Miscalculation

“ Deprived of ordinary resources, the analyst throws himself into the spirit of his opponent, identifies himself therewith, and not unfrequently sees thus, at a glance, the sole methods (sometimes indeed absurdly simple ones) by which he may seduce into error or hurry into miscalculation.
Whist has long been noted for its influence upon what is termed the calculating power; and men of the highest order of intellect have been known to take an apparently unaccountable delight in it, while eschewing chess as frivolous. ”

“ For, in respect to the latter branch of the supposition, it should be considered that the most trifling variation in the facts of the two cases might give rise to the most important miscalculations, by diverting thoroughly the two courses of events; very much as, in arithmetic, an error which, in its own individuality, may be inappreciable, produces, at length, by dint of multiplication at all points of the process, a result enormously at variance with truth. ”

“ Another line of fallacies comes from the misuse of averages. Mr. W. L. Sargant, in his Essays published in London in 1870, has an exceedingly interesting chapter entitled The Lies of Statistics, and I am indebted to him to some extent for an illustration as to averages. The frequent fallacies in the practice of striking averages add greatly to the disturbing influences resulting from inaccurate enumerations, the perplexity and differences in international trade accounts, the miscalculations by individual inquirers, and the inadequate consideration of all the elements of tabular statements. ”

“ All that can, or in fact need, be said in defence of the law is, that it may be made to represent the errors in any special case to a very close approximation, and that the probability of large and practically impossible errors, as given by the law, will be so small as to be entirely inconsiderable. And as we are dealing with error itself, and our results pretend to nothing more than approximation and probability, an indefinitely small error in our process of approximation is of no importance whatever. ”

“ 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. ”

“ The formation of a final equation by means of the equation of condition amounts to multiplying each one of these by an indeterminate factor and by uniting the products; it is necessary to choose the system of factors which gives the smallest opportunity for error. But it is apparent {75} that if we multiply the possible errors of an element by their respective probabilities, the most advantageous system will be that in which the sum of these products all, taken, positively is a minimum; for a positive or a negative error ought to be considered as a loss. ”

“ Suppose that one man has collected statistics including 1000 instances, and another has collected 4000 similar instances. Common sense can recognize that the latter are better than the former; but it has no idea how much better they are. Here, as elsewhere, quantitative precision is the privilege of science. The answer we receive from this quarter is that, in the long run, the modulus,—and with this the probable error, the mean error, and the error of mean square, which all vary in proportion,—diminishes 486 inversely as the square root of the number of measurements or observations. ”

“ As a general rule we can determine a quantity with less inaccuracy as it is smaller, so that if the error itself be small the error in determining that error will be of a still lower order of magnitude. But in some cases the absolute amount of an error is of no consequence, as in the index error of a divided circle, or the difference between a chronometer and astronomical time. Even the rate at which a clock gains or loses is a matter of little importance provided it remain constant, so that a sure calculation of its amount can be made. ”

“ The law of errors admitted by all calculators is Gauss's law, which is represented by a certain transcendental curve known under the name of 'the bell.'
But first it is proper to recall the classic distinction between systematic and accidental errors. If we measure a length with too long a meter, we shall always find too small a number, and it will be of no use to measure several times; this is a systematic error. If we measure with an accurate meter, we may, however, make a mistake ”

“ The probability of a positive error of exactly one inch is 210/1024, in which fraction the numerator is the number of combinations giving one inch positive error, and the denominator the whole number of possible errors of all magnitudes. I can also, by adding together the appropriate numbers get the probability of an error not exceeding a certain amount. Thus the probability of an error of three inches or less, positive or negative, is a fraction whose numerator is the sum of 45 + 120 + 210 + 252 + 210 + 120 + 45, and the denominator, as before, giving the result 1002/1024. ”

“ 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. ”

“ The probable error is the measurement that divides the deviations from the average in either direction into halves, so that there are as many events having a greater deviation as there are events having a less deviation. ”

“ As the measure of the detriment of error, Laplace [233] takes “la Valeur moyenne de l'erreur à craindre,” the mean first power of the errors taken positively on each side of the real point. The mean spquare of errors is proposed by Gauss as the criterion. [234] Any mean power indeed, the integral of any function which increases in absolute magnitude with the increase of its variable, taken as the measure of the detriment, will lead to the same conclusion, if the normal law prevails. ”

“ This is the answer to those who think that the mathematicians treat the arithmetical squarers too lightly, and that as some person may succeed at last, all attempts should be examined. Those who have so thought, not knowing that there is demonstration on the point, will probably admit that a person who contradicts a theorem of which the demonstration has been acknowledged for a century by all who have alluded to it as read by themselves, may reasonably be required to point out the error before he demands attention to his own result. ”

“ It can readily be shown that this problem of the measurements of ancient buildings, is nothing more than the reopening of the same question, in a slightly more complex form, in reference to the question of the functions of an average.
Take a simple example. Suppose an instance in which great errors, of a certain approximate magnitude, are distinctly more likely to be committed than small ones, so that the curve of facility, instead of rising into one peak towards the centre, as in that of the familiar law of error, shows a depression or valley there. ”

“ When we have made a number of determinations with a certain apparatus or method of measurement, there is a great advantage in altering the arrangement, or even devising some entirely different method of getting estimates of the same quantity. The reason obviously consists in the improbability that the same error will affect two or more different methods of experiment. If a discrepancy is found to exist, we shall at least be aware of the existence of error, and can take measures for finding in which way it lies. ”

“ Such instances of error to which mathematicians, like all other human beings, are ever liable, enforce the necessity of continual verification of mathematical conclusions by comparing them with facts revealed by observation and experiment. ”

“ According to Gauss the Law of Error expresses the comparative probability of errors of various magnitude, and partly from experience, partly from à priori considerations, we may readily lay down certain conditions to which the law will certainly conform. It may fairly be assumed as a first principle to guide us in the selection of the law, that large errors will be far less frequent and probable than small ones. We know that very large errors are almost impossible, so that the probability must rapidly decrease as the amount of the error increases. ”

“ If we consult history, we shall find them increasing, in every state, with the extent of dominion. In political arithmetic, it is necessary to substitute a calculation of probabilities, to mathematical exactness. ”

“ Even Cauchy, who was or became something of a purist according to the standards of his day, did not shrink on occasion from handling [Pg 39] divergent integrals, but managed always to come right in the end. There is this to be said about mathematical work, in any but quite incompetent hands, that a too careless induction sooner or later betrays itself, and leads to a revision of the whole calculation. The great mathematicians, whatever licence they may have allowed themselves, have always had a sure instinct to save them from logical disaster. ”

“ Considering, however, in how accelerating a ratio the uncertainty of our conclusions increases, as we attempt to take the effect of a greater number of concurrent causes into our calculations; the hypothetical combinations of circumstances on which we construct the general theorems of the science, cannot be made very complex, without so rapidly-accumulating a liability to error as must soon deprive our conclusions of all value. ”

“ In other words, if we suppose the errors to be produced by a limited number of finite, equal and independent causes, we have an approximation to the exponential Law of Error, which merges into identity as the causes are increased in number and diminished in magnitude without limit. ”

“ In almost every case in which we have to make measurements we shall find that large errors are much more mischievous than small ones, that is, mischievous in a greater ratio than that of their mere magnitude. ”

“ Either because they do not know any mathematics, or because the men who figured things out mathematically for them had frequently consciously or unconsciously deceived them, the opinion that mathematics was inadequate and useless for the calculation of measures has established itself as an undoubted truth which they think it is even unnecessary to prove. ”

“ The case in point lies without the operation of our rules.
It is always safe for the beginner to proceed on the assumption that the first of the above alternatives is the cause of the error; in other words, that his own calculations are at fault. ”

“ There is a recognized method of estimating the probable magnitudes of errors in physics,—the method of least squares. It is universally admitted that this method makes the errors smaller than they really are; 189yet even according to that theory an error indefinitely small is indefinitely improbable; so that any statement to the effect that a certain continuous quantity has a certain exact value, if well-founded at all, must be founded on something other than observation. ”

“ The abstruse truths of mathematics have found, in our days, some zealous and estimable followers, who have devoted themselves to the elucidation of what is held as most impenetrable in the infinite secrets and profound obscurities of that science; but these are rare and singular men; the infection was little to be dreaded, nor, in truth, has it spread; they are admired, but few would wish to follow their example. ”

“ Such sentiment—for the half-credences of which I speak have never the full force of thought—such sentiments are seldom thoroughly stifled unless by reference to the doctrine of chance, or, as it is technically termed, the Calculus of Probabilities, Now this Calculus is, in its essence, purely mathematical; and thus we have the anomaly of the most rigidly exact in science applied to the shadow and spirituality of the most intangible in speculation. ”

“ And for want of such precaution as this, the arithmetician is at sea the moment he steps out of the narrow path of mechanical routine. Who is not aware of the failures of calculation when applied to the general business of life—to statistics, moral and political? ”

“ Mathematics of the higher order expressed in elaborate formulæ do very well in communications between college professors—that is, if they happen to be agreed. When, however, these calculations are so intricate as to require a clever mathematician a whole day to study out the meaning of a single page, and if when the riddle is solved, we find that these calculations are based on a fallacy, and the results in conflict with facts, it becomes quite evident to the actual experimenter that they are of little value. ”

“ The comparison between the numerical results obtained at one time and place and another, and between those presented in different countries, is becoming a political method increasing in prevalence and repute. In many quarters, indeed, the value of purely numerical estimates has been much exaggerated, and its peculiar liability to error, when made a basis of political reasoning, has been too much ignored. ”