Normal distribution

“ In situations where the uncertainty of the observation is due to measurement error or to a multiplicity of factors, the distribution obtained often assumes a normal form or a form sufficiently normal so that the normal distribution can be used as an approximation. ”

“ The law can be derived from the following properties of random errors: (1) positive and negative errors of the same magnitude are about equal in number, (2) small errors occur more frequently than large errors, and (3) extremely large errors rarely occur. One immediate consequence of these properties is that the average or mean value of a large number of observations of a given quantity is zero. Also called GAUSSIAN DISTRIBUTION. See also SINGLE-AXIS NORMAL DISTRIBUTION, CIRCULAR NORMAL DISTRIBUTION, STANDARD DEVIATION.
normal distribution curve. ”

“ The studies of biological traits suggest that a unit of the standard deviation is the most important measure we have for equivalent degrees of any trait which distributes normally. It measures the same portion of the total distance from the lowest to the highest ability on any objective scale so long as the distribution of measurements is in the normal form. It thus affords the best interchangeable unit from measurements at one life-age to those at another, provided that the distributions keep close to the form of the normal probability curve. ”

“ A method of rejecting data analogous to the use of percentiles in one dimension is practised when, given the frequency of observations for each increment of area, e.g. each ∆x ∆y, we utilize only the frequency for integral areas. Mr Sheppard has given an elegant solution of the problem: to find the correlation between two attributes, given the medians L, and M, of a normal group for each attribute and the distribution of the total group, as thus. [261] ”

“ The lack of exactness might be due to the crudeness of the various measurements involved or perhaps to the fact that population depends on more than one such factor. To account in some way for the uncertainty, we might make a further assumption and propose the following relationships:
population = a + b (ln area) + X
population = a + b (ln fishing miles) + X
where X has a normal probability distribution with mean = 0 and some unknown variance = σ2. ”

“ These graphs represent the normal form of distribution usually expected when any fundamental ability is measured in a random group. If the number of cases at each unit of measurement are plotted by a point placed relatively as far above the scale, used as a base line, as the number of cases found at that unit of the scale, it will be discovered that these points arrange themselves in the form of a symmetrical curve high at the middle and flaring out 257along the base-line scale. ”

“ Theorem VII. If a system which in its different phases constitutes an ensemble consists of two parts, and we consider the average index of probability for the whole system, and also the average indices for each of the parts taken separately, the sum of the average indices for the parts will be either less than the average index for the whole system, or equal to it, but cannot be greater. The limiting case of equality occurs when the distribution in phase of each part is independent of that of the other, and only in this case. ”

“ In the following theorems we suppose, as always, that the systems forming an ensemble are identical in nature and in the values of the external coördinates, which are here regarded as constants.
Theorem I. If an ensemble of systems is so distributed in phase that the index of probability is a function of the energy, the average value of the index is less than for any other distribution in which the distribution in energy is unaltered. ”

“ If a distribution, which is not one of statistical equilibrium, should be given for a time , and the distribution at an earlier time should be defined as that given by the corresponding phases, if we increase the interval leaving fixed and taking at an earlier and earlier date, the distribution at will in general approach a limiting distribution which is in statistical equilibrium. ”

“ If an ensemble, whether grand or petit, is identical so far as generic phases are concerned with one canonically distributed, we shall say that its distribution is canonical with respect to generic phases. Such an ensemble is evidently in statistical equilibrium with respect to generic phases, although it may not be so with respect to specific phases. ”

“ Where is negligible, we may write Now, by Theorem III of Chapter XI, the quantity
has a minimum value when the ensemble to which and relate is distributed canonically with the modulus . If the ensemble had originally this distribution, the sign in (472) would be impossible. In fact, in this case, it would be easy to show that the preceding formulae on which (472) is founded would all have the sign . ”

“ Let us consider three attributes, say A, B and C, of the individuals of a population and suppose that the attribute C is positively correlated to the attributes A and B, so that to great or small values of A or B correspond respectively great or small values of C. Now if the individuals in the population are statistically selected in such a way that we choose out individuals having great values of the attributes A and small values of the attribute B, then we get a statistical series regarding the attribute C, which consists of two seemingly distinct normal frequency distributions. ”

“ We shall find however, in certain cases, that for certain values of the energy, viz., for those for which is infinite, this process fails to define a limiting distribution in any such distinct sense as for other values of the energy. ”

“ The symmetry of the distribution leaves something to be desired. The values below the average preponderate a little in number, those above preponderate a little in amount of deviation: only two of the largest eight deviations are below the mean value. The influence of attention referred to above, the fluctuations of which in the separate series show greater deviations toward the upper limit than toward the lower, has not, therefore, been quite compensated by the combination of several series. ”

“ Such is the great complexity of our mental life that it is not possible to deny that constant mean values, when obtained, are of the nature of such statistical constants. To test that, I examine the distribution of the separate numbers represented in an average value. If it corresponds to the distribution found everywhere in natural science, where repeated observation of the same occurrence furnishes different separate values, I suppose—tentatively again—that the repeatedly examined psychical process in question occurred each time under conditions sufficiently similar for our purposes. ”

“ By simple induction we know that the median of a large sample of observations is a probable value for the true median; how probable is determined as follows from a selection of our data. First suppose that all the observations are of the same weight. If x′ were the true median, the probability that as many as 1⁄2n + r of the observations should fall on either side of that point is given by the normal law for which the exponent is -2r2/n. ”

“ When the word "significantly" is used in comparisons, it is meant to show that there is a significant statistical difference between two or more samples. Whenever eight or more specimens from one locality were available, the mean, range, standard deviation, standard error of the mean, and coefficient of variability were calculated. ”

“ DISTRIBUTION OF GROUP AVERAGES FOR NEW WORLD NATIVES
It should be noted that we are here treating the successive social groups like individuals. Thus, if we wish to arrive at the head form for a social group, we take individuals at random and calculate from these observations the most probable average value. In exactly the same manner, we may take these values for groups in the New World and place them in a series. When we do this, we again find a typical distribution suggesting homogeneity. ”

“ If the present distribution of population be represented by the line numbered 1, the distribution a hundred years hence may be represented by the dotted line numbered 2, and that which will exist after five hundred years shall have passed may be represented by the dotted line numbered 3. Even within the economic center the comparative density of population in different divisions is therefore not to be treated as strictly [Pg 227] permanent, and it is not to be treated as in any sense permanent when we are forecasting effects that will be realized several centuries hence. ”

“ The median step of a distribution is the step which contains within it the median point. Similarly, the median measure in any distribution is the measure which contains the median point. In a distribution containing 25 measures, the 13th measure is the median measure, because 12 measures are greater and 12 are less than the 13th, while the 13th measure is itself divided into halves by the median point. Where a distribution contains an even number of measures, there is in reality no median measure but only a median point between the two halves of the distribution. ”

“ The distribution may be called regular if the classes increase to a maximum, the mode, and then decrease without interruption. (See Fechner, Collectivmaasslehre, pp. 121-122.)
[132] The values of the means and variabilities was determined to the second decimal for the females, but only to the first for the males; consequently the relative variability could be computed exactly to the first decimal for the females and to the unit only for the males. ”

“ To analyze statistically significant differences between variables, we selected appropriate statistical tests based on the distribution of data including the Kruskal Wallis test, chi square tests, Wilcoxon Rank sum test, and Spearman’s correlations for non-parametric data and one-way analysis of variance and Pearson’s correlations for continuous data. ”

“ The results of the distribution are given in Table B. As a test of the smoothness of the distribution, the results were drawn to scale on large diagrams, of which Plates I. and II. are reduced copies. ”

“ If, however, the form of distribution remains uniform, it would first be necessary for those advocating the use of any of the other quantitative definitions to show that the units of their scales are equal under some reasonable hypothesis. A ratio or a deviation statable only in scale units which are not demonstrably equal is a hazard, with the chances badly weighted against its reliability. So far as both the Binet and the Point scales are concerned we have found that the units are not equal. ”

“ Such a method, analogous to the one employed by statisticians for following the social phenomena in a population, is all the more legitimate the greater the number of individuals counted in the averages; now, the number of molecules contained in a limited space—for example, in a centimetre cube taken in normal conditions—is such that no population could ever attain so high a figure. ”

“ If treating the elements of volume as constant, we continue the stirring indefinitely, we get a uniform density, a result not affected by making the elements as small as we choose; but if treating the amount of stirring as finite, we diminish indefinitely the elements of volume, we get exactly the same distribution in density as before the stirring, a result which is not affected by continuing the stirring as long as we choose. ”

“ The same phenomena of distribution appear if a thousand wild horses run a race, or if a hundred unselected swimmers swim in competition. They appear whenever non-select organisms of a single species are submitted to an adequate test or measure of any function of endowment. The curve approximates that form which mathematicians tell us results when an infinite number of factors act together in an infinite number of ways. ”

“ The distributions to be considered are such that any number of objects may be in any one class subject to the restriction that no class is empty. Two cases arise. If the order of the objects in a particular class is immaterial, the class is termed a parcel; if the order is material, the class is termed a group. The distribution into parcels is alone considered here, and the main problem is the enumeration of the distributions of objects defined by the partition (pqr . . . ) of the number n into parcels defined by the partition (p1q1r1 . . . ) of the number m. ”

“ The median point of any distribution of measures is that point on the scale which divides the distribution into two exactly equal parts, one half of the measures being greater than this point on the scale, and the other half being smaller. When the scales are very crude, or when small numbers of measurements are being considered, it is not worth while to locate this median point any more accurately than by indicating on what step of the scale it falls. ”

“ The nature of the difficulty is at once apparent if we try to distribute a finite number of material points with this particular value of the energy as nearly as possible in statistical equilibrium, or if we ask: What is the probability that a point taken at random from an ensemble in statistical equilibrium with this value of the energy will be found in any specified part of the circle? ”

“ If we had smaller grouping the zero values of the standard deviation might be expected to increase, and a similar consideration will show that the smaller values of the standard deviation would also be likely to increase, such as .436, when 3 fall in one group and 1 in an adjacent group, or .50 when 2 fall in two adjacent groups. ”