Deviation From The Gaussian Law Of Error Distribution
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more common than the rates (one time in twenty and one time in four hundred) predicted by the Gaussian, or "normal" error distribution. As you recall, normal distribution deviation we justified the whole least-squares approach by the fact that it gives
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the most likely set of answers, if the distribution of random errors is Gaussian. The official reason why people normal distribution deviation table always assume a Gaussian error distribution goes back to something called the Central Limit Theorem. The Central Limit Theorem says that whenever a measurement is subject to a very large number of
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very small errors, the probability distribution for the total error is driven toward the Gaussian distribution. This is true regardless of the form of the original probability distributions of the individual errors. A proof - and it is a pretty one - can be found in any book on the theory of statistics. The real reason why people always assume a Gaussian error normal distribution standard deviation percentile distribution is that, having made that assumption, we can then easily derive (and have derived!) exact mathematical formulae which allow us to compute directly the "best" values for the unknown parameters. This is not necessarily possible for other probability distributions. What would happen if, for instance, the error distribution for your data were not Gaussian, but Lorentzian, With the Gaussian, when you go to maximize the likelihood you discover that you must minimize the sum of the squares of the residuals. This leads to a very simple and straightforward set of simultaneous linear equations. With the Lorentz function, you get Try differentiating the right side of this equation with respect to each of the unknown parameters, and see where it gets you. Pretending that the error distribution is Gaussian even if it isn't makes life a lot simpler. The fact is, with real data you don't know what the probability distribution of the errors is, and you don't even know that it has any particular mathematical form that is consistent from one experiment to another. Most likely, some formula like the Lorentz function - with a well-de
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for accurately describing the nature of measurement distributions. In practice, one must deal with a finite set of values, so the nature of their distribution is never known precisely. As always, one proceeds on the basis of reasonable assumptions. Consider a large number of repeated measured values of a physical quantity. Suppose the number of values is very large, and a bar graph (Fig. 5.1) is made of the number of occurrences of each value. The tops of the bars are connected with a smooth curve. Such a curve is called an error distribution curve. Such curves come in an infinite variety of shapes, as the four examples in Fig. 5.1 illustrate. Bar graph representationof an error distribution. A bimodal distribution. A distribution with a flattened top. Gaussian (normal) distribution very accurately drawn from computer generated data. Fig. 5.1 Error distributions. One can often guess the shape of the curve, even with a finite set of values, especially such features as symmetry and spread. Just as we represent a set of values by one value (some kind of average), so also we can represent the shape of the distribution curves by measures of dispersion (spread), skewness, etc. We can describe the measurement and its uncertainty by just a few numbers. The mathematical discipline of statistics has developed systematic ways to do this. 5.2 MEASURES OF CENTRAL TENDENCY OF DATA Some of the "measures of central tendency" commonly used are listed here for reference: ARITHMETIC MEAN. (or simply the MEAN, or the AVERAGE): The sum of the measurements divided by the number of measurements. GEOMETRIC MEAN. The nth root of the product of n positive measurements. HARMONIC MEAN. The reciprocal of the average of the reciprocals of the measurements. MEDIAN. The middle value of a set of measurements ranked in numerical order. MODE The most frequent value in a set of measurements. (more precisely: the value at which the peak of the distribution curve occurs.) 5.3 MEASURES OF DISPERSION OF DATA The difference between a measurement and the mean of its distribution is called the DEVIATION (or VARIATION) of that measurement. Measures of dispersion are defined in terms of the deviations. Some commonly used measures of dispersion are listed for reference: AVERAGE DEVIATION FROM THE MEAN. (usually just AVERAGE DEVIATION, abbreviated lower case, a. d.): The average of the absolute values of the deviations. [5-1] MEAN SQUARE DEVIATION. The average of the sum of the squares of the