Error (or uncertainty) is defined as the difference
between a measured or estimated value for a quantity and its true
value, and is inherent in all measurements. Knowledge of the type
and degree of error likely to be present is essential if data are
to be used wisely, whether the data being considered were measured
personally or merely read from manufacturer's data sheets for a
material or component. In medical research, biology, and the social
sciences, the plan for the data acquisition and analysis is the
heart of the experiment. Engineers also need to be careful; although
some engineering measurements have been made with fantastic accuracy
(e.g., the speed of light is 299,792,458 1 m/sec.), for most an
error of less than 1 percent is considered good, and for a few one
must use advanced experimental design and analysis techniques to
get any useful data at all. Making measurements and analyzing them
is a key part of the engineering process, from the initial characterization
of materials and components needed for a design, to testing of prototypes,
to quality control during manufacture, to operation and maintenance
of the final product.
Reported experimental results should always include
a realistic estimate of their error, either explicitly, as plus/minus
an error value, or implicitly, using the appropriate number of significant
figures. Furthermore, you need to include the reasoning and calculations
that went into your error estimate, if it is to be plausible to
others. An explicit estimate of the error may be given either as
a measurement plus/minus an absolute error, in the units of the
measurement; or as a fractional or relative error, expressed as
plus/minus a fraction or percentage of the measurement. The advantage
of the fractional error format is that it gives an idea of the relative
importance of the error. A 10-gram error is a tiny 0.0125% of the
weight of an 80-kg man, but is 33.3% of the weight of a 30-g mouse.
Errors may be divided roughly into two categories: Systematic error
in a measurement is a consistent and repeatable bias or offset from
the true value. This is typically the result of miscalibration of
the test equipment, or problems with the experimental procedure.
On the other hand, variations between successive measurements made
under apparently identical experimental conditions are called random
errors. Random variations can occur in either the physical quantity
being measured, the measurement process, or both. We will outline
statistical procedures for handling this type of error.
In reporting experimental results, a distinction should be made
between "accuracy" and "precision." Accuracy
is a measure of how close the measured value is to the true value.
A highly accurate measurement has a very small error associated
with it. Note that in experimental work the true value is often
not known, and thus what is reported must be an estimated accuracy
or error. Precision is a measure of the repeatability and resolution
of a measurement -- the smallest change in the measured quantity
that can be detected reliably. Highly precise experimental equipment
can consistently measure very small differences in a physical quantity.
Note that a highly precise measurement may, nevertheless, be quite
inaccurate. High precision in a measurement is a necessary but insufficient
condition for high accuracy.
EXAMPLE: a typical 3 1/2 digit voltmeter can measure voltages between
-199.9 volts and 199.9 volts, when set to its 200 volt range. Suppose
that in measuring a voltage an engineer obtains an average reading
of 47.1 volts, and that the right-most digit flickers up and down
between 0, 1, and 2 in an apparently random way. Thus, the precision
is approximately volts in absolute terms. The fractional or percentage
precision is volts/ 47.1 volts, or . The precision can be estimated
from the measurements obtained; the accuracy must be found by comparison
to an accepted voltage standard. Looking at the manufacturer's specification
sheet, which includes the results of such a comparison, the engineer
finds that the rated accuracy of the voltmeter is . Thus, the accuracy
is an order of magnitude worse than the precision. (The accuracy
of a voltmeter is only as good as its resistors and other components.
Resistor values drift as the resistors get older, and they vary
with temperature.) In terms of absolute error, the measurement should
be reported as 47.1 (0.2)(47.1) volts, or 47.1 1 volts (not 0.1
volts, the estimated precision).