Page 23 - Mechanical Engineers' Handbook (Volume 2)

12 Instrument Statics
2 4 2
2
2
W
W
W
W
D
L
R
D R L
4 2
2
2
2.5
0.001
0.0001
0.10 0.0959 250
4.00 10 4 1.09 10 6 1.00 10 4
5.01 10 4
The resulting resistivity and its precision W are

W (5.01)10 4 6.74 10 8

(3.01 0.07) 10 6 cm
3.5 Uncertainty Interval
When several measurements of a variable have been obtained to form a data set (multisample
data), the best estimates of the most representative value (mean) and dispersion (standard
deviation) are obtained from the formulas in Section 3.2. When a single measurement exists
(or when the data are taken so that they are equivalent to a single measurement), the standard
deviation cannot be determined and the data are said to be ‘‘single-sample’’ data. Under
these conditions the only estimate of the true value is the single measurement, and the
uncertainty interval must be estimated by the observer. It is recommended that the precision
12
index be estimated as the maximum reasonable error. This corresponds approximately to the
99% confidence level associated with multisample data.
Uncertainty Interval Considering Random Error
Once the unbiased estimates of mean and variance are determined from the data sample, the
uncertainty interval for is
ˆ
ˆ W ˆ k( , )ˆ (22)
ˆ
where ˆ represents the most representative value of from the measured data and W is the
uncertainty interval or precision index associated with the estimate of . The magnitude of
the precision index or uncertainty interval depends on the confidence level (or probability
chosen), the amount of data n, and the type of probability distribution governing the distri-
bution of measured items.
ˆ
ˆ
The uncertainty interval W can be replaced by k , where ˆ is the standard deviation
(measure of dispersion) of the population as estimated from the sample and k is a constant
that depends on the probability distribution function, the confidence level , and the amount
ˆ
of data n. For example, with a Gaussian distribution the 95% confidence limits are W
1.96 , where k 1.96 and is independent of n. For a t-distribution, k 2.78, 2.06, and
1.96 with a sample size of 5, 25, and , respectively, at the 95% level of confidence prob-
ability. Note that n 1 for the t-distribution. The t-distribution is the same as the
Gaussian distribution as n → .
Uncertainty Interval Considering Random Error with Resolution, Truncation, and
Significant Digits
ˆ
The uncertainty interval W in Eq. (22) assumes a set of measured values with only random
error present. Furthermore, the set of measured values is assumed to have unbounded sig-
nificant digits and to have been obtained with a measuring system having infinite resolution.
When finite resolution exists and truncation of digits occurs, the uncertainty interval may be
larger than that predicted by consideration of the random error only. The uncertainty interval
can never be less than the resolution limits or truncation limits of the measured values.