Page 135 - Mechanical Engineers' Handbook (Volume 2)

124 Measurements

, where C
estimating the error in a measurement. The first is the external estimate, or C E
e/q. This estimate is based on knowledge of the experiment and measuring equipment and

to some extent on the internal estimate C . I
The internal estimate is based on an analysis of the data using statistical concepts.
3.1 Internal Estimates
If a measurement is repeated many times, the repeat values will not, in general, be the same.
Engineers, it may be noted, do not usually have the luxury of repeating measurements many
times. Nevertheless the standardized means for treating results of repeated measurements are
useful, even in the error analysis for a single measurement. 8
If some quantity is measured many times and it is assumed that the errors occur in a
completely random manner, that small errors are more likely to occur than large errors, and
that errors are just as likely to be positive as negative, the distribution of errors can be
represented by the curve
Ye (X U)
F(X) o (14)
2 2
where F(X) number of measurements for a given value of (X U)
Y maximum height of curve or number of measurements for which X U
o
U value of X at point where maximum height of curve occurs determines
lateral spread of the curve
This curve is the normal, or Gaussian, frequency distribution. The area under the curve
between X and X represents the number of data points which fall between these limits and
the total area under the curve denotes the total number of measurements made. If the normal
distribution is defined so that the area between X and X X is the probability that a data
point will fall between those limits, the total area under the curve will be unity and
exp (X U)/2 2
2
F(X) (15)
2
and
P exp (X U)/2 2
2
x
2 dx (16)
Now if U is defined as the average of all the measurements and s as the standard deviation,
(X U) 2 1/2

N (17)
where N is the total number of measurements. Actually this definition is used as the best
estimate for a universe standard deviation, that is, for a very large number of measurements.
For smaller subsets of measurements the best estimate of is given by
(X U) 2 1/2

n 1 (18)
where n is the number of measurements in the subset. Obviously the difference between the
two values of becomes negligible as n becomes very large (or as n → N).
The probability curve based on these definitions is shown in Fig. 5.