CHAPTER 



1 



The Mathematical 

 Treatment of Data 



INTRODUCTION 



With a given set of data, the problem is always to extract the maximum 

 amount of information. This in no sense relieves the investigator of the 

 problem of designing better and more extensive and inclusive experi- 

 ments. But the existing data surely deserve to be analyzed as fully as 

 warranted. The word warranted is the problem word, and it is to this 

 problem that the mathematical analysis of data is directed. We would 

 like to know the information that can be gotten from some data includ- 

 ing the most likely values of the various quantities measured and, 

 equally importantly, the probable uncertainty of the values thus ob- 

 tained. 



A measurement of any property of a system is, by itself, almost with- 

 out significance, because we do not know the uncertainty in the measure- 

 ment. A statement as extreme as this needs some justification. We feel 

 intuitively that the purpose of a measurement is to know something 

 accurate about a property. If we are greatly uncertain about the result, 

 then we have learned little. For we then have to say that the measure- 

 ment might be some particular number but that it also might be, say, 

 1000 times greater or less than that number. Thus, not even knowing 

 the uncertainty in the measurement is equivalent to changing the number 

 1000 to any number you may wish to insert. The whole purpose of 

 statistical analysis is to show us how to maximize the relevance of the 

 measurements we make. 



MEASUREMENTS AND THEIR VARIATION 



As a simple first example, let us take a series of n individual measure- 

 ments of something: x 1 ,x 2 , . . . , x n . Why do we take more than one 

 measurement? What have we gained by taking, as we usually do, the 

 arithmetic average of these n measurements? 



If we consider that the sources of uncertainties in measurements act 

 randomly, then in a set of measurements we are as likely to get an 

 individual result higher than the "true" value as to get one equally much 

 lower than the "true" value. Indeed, unless measurements somehow 



