Page 21 - Mechanical Engineers' Handbook (Volume 2)

10 Instrument Statics
ˆ S
n
2
2
n 1
2
where S is the sample variance
(x x) 2
S i (13)
2
n
and
ˆ 2
ˆ (14)
2
x
n
where ˆ 2 x is the variance of the mean.
Sampling without Replacement (Usual Case)
ˆ S (15)
ˆ x
N 1
n
2
2
n 1 N (16)
where N is the population size and n the sample size, and
ˆ
N n
2
ˆ
2
x
n N 1 (17)
Note that sampling without replacement from an extremely large population is equivalent to
random sampling.
3.3 Types of Errors
There are at least three types of errors that one must consider in making measurements.
They are systematic (or fixed) errors, illegitimate errors (or mistakes), and random errors:
Systematic errors are of consistent form. They result from conditions or procedures that
are correctable. This type of error may generally be eliminated by calibration.
Illegitimate errors are mistakes and should not exist. They may be eliminated by using
care in the experiment, proper measurement procedures, and repetition of the mea-
surement.
Random errors are accidental errors that occur in all measurements. They are charac-
terized by their inconsistent nature, and their origin cannot be determined in the
measurement process. These errors are estimated by statistical analysis.
If the illegitimate errors can be eliminated by care and proper measurement procedures and
the systematic errors can be eliminated by calibrating the measurement system, then the
random errors remain to be determined by statistical analysis to yield the precision of the
measurement.

3.4 Propagation of Error or Uncertainty
In many cases the desired quantity cannot be measured directly but must be calculated from
the most representative value (e.g., the mean) of two or more measured quantities. It is
desirable to know the uncertainty or precision of such calculated quantities.
Precision is specified by quantities called precision indexes (denoted by W ) that are
x
calculated from the random errors of a set of measurements. A W should be specified for
x