xlii DEFINITIONS AND KX TLA NATIONS. 



Example. 426018. has six places of figures and of significant figures 

 3479 100. has seven places of figures, luit whether its number of places of signifi- 

 cant lijrures is nmre than live is indeterminate. 0.027680 has seven phu-cs of 

 figures with live places of significant figures. 2.7680-10-- has five places of 

 liiaires and live .significant figures. 



Places of Decimals. These, of course, are the places following the decimal 

 point as the number happens to be written. E.g. 0.027680 has six decimal 

 places. 2.768o-io~ 2 has four decimal places, although its magnitude is the 

 same as that of the preceding quantity. 



From the foregoing six examples it will be seen that the number of deci- 

 mal places and the number of places of significant figures have no necessary 

 mutual relation. 



Accuracy ; Reliability. To know the exact accuracy of a given quantity 

 it would be necessary, of course, to know the true value of that quantity. In 

 the case of a few mathematical constants (such as TT = 3. 14 159 265 ) the true 

 value is known, at least far beyond ordinary requirements. But in case of all 

 measurements the same is obviously not true, for if the true value were known, 

 measurements would be unnecessary. Some approximate expression of the accu- 

 racy of a measured result can, however, usually be obtained, and is necessary. 

 Sometimes this is afforded by a knowledge of the instrument used and the degree 

 of care employed. Thus, suppose the distance of about 3 feet 6* f inches 

 = 3-539 f eet between two marks to have been carefully once measured with a 

 good foot-rule, it could safely be assumed from our previous knowledge of such 

 work that if a series of these measurements were undertaken, the results would 

 vary from their mean by less than ^ of an inch ; also, that the error in the 

 rule itself, and from other unavoidable or unavoided sources, would not on the 

 average materially increase the error of measurement. Then -., inch 

 0.0026 ft. would be taken as the estimated measure of accuracy of the result. 

 Instead of expressing the accuracy in units, e.g. inches or feet, it is usually 

 more convenient or intelligible to express it as a fraction ; or, better still, in per- 

 centage. Thus, the foregoing will be 



g^ = o.oo 074, i.e. -I*, or ioof ao 26 W 0.074 per cent. 

 3-5 loo ooo \ 3-5 / 



If in this example the reliability had not been estimated at ^ inch, but if 

 several measurements had been made, and these had been found upon inspec- 

 tion to deviate from .their mean by about j. s of an inch, then, other things being 

 the same, the measure of accuracy of any single measurement taken without 

 knowledge of the others would be regarded as -fa inch, or 0.074 per cent. 



Mean ; Average ; Deviation Measure. When the result is the arithmetical 

 mean or average of several separate measurements of the same quantity, its 

 reliability or accuracy is taken. to be in proportion to the square root of the 

 number of such observations. Thus, in the last preceding example, if there had 

 been n 9 single observations made, the measure of accuracy of the mean of 

 these would be 



-f- Vn = -^ = = o.oio inch, or 0.025 P er cent - 



The differences of the single observations taken under like conditions from 

 their mean will be called deviations, and their numerical average (i.e. their sum 



