Uncertainty Calculations

The purpose of this section is to outline the fundamental methods
of measurement uncertainty analysis for use as an objective
estimator of data quality. These methods apply to all test, evaluation,
and process data secured by a measurement instrument
or system. Some examples are given to clarify the application
of the principles presented.
UNCERTAINTY AND ERROR
Measurements are made so that the resulting data may be
used for decision-making. In fact, the most fundamental definition
of “good” data is “data that are applicable, or useful,
for drawing conclusions or making decisions.” Because of
this, no test or evaluation data should be presented or used
without including its measurement uncertainty. It is a properly
evaluated measurement uncertainty that provides the information
needed to properly assess the usefulness of data. For
data to be useful, it is necessary that their measurement errors
be small in comparison to the changes or effect under evaluation.
The actual measurement error is unknown and unknowable.
Measurement uncertainty estimates its limits with some
confidence.
Therefore,
measurement uncertainty
may be defined as
the limits to which a specific error or system error may extend
with some confidence. The most commonly used confidence
in uncertainty analysis is 95%, but other confidences may be
employed where appropriate. In this section, all examples
will be at 95% confidence.
Error
is most often defined as the difference between the
measured value of one data point and the true value of the
measurand. That is:
E
=
(measured)
−
(true)
1.5(1)
where
E
=
measurement error
measured
=
value obtained by a measurement
true
=
true value of the measurand
It is possible to estimate only the expected limits to an
error at some confidence. The most common method for
estimating those limits is to use the
normal distribution
.
1
For an infinite population (
N
=
∞
), the standard deviation,
σ
, would be used to estimate the expected limits of a particular

error with some confidence. That is, the average, plus or minus
2
σ
divided by the square root of the number of data points,
would contain the true average,
µ
, 95% of the time.
However, in test measurements, one typically cannot sample
the entire population and must make do with a sample of
data points. The sample standard deviation,
S
X
, is then used
to estimate
σ
X
. For a large data set (defined as having 30 or
more degrees of freedom
1
)
±
2
S
X
divided by the square root
of the number of data point e reported average contains the
true average,
µ
, 95% of the time. That
S
X
divided by the square
root of the number of data points in the reported average,
M
,
is called the
standard deviation
of the average (sometimes
also called the
random uncertainty
) and is written as
1.5(2)
where
=
standard deviation of the average; the sample standard
deviation of the data divided by the square root of
M
S
X
=
sample standard deviation
=
sample average, that is,
1.5(3)
X
i
=
i
th data point used to calculate the sample
standard deviation and the average
N
=
number of data points used to calculate the
sample standard deviation
(
N
−
1)
=
degrees of freedom of
S
X
and
M
=
the number of data points in the reported
average test result
Note in Equation 1.5(3) that, usually,
N
=
M
. This is not
a requirement, however.
N
does not necessarily equal
M
. It is
possible to obtain
S
X
from historical data with many degrees
of freedom ((
N –
1) greater than 30) and to take only
M
data
points in a specific test. The test result, or average, would
therefore be based on
M
measurements, and the standard deviation
of the average could still be calculated with Equation
1.5(3). In that case, there would be two averages. One average
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