Measurement error

“ 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. ”

“ It must, however, be remembered that in all practical applications of formulae the data have first to be ascertained by direct or indirect measurement; and this measurement involves a certain margin of error.
The two sources of error mentioned under (a) and (b) above are closely related. Suppose, for instance, that we require the area of a circular grass-plot of measured diameter. As a matter of fact, no grass-plot is truly circular; and it might be found that if the breadth in various directions were measured more accurately the want of circularity would reveal itself. ”

“ The ratio of two quantities can be determined with unlimited accuracy, if we can multiply both the object of measurement and the standard unit without error, and then observe what multiple of the one coincides or nearly coincides with some multiple of the other. Although perfect coincidence can never be really attained, the error thus arising may be indefinitely reduced. ”

“ The nature of measurement is well understood except in so far as it relates to psychological phenomena; in this realm metaphysicians seem wholly to misconceive its nature. I cannot measure the number of the ultimate particles of a body by counting them, but I may measure the relative number of its atoms by weighing it. I do not determine its force, but its mass only, when I weigh the body, for the total force in the body is the sum of its motion in all its incorporations. A pound of powder has much more force than that which is measured as a pound. ”

“ 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. ”

“ We do not search with the microscope for atoms and count them, but we consider their forms as extensions or their forces as masses and reason about the artificial numbers derived therefrom by measurement with the same degree of certainty that we would have if we should actually count the particles. Thus measure is devised in order that we may consider numbers when the actual numbers are concealed from observation. That every property is concomitant with all others is thus assumed as the fundamental doctrine of mathematics where quantitative reasoning is held to be exact and irrefragable. ”

“ All judgments are liable to error, and cognition comes only with verification. In quantitative judgments the liability to error is infinite as that term is used by mathematicians, and all judgments must be verified unless the amount of error may be neglected.
In scientific research verification is often by measurement. Counting itself is measuring, and the sum is the number of units which the measured body contains, and these units are units of a kind. It is only in counting that the units are natural ”

“ In the second place, her estimate is so rough that an error must be very great in order to have any meaning. If I say that a man is six feet and two inches tall, it is easy enough to apply a measuring stick and prove the correctness or incorrectness of my assertion. ”

“ 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 ”

“ 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. ”

“ 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. ”

“ 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. ”

“ It is obvious that anything which alters the weight of a single piece in an analytical set will introduce an error in every weighing made in which that piece is used. This source of error is often extremely obscure and difficult to detect. The only safeguard against such errors is to be found in scrupulous care in handling and protection on the part of the analyst, and an equal insistence that if several analysts use the same set of weights, each shall realize his responsibility for the work of others as well as his own. ”

“ Even so, when a number of measurements are made of the same quantity, it regularly happens that different values are obtained, the difference between a value as obtained and the true value being called an error of observation. ”

“ 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 most elaborate researches, such as those undertaken in connection with standard weights and measures, always render it apparent that complete coincidence is out of the question, and that the more accurate our modes of observation are rendered, the more numerous are the sources of minute error which become apparent. We may look upon the existence of error in all measurements as the normal state of things. It is absolutely impossible to eliminate separately the multitude of small disturbing influences, except by balancing them off against each other. ”

“ In many experiments and measurements we know that there is a preponderating tendency to error in one direction. The readings of a thermometer tend to rise as the age of the instrument increases, and no drawing of means will correct this result. Barometers, on the other hand, are likely to read too low instead of too high, owing to the imperfection of the vacuum and the action of capillary attraction. If the mercury be perfectly pure and no appreciable error be due to the measuring apparatus, the best barometer will be that which gives the highest result. ”

“ The measuring instrument of to-day is an instrument of comparison constructed with meticulous care, which enables us to do away with causes of error formerly ignored, to eliminate the action of external phenomena, and to withdraw the experiment from the influence of even the personality of the observer. This standard is no longer, as formerly, a flat rule, weak and fragile, but a rigid bar, incapable of deformation, in which the material is utilised in the best conditions of resistance. ”

“ 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. ”

“ Again, within the aspects selected for attention, all that experiment can establish is that the deviation from uniform law, if there is any deviation, is not sufficiently great to affect our measurements and calculations. But how far our standards of measurement are from rigid precision may be readily learned from the chapter on physical standards in any good work on the logic of the inductive sciences. ”

“ In these cases, and only in these cases, it will be desirable to invent some artificial mode of counterpoising the variable error against an equal error subject to exactly the same variation.
We cannot weigh an object with great accuracy unless we make a correction for the weight of the air displaced by the object, and add this to the apparent weight. In very accurate investigations relating to standard weights, it is usual to note the barometer and thermometer at the time of making a weighing, and, from the measured bulks of the objects compared, to calculate the weight of air displaced ”

“ The nature and number of blank measurements will depend on the method and circumstances, but the purpose of all blank measurements is to help prevent errors in identification and quantitation of field samples. [3] d. Blind Samples Blind samples are prepared by someone other than the analyst performing the measurement and are to provide an independent check on the accuracy and precision of the measurement. If blind samples cannot be prepared with confidence, their use should be avoided. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ In taking the various test measurements for the purpose of calculating one's personal error, it is well to use the precaution of not taking them twice at the same sitting, but after an interval of time, not only so that all marks will have disappeared that may have been left upon the skin by the instrument in the act of measuring, but also that the preceding figure will have faded from our memory. ”

“ Those quantities, which are called errors in one case, may really be most important and interesting phenomena in another investigation. When we speak of eliminating error we really mean disentangling the complicated phenomena of nature. The physicist rightly wishes to treat one thing at a time, but as this object can seldom be rigorously carried into practice, he has to seek some mode of counteracting the irrelevant and interfering causes.
The general principle is that a single observation can render known only a single quantity. ”

“ 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. ”