02 Records of the Indian Museum. [Vol. XXIII, 



is not at all suitable for this purpose. The inclusion in the sample 

 of a single abnormal " dwarf" or ''giant" will completely upset 

 the value of the range. A measure so radically affected by stray 

 items at the extremes is practically useless for scientific purposes } 



In current statistical practice it is usual to measure variability 

 by the Standard Deviation. The deviation of each measurement 

 from the Mean (or Average) is squared. The sum of all such 

 squares divided by their total number gives the second moment f*g, 

 which is thus the average squared-deviation of all the measurements. 

 The square-root of f*a finally, gives the Standard Deviation. It 

 is the average root-square deviation of all the measurements, 

 and is a precise mathematical measure of the variability of the 

 sample. One great advantage in using the Standard Deviation is 

 this that it uniquely defines the corresponding Gaussian curve, so 

 that the Gaussian can be found as soon as the Standard Deviation 

 is determined. Standard Deviation (or S.D.) is usually represent- 

 ed by < r . 



Probable Errors. — The Gaussian distribution is also known 

 as the " normal curve of errors," since it is assumed that this 

 curve gives the distribution of ''errors" made in physical 

 measurements. 2 The greater the diversity in any set of measure- 

 ments the greater will be the Standard Deviation of the set. 

 Accuracy or reliability depends on the uniformity of the set of 

 measurements, ithat is, on the smallness of the Standard Deviation 

 The " probable error," which measures the accuracy or reliability 

 of any set of measurements, is hence suitably defined by a parti- 

 cular sub multiple of the Standard Deviation. 



If a is adopted as the unit of measurements (that is, all 

 measurements in terms of ordinary units are divided by v) } then the 

 curve of errors becomes the standard curve of probability. The 

 mathematical theory of probability then enables us to find the 

 probability of any given deviation from the Mean occurring in the 

 sample. 



For example, a deviation half of the Standard Deviation will 

 occur no less than 62 times in 100 samples. A deviation as great 

 as the Standard Deviation will occur in 32% instances, while a 

 deviation four times as great will not happen more than once in 

 17, 000 instances. The Probable Error is defined to be such a 

 deviation as will be exceeded by half the total deviations, or in 

 other words the ehanees are even that any deviation will be great- 

 er than or less than the Probable Error. 



We must now come back to Anthropology. It is well known 

 that almost all anthropometric measurements have an approxim- 

 ately Gaussian distribution. This was originally pointed out by 

 Quateletj and since then has been confirmed b\ many different 



I • ' 1 .1 simple non-technical account ol the different measures o\ dispersion, 

 King: " Elements of Statistical Theory " (MacMHlan, 1919)1 p. 141. 

 * This assumption itself isnol always strictly true. See Pearson's memoir on, 

 rors of Judgement, etc." Phil. Trans, Roy, Soc. n>s\ (1902). 



