Standard deviation

“ But, as we decrease the number of experiments, the value of the standard deviation found from the sample of experiments becomes itself subject to an increasing error, until judgments reached in this way may become altogether misleading.
In routine work there are two ways of dealing with this difficulty: (1) an experiment may be repeated many times, until such a long series is obtained that the standard deviation is determined once and for all with sufficient accuracy. This value can then be used for subsequent shorter series of similar experiments. ”

“ This sphere of subjective probability is widened by the latest developments of the science [47] so far as they add to the number of constants for which it is important to determine the probable—and improbable—error. For instance, a measure of the deviation of observations from an average or mean value was required by the older writers only as subordinate to the determination of the mean, but now this “standard deviation” (below, par. 98) is often treated as an entity for which it is important to discover the limits of error. ”

“ Is the American stock more or less variable? [Pg 428] Can a "melting pot" influence be seen, tending to produce homogeneity, or has change of environment rather produced greater variability, as is sometimes said to be the case?
The amount of variability is most conveniently measured by a coefficient known as the standard deviation σ, which is small when the range of variation is small, but large when diversity of material is great. The following comparisons of the point at issue may be made. [202]
For the men, little difference is discernible. ”

“ The area being constant (viz. unity) , the curve is furled up when c is small, spread out when c is large. There is added a table of integrals, corresponding to areas subtended by the curve; in a form suited for calculations of probability, the variable, τ, being the length of the abscissa referred to (divided by) the modulus. [164] It may be noted that the points of inflexion in the figure are each at a distance from the origin of 1/√2 modulus, a distance equal to the square foot of the mean square of error—often called the “standard deviation.” ”

“ Doubtless different parameters are suited to different purposes and contexts; c when we consult the common tables, and in connexion with the operator, as below, par. 160; k ( = 1⁄2c2) when we investigate the formation of the probability-curve out of independent elements (below, par. 104) ; h ( = 1/c2) when we are concerned with weights or precisions (below, par. 134) . If one form of the coefficient must be uniformly adhered to, probably, σ ( = c/√2) , for which Professor Pearson expresses a preference, appears the best. It is called by him the “standard deviation.” ”

“ It would seem, therefore, that we might find a sort of measure of the deviation of an ensemble from statistical equilibrium in the excess of the average index above the minimum which is consistent with the condition of the invariability of the distribution with respect to the constant functions of phase. But we have seen that the index of probability is constant in time for each system of the ensemble. The average index is therefore constant, and we find by this method no approach toward statistical equilibrium in the course of time. ”

“ 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. ”

“ It will be recalled that the coefficient of correlation measures the resemblance between two variables on a scale between 0 and-1 or +1. If the correlation is zero, there is no constant relation; if it is unity, any change in one must result in a determinate change in the other; if it is 0.5, it means that when one of the variables deviates from the mean of its class by a given amount, the other variable will deviate from the mean of its class by 50% of that amount (each deviation being measured in terms of the variability of its own class, in order that they may be properly comparable.) ”

“ The Law of Error.—Of the propositions respecting average with which Probabilities is concerned the most important are those which deal with the relation of the average to its constituents, and are commonly called “laws of error.” Error is defined in popular dictionaries as “deviation from truth”; and since truth commonly lies in a mean, while measurements are some too large and some too small, the term in scientific diction is extended to deviations of statistics from their average, even when that average—like the mean of human or barometric heights—does not stand for any real objective thing. ”

“ One is that the result formulated in the statistical law is always one to which the actual course of events may reasonably be expected to conform within certain limits of deviation, never one to which we have a right to expect absolute conformity. Not only is the actual number of marriages, e.g., in any one year, usually slightly above or below the average percentage computed, e.g., for a ten years’ period, but as we compare one longer period with others, the average percentage for the longer period itself fluctuates. ”

“ 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. ”

“ Provide Assessments of Variability Nearly all biological measurements, when repeated, exhibit substantial variation, and this creates uncertainty in the result of every calculation based on the data. A basic purpose of statistical analysis is to help assess uncertainty, often in the form of a standard error or confidence interval, and one of the great successes of statistical modeling and inference is that it can provide estimates of standard errors from the same data that produce estimates of the quantity of interest. ”

“ Therefore, the distribution in phase which was one of statistical equilibrium for the old values of the external coördinates, will not be such for the new values.
Now we have seen, in the last chapter, that when the distribution-in-phase is not one of statistical equilibrium, an ensemble of systems may, and in general will, after a longer or shorter time, come to a state which may be regarded, if very small differences of phase are neglected, as one of statistical equilibrium, and in which consequently the average value of the index () is less than at first. ”

“ It is usual, however, to assume a normal distribution, because, in a very large number of cases, this gives an approximation so close that a small sample will give no real information as to the manner in which the population deviates from normality: since some law of distribution must be assumed it is better to work with a curve whose area and ordinates are tabled, and whose properties are well known. ”

“ 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. ”

“ We have also shown that the average square of the anomalies of , that is, of the deviations of the individual values from the average, is in general of the same order of magnitude as the reciprocal of the number of degrees of freedom, and therefore to human observation the individual values are indistinguishable from the average values when the number of degrees of freedom is very great. ”

“ Since there is no limit to the diminution of the disturbance of equilibrium by division of the change in the external coördinates, we may suppose as a general rule that by diminishing the velocity of the changes in the external coördinates, a given change may be made to produce a very small disturbance of statistical equilibrium. ”

“ It is evident, therefore, that in close-breeding, and in the crossing of varieties of different degrees of divergence, there is, first, a less than average result, then an average, then better than an average, then this better result quickly reaches a maximum and again declines, crosses the line of average and becomes bad, and finally infinitely bad, or dies out. ”

“ Why are we so interested in the type of scale that measures a dependent variable? The crux of the matter is the relationship between the variable's level of measurement and the statistics that can be meaningfully computed with that variable. ”

“ A mere average, therefore, might lead to conclusions only the more misleading, for their deceptive semblance of accuracy. There would be less danger of error in taking the average of only a small number of years, and correcting it by a conjectural allowance for the character of the seasons, than in trusting to a longer average without any such correction. ”

“ It is evident that the canonical distribution is entirely determined by the modulus (considered as a quantity of energy) and the nature of the system considered, since when equation (92) is satisfied the value of the multiple integral (93) is independent of the units and of the coördinates employed, and of the zero chosen for the energy of the system.
In treating of the canonical distribution, we shall always suppose the multiple integral in equation (92) to have a finite value, as otherwise the coefficient of probability vanishes, and the law of distribution becomes illusory. ”

“ Distributions have different shapes; they don't all look like the normal distribution in Figure 4. For example, the normal probability density is higher in the middle compared to its two tails. Other distributions need not have this feature. There is even variation among the distributions that we call “normal.” For example, some normal distributions are more spread out than the one shown in Figure 4 (their tails begin to hit the X-axis further from the middle of the curve --for example, at 10 and 90 if drawn in place of Figure 4) . ”