MEASUREMENTS AND THEIR VARIATION 7 



a deviation of —2s or more (in the negative direction) will occur once in 

 40 measurements. 



This table will provide the basis for much of the analysis we will 

 present. With its presentation, we have answered one of the two ques- 

 tions we set out to answer about the standard deviation: what is its 

 significance? 



The other question concerns the utility of the averaging procedure 

 employed in obtaining the arithmetic average. We already know that 

 the clustering of measurements around this value is greatest, but still 

 more can be said. If we ask the mathematicians to compute the cluster- 

 ing of the averages themselves, we are asking what would be the distri- 

 bution of averages if we made a set of n measurements many, many 

 times. Another way of asking the same question is to ask for the standard 

 deviation of the average itself. The mathematician's reply to the latter 

 question is 



V 



or 



n 



V 



V = — 

 n 



That is, the average variation, or variance, of the average measurement 

 is only one-nth of the variance of the individual measurements. This is 

 the result we want. It says that the reason for taking the average is that 

 the average is roughly -\Jn times more likely to be the true value than is 

 any individual measurement. s a is called the standard error by statis- 

 ticians. 



X or a 



Fig. 4. A schematic representation of a set of individual measurements, x, 

 and of a set of averages, a, obtained by making many repeated sets of measure- 

 ments. The relative narrowness of the curve for a shows that the average ob- 

 tained in any single set of measurements is very likely to be close to the "true" 

 value. 



Still another way of indicating the same information is shown in the 

 schematic drawing in Fig. 4, which shows the distribution of individual 

 measurements and also the distribution of averages. The latter is so 

 narrow that it is very unlikely that we will ever get a much different 

 result, even by chance. 



