﻿Laws 
  of 
  Fluctuations 
  735 
  

  

  cients 
  of 
  the 
  binomium, 
  all 
  further 
  details 
  seem 
  

   to 
  depend 
  upon 
  them. 
  In 
  respect 
  to 
  the 
  average 
  

   this 
  is 
  no 
  doubt 
  the 
  case 
  ; 
  it 
  is 
  an 
  empirical 
  value 
  

   without 
  need 
  of 
  any 
  further 
  discussion. 
  The 
  

   more 
  the 
  number 
  of 
  the 
  observations 
  increases, 
  

   the 
  more 
  assured 
  and 
  the 
  more 
  correct 
  is 
  this 
  

   mean 
  value, 
  but 
  generally 
  it 
  is 
  the 
  same 
  for 
  

   smaller 
  and 
  for 
  larger 
  groups 
  of 
  observations. 
  

  

  This 
  however, 
  is 
  not 
  the 
  case 
  with 
  the 
  ex- 
  

   tremes. 
  It 
  is 
  quite 
  evident 
  that 
  small 
  groups 
  

   have 
  a 
  chance 
  of 
  containing 
  neither 
  of 
  them. 
  

   The 
  more 
  the 
  number 
  of 
  the 
  observations 
  in- 
  

   creases, 
  the 
  larger 
  is 
  the 
  chance 
  of 
  extremes. 
  As 
  

   a 
  rule, 
  and 
  excluding 
  exceptional 
  cases, 
  the 
  ex- 
  

   treme 
  deviations 
  will 
  increase 
  in 
  proportion 
  to 
  

   the 
  number 
  of 
  cases 
  examined. 
  In 
  a 
  hundred 
  

   thousand 
  beans 
  the 
  smallest 
  one 
  and 
  the 
  largest 
  

   one 
  may 
  be 
  expected 
  to 
  differ 
  more 
  widely 
  from 
  

   one 
  another 
  than 
  in 
  a 
  few 
  hundred 
  beans 
  of 
  the 
  

   same 
  sample. 
  Hence 
  the 
  conclusion 
  that 
  ex- 
  

   tremes 
  are 
  not 
  a 
  safe 
  criterion 
  for 
  the 
  discus- 
  

   sion 
  of 
  the 
  curves, 
  and 
  not 
  at 
  all 
  adequate 
  for 
  

   calculations, 
  which 
  must 
  be 
  based 
  upon 
  more 
  

   definite 
  values. 
  

  

  A 
  real 
  standard 
  is 
  afforded 
  by 
  the 
  steepness 
  

   of 
  the 
  slope. 
  This 
  may 
  be 
  unequal 
  on 
  the 
  two 
  

   sides 
  of 
  one 
  curve, 
  and 
  likewise 
  it 
  may 
  differ 
  for 
  

   different 
  cases. 
  This 
  steepness 
  is 
  usually 
  meas- 
  

   ured 
  by 
  means 
  of 
  a 
  point 
  on 
  the 
  half 
  curve 
  and 
  

  

  