THE MATHEMATICAL TREATMENT OF DATA 



a — 3s a — 2s a 



a + 2s a + 3s 



.r = the measured value of some property 



/=the frequency with which the value 

 x is obtained in a set of measurements 



Fig. 2. Gaussian distribution of measurements, showing the average value a 

 and the frequency of measurements equaling a or deviating from a by one, two, 

 or three standard deviations. 



Fig. 3. The shaded area is proportional to the fraction of all measurements 

 lying between a — s and a + s. 



at least ±s in one of every three measurements. By calculating in an 

 entirely similar way, the following table is obtained: 



Occurs in fraction 



Note that a deviation in only one direction will occur in half the given 

 fraction of measurements. That is, for example, a deviation of 3s or 

 more will occur, by chance alone, once in 800 measurements. Similarly, 



