Page 22 - Mechanical Engineers' Handbook (Volume 2)

3 Statistics in the Measurement Process 11

every measured variable. The confidence limits or probability for obtaining the range W x
is generally specified directly or is implied by the particular type of precision index being
used.
The precision index of a calculated quantity depends on the precision indexes of the
9
measured quantities required for the calculations. If the measured quantities are determined
independently and if their distribution about a measure of central tendency is approximately
symmetrical, the following ‘‘propagation-of-error’’ equation is valid :
11
W 2 2
R
2
R
x i W x t (18)
In this equation, R represents the calculated quantity and x , x , ..., x represent the
2
n
1
measured independent variables so that mathematically we have R ƒ(x , x ,..., x ). The
n
2
1
precision index is a measure of dispersion about the central tendency and is denoted by W
in Eq. (18). The standard deviation is often used for W; however, any precision index will
do as long as the same type of precision index is used in each term of the equation.
A simplified form of this propagation-of-error equation results if the function R has the
form
a
R kx x x x m (19)
c
b
1
2
3
n
where the exponents a, b,..., m may be positive or negative, integer or noninteger. The
simplified result for the precision W in R is
R
2 a 2 m 2
2
W
W
W
W R
x 2
2
2
x n
x 1
2
R x 1 b x 2 x n (20)
The propagation-of-error equation is also used in planning experiments. If a certain precision
is desired on the calculated result R, the precision of the measured variables can be deter-
mined from this equation. Then, the cost of a proposed measurement system can be deter-
mined as it is directly related to precision.
Example 3 Propagation of Uncertainty. Determine the resistivity and its uncertainty for
a conducting wire of circular cross section from the measurements of resistance, length, and
diameter. Given
2
L 4L DR
R or (21)
A D 2 4L
R 0.0959 0.0001 L 250 2.5 cm D 0.100 0.001 cm
where R wire resistance,
L wire length, cm
2
A cross-sectional area, D /4, cm 2
wire resistivity, cm
Solution: Thus the resistivity is
2
( )(0.100) (0.0959)
3.01 10 6 cm
4(250)
The propagation of variance (or precision index) equation for reduces to the simplified
form, that is,