Error of observation

The entire determination, from gas shutoff to printout from the computer, rarely exceeds 3 to 4 minutes for normal solids. And since all measurements are taken by the instrument, they are not subject to personal error of observation. The size of solids test sample is of the order of a kilogram or less. 

Topping, J. (1972). Errors of Observation and their Treatment. London, Chapman and Hall (4th edn.). 

A random variable, Z. may be constructed on a sample space by specifying its values Zi, Z2. on the sample points. Such variables are useful in modeling random errors of observations. The mean value of Z over the sample space is then given by the relations... 

Obviously the greater the value of the variance of f(random error in f(ay) originates from the random errors of observation of ay and provided that these are small so that [Pg.372]

Two types of errors of observation exist systematic and random. The total error associated with a measurement is a function of the systematic error, commonly called bias, and the random error. Systematic errors in environmental measurements can be divided into three general categories instrument errors (e g., nonideal functioning of an instrument) interference from envi-... 

Topping, J. Errors of Observation and Their Treatment. The Institute of Physics and The Physical Society Monographs for Students. Chapman Hall, London, 1969. 

The error mean squares, MSE, is an estimate of the variance of the error of observation, and is ... 

From the beginning it was clear to me that it would be of little significance for our purpose to investigate a few substances with the highest accuracy. It was undoubtedly of much more importance to secure, at first in its general lines, a picture as complete as possible of the behaviour of the specific heats at low temperature, avoiding, of course, as far as possible serious errors of observation. 

Errors of observation will not account for the fact that the difference in question is a little smaller at low temperatures, and at high temperatures a little greater, than corresponds with the above formula but the discrepancies are too small to affect the accuracy of the above equations appreciably. The experiments have certainly given good confirmation of the variation of the specific heats demanded by the foregoing formulae."... 

The values in the last column but one, calculated by means of the Heat Theorem, agree extremely well with the mean of Uj and Ua (last column but two) the differences, sometimes positive and sometimes negative, are all within the errors of observation which affect the values of Ux and U3, and also of the specific heats, and may easily amount to 200 cals, by accumulation. The mean value of U, U and U2 should give us the heat evolution in the re-spective reactions with an ac- > curacy hitherto unknown in such W cases. W... 

By no process of sound reasoning can a conclusion drawn from limited data have more than a limited application. Even when the comparison between the observed and calculated results is considered satisfactory, the errors of observation may quite obscure the imperfections of formulae based on incomplete or simplified premises. Given a sufficient number of if s, there is no end to the weaving of cobwebs of learning admirable for the fineness of thread and work, but of no substance or profit .2 The only safeguard is to compare the deductions of mathematics with observation and experiment for the very simple reason that they are only deductions, and the premises from which they are made may be inaccurate or incomplete. We must remember that we cannot get more out of the mathematical mill than we put into it though we may get it in a form infinitely more useful for our purpose. 8... 

This idea is of great practical use in the reduction of intricate expressions to a simpler form more easily manipulated. It is usual to reject magnitudes of a higher order than those under investigation when the resulting error is so small that it is outside the limits of the errors of observation peculiar to that method of investigation. 

This method may be used with any of the above formulae when an exact determination of the constants is of no particular interest, or when the errors of observation are relatively small. Y. H. Reg-nault used it in his celebrated M6moire sur les forces 61astiques de la vapeur d eau" (Ann. Ghim. Phys., [3],. 11, 273, 1844) to evaluate the constants mentioned in the formula, page 323 so did G. 0. Schmidt (Zeit. phys. Chem.f 7, 433, 1891) and A. Horst-mann (Liebig s Ann. Ergbd.t 8, 112, 1872). 

An event of this kind, produced by the composition of several events, is said to be a compound event. To throw three aces with three dice at one trial is a compound event dependent on the concurrence of three simple events. Errors of observation are compound events produced by the concurrence of several independent errors. 

For details of the calculations, the reader must consult the original memoirs. Most of the calculations are based upon the analysis in Laplace s old but standard Theorie (l.c.). An excellent r sumk of this latter work will be found in the Encyclopedia Metro-politana. The more fruitful applications of the theory of probability to natural processes have been in connection with the kinetic theory of gases and the law relating to errors of observation. 

The irregular deviations of the measurements from, say, the arithmetical mean of all are called accidental errors. In the following discussion we shall call them errors of observation unless otherwise stated. These deviations become more pronounced the nearer the approach to the limits of accurate measurement. Or, as Lamb1 puts it, the more refined the methods employed the more vague and elusive does the supposed magnitude become the judgment flickers and wavers, until at last in a sort of despair some result is put down, not in the belief that it is exact, but with the feeling that it is the best we can make of the matter . It is the object of the remainder of this chapter to find what is the best we can make of a set of discordant measurements. 

The greater the radius of the circle (Fig. 165), the cruder and less accurate the measurements and, vice versd, the less the measurements are affected by errors of observation, the smaller will be the radius of the circle. In other words, the less the skill of the shooter, the larger will be the target required to record his attempts to hit the centre. 

We must now submit our empirical law to the test of experiment. Bessel has compared the errors of observation in 470 astronomical measurements made by Bradley with those which should occur according to the law of errors. The results of this comparison are shown in the following table —1... 

The symbols y and P are convenient abbreviations for this cumbrous phrase. For a large number of observations affected with accidental errors, the probability of an error of observation having a magnitude a , is,... 

What is the best representative value of a series of measurements affected with errors of observations ... 

---