168 STANDARDIZATION OF EXPRESSIONS FOR ACCURACY, HALLERT
Irregular or accidental errors affect the results of the measurements with small
amounts according to the law of chance only. Evidently, such errors cannot individually
be determined in advance. Only concerning the relation between the size and the fre-
quency of the errors some assumptions can be made. Usually the errors are assumed to
be normally distributed which among other things means that the frequency is largest
for the error zero and then decreases symmetrically with increasing absolute values of
the errors according to the normal frequency function. The errors in this group are pri-
marily caused by the settings (coincidence) and the readings in connection with the meas-
urements but also the great number of small errors which remain after correction of
large and regular (constant) errors are assumed to belong to this group. For the as-
sumption that the sum of a great number of such residual errors is normally distributed
the central limit theorem is of basic importance (ref. 3, p. 231-232).
4.2 Repeated direct measurements of unknown quantities.
From n repeated measurements the arithmetic mean (the average) M is computed
and also the corrections v to the individual measurements in order to make them coin-
cide with M.
The accuracy of the individual measurements is then defined as the standard devia-
tion of one measurement according to the expression (ref. 1, p. 277 and 2, p. 36).
[vv]
8 = 4)
) n— 1 (
n is the number of measurements. n—1 is the number of redundant measurements or the
degrees of freedom.
The standard deviation of the arithmetic mean M is found from (ref. 2, p. 36).
1/ [vv]
y= V n(n —1) (3)
All measurements are assumed to have equal accuracy (weight).
4.3 Indirect measurements. (ref. 4, p. 235-242).
Unknowns (parameters) are to be determined from functions of measurements. In the
following example two unknowns x and » are assumed which are connected with the
measured quantity | by the linear or at least approximately linear function
[= ax + by (6)
where a and b are errorless coefficients. For the determination of x and y at least two
determinations (measurements) of / are necessary. If n determinations are available the
expression (6) is written
vy ax + by — 1
tn + E, (7)
v, ^ Ay t b,y — I,
where v denotes corrections to the measured / in order to make them coincide with unique
values of x and y via the function (6).
Under the assumption that the sum of the squares uuu u$... ui-lvw]
shall be a minimum the unknowns x and y are to be found from normal equations
[aa]x + [ably — [al] = 0
[ab]x + [bb]y — [bl] = 0
For the determination of the expression [vv] the equation system (8) is usually com-
pleted with the additional equation
—[al]e — [bl]y + [U] = [vv]
(8)
(9)
