The Normal Law of Error

The distribution function for the random walk gives us an immediate approach to the theory of random errors. In any physical experiment there may be a number of factors disturbing an observation, with each one contributing an error of magnitude, let us say, 8 (for the ith source) which may be positive or negative. The result of all of these individual errors is to produce a total error a = SS, such that our observed measurement, call it m, differs from what we presume is the true value m by the amount x 

If we assume that each of the individual errors may be assigned an equal probability of being positive or negative and all act independently, then the question of finding the distribution of possible errors is given precisely by the answer to the random walk problem with varied step. That is, the chance of making an error x is given by 

This Gaussian distribution function is symmetrical about the true value m (here choosen as the origin for x) and thus implies that positive errors are as probable as negative errors. It is normalized, since  

This Gaussian function is shown plotted in Fig. VI.2. The ma.ximum value of P(pc) [obtained by differentiating Eq. (VI.8.2) and solving the 

The half-width of the distribution is defined as the value of x at which P x) has fallen to half its maximum value. We find = o-(21n2) = 1.175 r. The mean width is given by the value of x at which P(x) has fallen to 1/e of its maximum value. We find xi/e = a 2 = 1.414(r. From a  

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The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

These are the relations which exist between the moments of the normal distribution and in this sense the mean concentration is ultimately distributed about a point moving with the mean speed of the stream according to the normal law of error, the variance being 2(1 + kh2)t. It should be noted that the approach to normality is as r 1, a very much slower process than the vanishing of terms in the expressions for the moments, which is as exp(-Air). 

Thus, the table forms a convenient reference to check observed data— that is, to test what would be expected if the normal law of errors applied. Note, too, how a measures the dispersion of our data. 

The Gaussian distribution and the normal law of error are both often expressed as the same relationship. The Gaussian distribution law is the theoretical frequency distribution for a set of data of any normal, repetitive function, due to chance, represented by a bell-shaped curve symmetrical about a mean. The relationship of the number of events occuring and frequency when the events occur are due to chance only. The probability for distributions that occur due to chance is ... 

NORMAL LAW OF ERROR A mathematical equation that in many cases describes the scatter of a collection of measurements around the average for the collection. 

DeMoivre did his work. The normal distribution is sometimes called the Gaussian distribution and sometimes the Law of Errors. 

If position error is governed by normal law of distribution, and the PAD is circle-shaped, then the equation (1) becomes ... 

Comparison and agreement with the calorimetric value verifies the assumption that So = 0. For example, we showed earlier that the entropy of ideal N2 gas at the normal boiling point as calculated by the Third Law procedure had a value of 152.8 0.4 J-K mol. The statistical calculation gives a value of 152.37 J K -mol-1, which is in agreement within experimental error. For PH3, the Third Law and statistical values at p 101.33 kPa and T— 185.41 K are 194.1 0.4 J K, mol 1 and 194.10 J-K 1-mol 1 respectively, an agreement that is fortuitously close. Similar comparisons have been made for a large number of compounds and agreement between the calorimetric (Third Law) and statistical value is obtained, all of which is verification of the Third Law. For example, Table 4.1 shows these comparisons for a number of substances. 

If we assume that the residuals in Equation 2.35 (e,) are normally distributed, their covariance matrix ( ,) can be related to the covariance matrix of the measured variables (COV(sy.,)= LyJ through the error propagation law. Hence, if for example we consider the case of independent measurements with a constant variance, i.e. 

The previous equations do not require that the isotopic ratio used for normalization (x-axis) and the ratio to be measured (y-axis) have to be for the same element. It is therefore possible to normalize the isotopic composition of Cu to that of a standard Zn solution without any assumption made on the particular mass-fractionation law. The original formulation of this property by Longerich et al. (1987) calls for identical isotopic fractionation factors for the two elements, but this is not at all a necessary constraint and Albarede et al. (2004) show that, in fact, this very assumption may lead to significant errors. For a Cu sample mixed with a Zn standard, in which the Zn/ Zn ratio of the standard solution is used for normalization, we obtain the expression ... 

In criminal proceedings caused by crimes that are considered by the German authorities to have caused major violations of law and order, the trial is held immediately on the District Court level, i.e., on what normally is supposed to be the appeal level (the first level is the County Court). In such cases, the accused has only one trial during which evidence can be presented, that is, there is no appeal possible to the verdict of this court Only a so-called application for a revision of the verdict with the German Federal Supreme Court is possible, but such an application can only criticize errors of form (matters of law). The factual assertions of the deciding court, i.e., description and evaluation of evidence (matters of fact), will not be discussed anymore. Furthermore, it is usually the case that applications for a revision will be denied by the German Federal Supreme Court, if the defense is the only party to request it. 

If we suppose that two arrays of time delays Mi and M2 correspond to the normal distribution law and they are not homogeneous then the attack is carried out using intermediate computers [12], Therefore, it is necessary to fulfill the appropriate analysis in order to determine the attack source status. For this purpose it is necessary to calculate mathematical expectations xv for an array Mi(f) and yv for an array M2(0, dispersions Dvx, Dvy and mean-square errors crvx and [Pg.197]

Even if all systematic error could be eliminated, the exact value of a chemical or physical quantity still would not be obtained through repeated measurements, due to the presence of random error (Barford, i985). Random error refers to random differences between the measured value and the exact value the magnitude of the random error is a reflection of the precision of the measuring device used in the analysis. Often, random errors are assumed to follow a Gaussian, or normal, distribution, and the precision of a measuring device is characterized by the sample standard deviation of the distribution of repeated measurements made by the device. [By contrast, systematic errors are not subject to any probability distribution law (Velikanov, 1965).] A brief review of the normal distribution is provided below to provide background for a discussion of the quantification of random error. 

Because part of the anomalous dispersion component is jt/2 out of phase with the isomorphous, real component, the net observable effect is a breakdown of Friedel s law regarding the perfect equality of the magnitudes of and If-h-k-i- That is, the two need not be absolutely equivalent but can demonstrate some slight difference A I anom = fhki — f-h-k-i- This difference will normally be imperceptible and within the expected statistical error of most X-ray diffraction intensity measurements, but with care in data collection, and judicious choice of X-ray wavelength, it can be measured and used to obtain phase information in conjunction with isomorphous replacement phase determination, or even independently, as described in Chapter 8. 

Add your results to those in the table, and plot a histogram similar to the one shown in Figure 6F-1. Find the mean and the standard deviation (see Section 6B-3) for your results and compare them with the values shown in the plot. The smooth curve in the figure is a normal error curve for an infinite number of trials with the s.ame mean and standard deviation as the data set. Note that the mean of 5.06 is very close to the value of 5 that you would predict based on the laws of probatdlity. As the number of trials increases, the histogram approaches the shape of the smooth curve and the mean approaches 5. 

Where no effective treatment exists at present for some inborn errors of metabolism, and where these disorders are associated with severe illness, mental retardation, or early death, the prenatal diagnosis of the diseased fetus by amniotic fluid enzyme analysis is now possible (MIO, Mil, M12). These new developments in the field of transabdominal amniocentesis, coupled with more liberal abortion laws, can serve to reassure families with a previous history of a fatal disease that their offspring will be physiologically normal. As applied to the field of amino acid disorders, enzyme analysis of cultured amniotic fluid cells has been used to diagnosis potential cases of homocystinuria (Fig. 51) and... 

On a chromatogram the perfect elution peak would have the same form as the graphical representation of the law of Normal distribution of random errors (Gaussian curve 1.1, cf Section 22.3). In keeping with the classic notation, p would correspond to the retention time of the eluting peak while a to the standard deviation of the peak (cr represents the variance), y represents the signal as a function of time X, from the detector located at the outlet of the column (Figure 1.3). 

The normal distribution law (bell-shaped gaussian curve) is the mathematical model which best represents the distribution of random or indeterminate errors due to hazards (Equation 22.10) ... 

A straight line supposes that the errors in y follow the law of Normal distribution. is the determination coefficient that conveys information about how the variations of x overlap variations in y. 

In C laussian statistics, tlio results ot replicate measurements arising from indeterminate errors arc assumed to be disiribuied according to the normal error law, which stales ihal the fraction of a population of observations. 

The binomial distribution law correctly expresses this probability, but it is common practice to use either the Poisson distribution or the normal Gaussian distribution fimctions since both approximate the first but are much simpler to use. If the average number of counts is high (above 100) the Gaussian function may be used with no appreciable error. The probability for observing a measured value of total count N is... 

Perhaps the central limit theorem justifies the rapturous remark of Sir Francis Galton, the inventor of linear regression Hardly will there be something as impressive for the imagination as the admirable form of cosmic order expressed by the Law of the Frequency of Errors that is, the normal distribution). Had the Greeks known of it, they would certainly have personified and deified it . 

From the standpoint of statistics, the transformation Eq. 2.2-19 into Eq. 2.3.b-l and the determination of the parameters from this equation may be criticized. What is minimized by linear regression are the (residuals) between experimental and calculated y-values. The theory requires the error to be normally distributed. This may be true for r, but not necessarily for the group /(Pa - PrPs/I Wa and this may, in principle, affect the values of k, K, K, Ks, — However, when the rate equation is not rearranged, the regression is no longer linear, in general, and the minimization of the sum of squares of residuals becomes iterative. Search procedures are recommended for this (see Marquardt [41]). It is even possible to consider the data at all temperatures simultaneously. The Arrhenius law for the temperature dependence then enters into the equations and increases their nonlinear character. 

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