ON CORRECTIONS FOR THE MOMENT-COEFFICIENTS OF 
LIMITED RANGE FREQUENCY DISTRIBUTIONS WHEN 
THERE ARE FINITE OR INFINITE ORDINATES AND 
ANY SLOPES AT THE TERMINALS OF THE RANGE. 
By ELEANOR PAIRMAN and KARL PEARSON, F.R.S. 
Part I. Noa-Asijinptotic Gtirves. 
(1) We liave in recent pnictice found the importance of full corrections for the 
monient-coeflficients in the case of singly and doubly curtailed blocks of frequency 
such as are indicated in the accompanying figure. It has not been adequately 
recognised that even the mean of such distributions is not correctly obtained by 
grouping at the midpoints of the subranges and merely finding the mean of 
these concentrated groups. Still less is this a correct process in the case of the 
higher momcnt-coefificients. The practical statisticians, aware possibly of the exist- 
ence of " Sheppard's corrections," have been warned that they are only exact for the 
case of high contact, and regarding this have in their doubt neglected all corrections 
whatever. Now Sheppard's corrections are still valid when there is no high con- 
tact, and they should therefore always be used, but they form only part of the full 
correction* and may indeed merely amount to some 50 of its value, although 
75 is a more usual average proportion, if the frequency block does not end in 
finite ordinates. We propose in the first part of this paper to deal with frequency 
* lu certain cases although part of the full correction they are in the wrong sense, and therefore if 
used alone would be worse than the raw moments. 
