12 SUMMARIZATION OF DATA Ch. 2 



2.1 THE ARITHMETIC MEAN AND THE STANDARD 



DEVIATION 



When it is necessary to analyze a population of statistical meas- 

 urements it often is desirable to calculate a single number which 

 will be typical of the general level of magnitude of the measurements 

 in the population. Logically, the first question is: What features 

 should averages have in order to be typical of the data in some useful 

 sense? Therefore, the following properties of averages are suggested 

 as being either required of averages, or desirable: 



(2.11) An average should be close, on the scale of measurement, 

 to the point (or interval) of greatest concentration of the measure- 

 ments in the population. 



(2.12) It should be as centrally located among the numbers as is 

 compatible with property (2.11). 



(2.13) An average should be simple to compute if that is achiev- 

 able under the circumstances. 



(2.14) It should be tractable to mathematical operations so that 

 useful theoretical information can be derived by means of mathe- 

 matical methods. 



(2.15) The average should be such that measures of the scatter 

 of the data about the average can be obtained and also have prop- 

 erties (2.13) and (2.14). 



A simple but crude average which sometimes is quite useful is the 

 midrange, MR. It is defined as that number (not necessarily one of 

 those being studied) which is halfway between the extreme numbers 

 in the set being summarized. For example, the extreme ACE scores 

 in Table 2.01 are 23 and 183. The difference is 160; hence 



23 4- 183 

 MR = 23 + 160/2 = 103, also = . 



because this is the number which is halfway between 23 and 183. 

 Among the desirable properties of averages listed above, the midrange 

 is centrally located (in the sense that it is midway between the 

 extremes), and it often is in the region of the greatest concentration 

 of the data. It also is easily calculated, but it does not possess the 

 other properties listed. In addition, the midrange does not appear 

 to be a very reliable average because its size depends on only two of 

 the numbers. 



