57 



Mean length =- 10-5, 11-5, 12-5, 13-5, etc., eras. 



Nos. caught = 2, 5, 13, 46, etc. 

 we obtam a series called a '"' frequency distribution." There 

 are manv ways of forming such distributions from the same 

 data. The "mean-lengths" 10-5, etc., represent the middle 

 points of the groups of measurements, that is 10 to 11 .cms., 



11 to 12 cms., etc., but these groups might have been 10 to 12, 



12 to 14, 9 to 1] , 11 to 13, etc., or they might also have been 



4 to 4| inches, 4| to 5 inches, etc., or even 4 to 5, 



5 to 6 inches, etc. If we were to make such alternative series 

 of measurements from the same sample of fish and then plot 

 curves from the various distributions we should not get 

 graphs of the same form. Nor should we get quite the same 

 averages and other statistical results. It is convenient to 

 measure the fish in centimetre groups, but such a method has 

 no superiority over any other arrangement except its con- 

 venience. 



If the series of measurements is a very big one — say 

 several thousands of fish — it will not matter nuicli what way we 

 express the data. But every now and then, small samples, 

 50 or 100 fish, say, must be studied. Therefore we require 

 some way of avoiding the errors which arise because of the 

 alternative methods of grouping the measurements. 



This means that the crude distributions must be 

 " smoothed " in some manner. In general, a series such as 

 the above one is irregular and these irregularities aftect 

 whatever form of average we adopt. If we calculate the 

 latter ani then re-measure the fish and arrange them in a 

 new way we may get different irregularities which affect our 

 averages (or other statistical conclusions) in dift'erent ways. 

 Which results are we to accept ? In social and political 

 controversies we do all these things and then accept the 

 results that are the most welcome ones ! But this kind of 

 statistics is that which " can be made to prove anything," and 

 we must avoid it " like the plague." 



