Probability

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By Henri Poincaré

Excerpt from Calcul des Probabilités

Le ideal exemple que nous allons choisir est celui de l'équilibre instable; si un cône repose sur sa pointe, nous savons bien qu'il va tomber, mais nous ne savons pas de quel côté; il nous semble que le hasard seul va en décider. Si le cône était parfaitement symétrique, si son awl était_ parfaitement vertical, s'il n'était soumis à aucune autre strength que l. a. pesanteur, il ne tomberait pas du tout. Mais le moindre défaut de symétrie va le faire pencher légèrement d'un côté ou de l'autre, et dès qu'il penchera, si peu que ce soit, il tombera tout à fait de ce côté. Si même l. a. symétrie est parfaite, une trépidation très légère, un soufi°le d'air pourra le faire incliner dequelques secondes d'arc; ce sera assez pour déterminer sa chute et même le sens de sa chute qui sera celui de l'inclinaison initiale.

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4 Measures of Variability 15 Sample Range and Sample Standard Deviation Just as there are many measures of central tendency or location, there are many measures of spread or variability. Perhaps the simplest one is the sample range Xmax − Xmin . The range can be very useful and is discussed at length in Chapter 17 on statistical quality control. The sample measure of spread that is used most often is the sample standard deviation. We again let x1 , x2 , . . , xn denote sample values. 3: The sample variance, denoted by s2 , is given by n s2 = i=1 (xi − x ¯ )2 .

C) Comment on what the dot plot indicates regarding whether or not the population emissions changed from 1980 to 1990. Use the concept of variability in your comments. 24 The following are historical data on staff salaries (dollars per pupil) for 30 schools sampled in the eastern part of the United States in the early 1970s. 37 (a) Compute the sample mean and sample standard deviation. (b) Construct a relative frequency histogram of the data. (c) Construct a stem-and-leaf display of the data. 24.

So here we have sample estimates used to draw scientific conclusions regarding population parameters. 3, this is the general theme in many practical problems using statistical inference. 6 Statistical Modeling, Scientific Inspection, and Graphical Diagnostics Often the end result of a statistical analysis is the estimation of parameters of a postulated model. This is natural for scientists and engineers since they often deal in modeling. A statistical model is not deterministic but, rather, must entail some probabilistic aspects.

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