicative factors. […] Showing data on a logarithmic scale can cure skewness toward large values." (Naomi B Robbins, "Creating More effective Graphs", 2005) 

"Outliers or influential data points can be defined as data values that are extreme or atypical on either the independent (X variables) or dependent (Y variables) variables or both. Outliers can occur as a result of observation errors, data entry errors, instrument errors based on layout or instructions, or actual extreme values from self-report data. Because outliers affect the mean, the standard deviation, and correlation coefficient values, they must be explained, deleted, or accommodated by using robust statistics." (Randall E Schumacker & Richard G Lomax, "A Beginner’s Guide to Structural Equation Modeling" 3rd Ed., 2010)

"Comparisons are the lifeblood of empirical studies. We can’t determine if a medicine, treatment, policy, or strategy is effective unless we compare it to some alternative. But watch out for superficial comparisons: comparisons of percentage changes in big numbers and small numbers, comparisons of things that have nothing in common except that they increase over time, comparisons of irrelevant data. All of these are like comparing apples to prunes." (Gary Smith, "Standard Deviations", 2014)

"It is not enough to give a single summary for a distribution - we need to have an idea of the spread, sometimes known as the variability. [...] The range is a natural choice, but is clearly very sensitive to extreme values [...] In contrast the inter-quartile range (IQR) is unaffected by extremes. This is the distance between the 25th and 75th percentiles of the data and so contains the ‘central half’ of the numbers [...] Finally the standard deviation is a widely used measure of spread. It is the most technically complex measure, but is only really appropriate for well-behaved symmetric data since it is also unduly influenced by outlying values." (David Spiegelhalter, "The Art of Statistics: Learning from Data", 2019)

Previous Post <<||>> Next Post

On Numbers: Large Numbers I

"The calculation of probabilities is of the utmost value, […] but in statistical inquiries there is need not so much of mathematical subtlety as of a precise statement of all the circumstances. The possible contingencies are too numerous to be covered by a finite number of experiments, and exact calculation is, therefore, out of the question. Although nature has her habits, due to the recurrence of causes, they are general, not invariable. Yet empirical calculation, although it is inexact, may be adequate in affairs of practice." (Gottfried W Leibniz [letter to Bernoulli], 1703)

"Further, it cannot escape anyone that for judging in this way about any event at all, it is not enough to use one or two trials, but rather a great number of trials is required. And sometimes the stupidest man - by some instinct of nature per se and by no previous instruction (this is truly amazing) - knows for sure that the more observations of this sort that are taken, the less the danger will be of straying from the mark." (Jacob Bernoulli, "The Art of Conjecturing", 1713)

"If thus all events through all eternity could be repeated, by which we would go from probability to certainty, one would find that everything in the world happens from definite causes and according to definite rules, and that we would be forced to assume amongst the most apparently fortuitous things a certain necessity, or, so to say, FATE." (Jacob Bernoulli, "The Art of Conjecturing", 1713)

"And thus in all cases it will be found, that although Chance produces Irregularities, still the odds will be infinitely great that in the process of time, those Irregularities will bear no proportion to the recurrency of that Order which naturally results from ORIGINAL DESIGN." (Abraham de Moivre, "The Doctrine of Chances", 1718)

"Things of all kinds are subject to a universal law which may be called the law of large numbers. It consists in the fact that, if one observes very considerable numbers of events of the same nature, dependent on constant causes and causes which vary irregularly, sometimes in one direction, sometimes in the other, it is to say without their variation being progressive in any definite direction, one shall find, between these numbers, relations which are almost constant." (Siméon-Denis Poisson, "Poisson’s Law of Large Numbers", 1837)

"Huge numbers are commonplace in our culture, but oddly enough the larger the number the less meaningful it seems to be." (Albert Sukoff, "Lotsa Hamburgers", Saturday Review of the Society, 1973)

"We know the laws of trial and error, of large numbers and probabilities. We know that these laws are part of the mathematical and mechanical fabric of the universe, and that they are also at play in biological processes. But, in the name of the experimental method and out of our poor knowledge, are we really entitled to claim that everything happens by chance, to the exclusion of all other possibilities?" (Albert Claude, [Nobel Prize Lecture], 1974)

"The logarithm is an extremely powerful and useful tool for graphical data presentation. One reason is that logarithms turn ratios into differences, and for many sets of data, it is natural to think in terms of ratios. […] Another reason for the power of logarithms is resolution. Data that are amounts or counts are often very skewed to the right; on graphs of such data, there are a few large values that take up most of the scale and the majority of the points are squashed into a small region of the scale with no resolution." (William S. Cleveland, "Graphical Methods for Data Presentation: Full Scale Breaks, Dot Charts, and Multibased Logging", The American Statistician Vol. 38 (4) 1984)

"The trouble with integers is that we have examined only the small ones. Maybe all the exciting stuff happens at really big numbers, ones we can’t get our hand on or even begin to think about in any very definite way. So maybe all the action is really inaccessible and we’re just fiddling around. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions." (Paul Hauffman, "The Man Who Loves Only Numbers", The Atlantic Magazine, Vol 260, No 5, 1987)

"The law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen." (Frederick Mosteller, "Methods for Studying Coincidences Journal of the American Statistical Association, Volume 84, 1989)

Previous Post <<||>> Next Post