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UCRL-6252
The normal equations to be solved for a and b were obtained by minimizing with respect to both a and b the weighted sum of the squares of the
deviations from the proposed line.
The weight of each point is chosen to be
the reciprocal of the variance of that particular deviation, taking into account
errors in both xand y.
Ifthe weights are assigned in this fashion,then re-
gardless of whether we choose to minimize deviations in the y direction, deviations in the x direction, or perpendicular distances from the pointto the
line, the expression for the weighted sum of squares becomes S = » (y,-a-bx,)°
Nw? tb” wu’).
In this expression u, is the standard deviation of xy and v. is
the standard deviation of y,;-
Because of the way in which the parameter b
appears in the sum S. we could not solve the normal equations directly fora
and b.
A method of successive approximations was used, with an IBM 650
computer.
The values ofa and b determined for each of the lines shown in.
Figs. 1 and 2 are given in Tables LIlI and IV. Also given in Tables III and IV are some statistics which can serve as
rough measures of how well the calculated lines y = a + bx fit the experimental data.
Listed are the number of experimental points for each line, the
|
mean percentage deviation of the calculated and measured values of y, and
the value for the minimized sum of squares, S = » (y; -a- bx,)° Av? + pe uw).
The mean percentage deviation of the calculated and measured y values was
. computed for each line by the expression
100% x
]
n-2
7
Ymeas” ‘calc
¥
where nis the number of experimental points.
meas
é
1/2
,
This "average! deviation can
be compared to the standard deviations given for the experimental data.
The
values for the minimized sum of squares, S, appear to be too low, despite the
fact that no data points were excluded.
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This can be attributed, in part at least,