MEASUREMENTS AND THEIR VARIATION 



Xi 



xi = the measured value of some property 



/i = the frequency with which the value Xi 

 is obtained in the set of measurements 



Fig. 1. A schematic representation of the two kinds of sets of measurements 

 usually obtained experimentally. 



mental curves are reasonable because their relevance has been proven 

 by the mathematicians. In fact, of course, there is an inseparable mutual 

 interaction between the two approaches. 



To derive the symmetrical curve, the mathematicians assume that 

 there exists a large number of small, randomly occurring, positive and 

 negative contributions of error to the individual measurements. Thereby 

 they derive the following expression for the frequency, f u of deviations 

 from the average value : 



fi = "4= e~ (Xi 



-a) 2 /2s 2 



Here the pi and 2's result from the requirement that the individual 

 fractional frequencies add up to unity. That means that the fractions of 

 cases with all possible deviations from the average must includ 3 all cases, 

 so that the sum is unity (or 100%, if expressed in percentage terms). 



This distribution of deviation frequencies is called, variously, the 

 Normal Error Curve, the Normal Distribution, or the Gaussian Distribu- 

 tion, after the great 19th-century scientist Gauss. The bell-shaped distri- 

 bution is plotted in Fig. 2, which also indicates the frequencies for 

 deviations equal to zero, ±s, ±2s, and ±3s. 



We can now begin to answer some of the many questions we have 

 been accumulating. First, what fraction of the deviations lies within 

 ±s of the average value a? This amounts to finding the fraction of cases 

 in the shaded area of Fig. 3. (For those who understand the calculus, 

 we can say that the fraction is found mathematically by integrating the 

 Gaussian Distribution from a — s to a + s.) The result is that about 

 % of the deviations fall in this range; conversely, of course, % fall out- 

 side this range. This means that if the average value of a measurement is 

 a, then by chance alone one will have a measurement deviating from a by 



