Biomedical Engineering Reference
In-Depth Information
Table 11.2
One-Tail Areas
α
Under the Standard Normal Distribution from
z
=
k
α
to
∞
Area,
α
k
α
0.5000
0.000
0.2500
0.675
0.1587
1.000
0.1000
1.282
0.0500
1.645
0.0250
1.960
0.0228
2.000
0.0100
2.326
0.0050
2.576
0.0013
3.000
0.0002
3.500
11.7
Error and Error Propagation
As we have seen, the standard deviation of the values observed for a random vari-
able provides a measure of the uncertainty in the knowledge of the mean of that
variable. The uncertainty is often expressed as the probable error, which is the sym-
metric range about the mean within which there is a 50% chance that a measure-
ment will fall. For a normal distribution, the probable error is thus
±0.675
σ
(Ta-
ble 11.2).
Another measure of uncertainty is the fractional standard deviation, defined as
the ratio of the standard deviation and the mean of a distribution,
σ
µ
. This di-
mensionless quantity, which is also called the coefficient of variation, expresses the
uncertainty in relative terms. For the Poisson distribution, the fractional standard
deviation is simply
/
√
µ
µ
σ
µ
=
1
√
µ
=
.
(11.45)
In
F
ig. 11.1, for example, the standard deviation of th
e P
oisson distribution is
√
µ
=
√
10
3.16
. The fractional standard deviation is
1/
√
10
0.316
.
Often in practice one has only a single measurement of a random variable, such
as a number
n
of counts, and wishes to express an uncertainty associated with it.
The best estimate of the mean of the distribution from this single measurement is
that result: namely,
n
. If one assumes that the distribution
sa
mpled is Poisson or
normal, then the best estimate of the standard deviation is
√
n
.Thesignificanceof
the m
e
asurement, then, is that the true mean is estimated to lie within the interval
n
=
=
±
√
n
, with a probability (confidence) of 0.683.
Many measurements involve more than one random variable. For example, the
activity of a source can be obtained by counting a sample and then subtracting the
number of background counts measured with a blank. Both the number of gross
counts with the sample present and the number of background counts are subject
