^f\^-'''^'Adl  (202)

Vtt  Jo

-^  r    c'-"'^'(-2/i2A)r/A  (203)


:— .  J    r---^  (204)


— -  (205)


158  PRACTICAL  LEAST  SQUARES

The  ordinate  of  the  average  error  passes  through  the  center
of  gravity  of  the  area  between  the  curve  of  error  and  the  horizontal
axis  on  either  side  of  the  vertical  axis.  For,  if  Ao  represent  the
abscissa  of  the  center  of  gravity,  by  considering  vertical  strips  of
width  dA  and  length

Vtt
and  taking  moments  about  the  origin,  we  have.
~  re-'^^'dA^-^  Ce-"'^'AdA
ttJo  'Vtt.  Jo


Ao4=  I     e-'"^'dA^-^\     e-'"^'AdA  (206)

Vti


But  since  the  total  probability  area  is  equal  to  unity,  the  area
on  one  side  of  the  vertical  axis  is  1/2,  that  is


(207)


Hence,


2h    C"^

Ao  =  -iL(     e-'-'^'Af/A      =77,  from  (202).  (208)


138.  Comparison  of  the  Indices  of  Precision.  From  (186),
(196),  and  (205),  we  obtain  directly,

eV2  '"  tjVtt

-  =  1 . 4 142  6  =  2 .  09()6r  =  1 .  7726r?  (2 1 0)

e=1.4826r=1.2533r7  ]

r  =  0.07456  =  0.84587?  \  (211)

r?  =  0.7979e=1.1829r  J

Thus  it  is  seen  that  the  mean  square  error,  the  probable  error,
and  the  average  error  are  related  b}-  constant  factoi-.'^.  Therefore,
they  may  be  used  interchangeably  in  various  formulas  and  math-
ematical investigations  b,y  simply  providing  for  the  numerical
factors.
In  Fig.  40,  these  quantities  are  shown  in  their  correct  relative
positions    and    magnitudes.     The    abscissa'   rcpi'o.sent    the   erroi-s


PRECISION  OF  OBSERVATIONS  AND  RESULTS


159


and  the  ordinates  their  probabihties.     It  will  be  recalled  that  the

intercept  on  the  vertical  axis  is  -— =.
Vtt

The  quantity,  h,  is  directly  proportional  to  the  precision.
However,  it  is  inconvenient  in  practice  and  is  not  generally  used.
