Page 244 - Civil Engineering Formulas

178 CHAPTER SEVEN

1 rad 57 17 44.8 or about 57.30
1 1 4
1 grad (grade) 400 circle 100 quadrant 100 centesimal min 10 cen-
tesimals (French)
1
1 mil 6400 circle 0.05625
1
1 military pace (milpace) 2 2 ft (0.762 m)
THEORY OF ERRORS

When a number of surveying measurements of the same quantity have been
made, they must be analyzed on the basis of probability and the theory of
errors. After all systematic (cumulative) errors and mistakes have been elimi-
nated, random (compensating) errors are investigated to determine the most
probable value (mean) and other critical values. Formulas determined from
statistical theory and the normal, or Gaussian, bell-shaped probability distrib-
ution curve, for the most common of these values follow.
Standard deviation of a series of observations is

d 2
s (7.1)
Bn 1
where d residual (difference from mean) of single observation and n num-
ber of observations.
The probable error of a single observation is
(7.2)
PE s 0.6745 s
(The probability that an error within this range will occur is 0.50.)
The probability that an error will lie between two values is given by the
ratio of the area of the probability curve included between the values to the total
area. Inasmuch as the area under the entire probability curve is unity, there is a
100 percent probability that all measurements will lie within the range of the
curve.
The area of the curve between is 0.683; that is, there is a 68.3 percent
s
probability of an error between in a single measurement. This error range
s
is also called the one-sigma or 68.3 percent confidence level. The area of the
curve between 2 is 0.955. Thus, there is a 95.5 percent probability of an
s
error between 2 and 2 that represents the 95.5 percent error (two-
s
s
sigma or 95.5 percent confidence level). Similarly, 3 is referred to as the
s
99.7 percent error (three-sigma or 99.7 percent confidence level). For practical
purposes, a maximum tolerable level often is assumed to be the 99.9 percent
error. Table 7.1 indicates the probability of occurrence of larger errors in a sin-
gle measurement.
The probable error of the combined effects of accidental errors from differ-
ent causes is
2
E sum 2 E 1 E 2 E 3 (7.3)
2
2