Information Technology Reference
In-Depth Information
1500
1200
900
600
300
0
56
58
60 62
64
Stature in inches
66
68
70 72
74
76 78 80
Figure 1.1.
The dots denote the relative frequency of the heights of adult males in the British Isles [
41
]. The
solid curve is the normal distribution with the same mean and variance as that of the data points.
In the continuum limit, that is, the limit in which the number of independent observa-
tions of a web variable approaches infinity, the characteristics of any measured quantity
are specified by means of a distribution function. From this perspective any particular
measurement has little or no meaning in itself; only the collection of measurements, the
ensemble, has a scientific interpretation that is manifest through the distribution func-
tion. The distribution function is also called the probability density and it associates a
probability with the occurrence of an event in the neighborhood of a measurement of a
given size. For example in Figure
1.1
is depicted the frequency of occurrence of adult
males of a given height in the general population of the British Isles. From this distribu-
tion it is clear that the likelihood of encountering a male six feet in height on your trip
to Britain is substantial and the probability of meeting someone more than ten feet tall
is zero.
Quantitatively, the probability of meeting someone with a height
Q
in the interval
(
q
,
q
+
q
) is given by the product of the distribution function and the size of the
interval
P
q
. The solid curve in Figure
1.1
is given by a mathematical expression
for the functional form of
P
(
q
)
. Such a bell-shaped curve, whether from measurements
of heights or from errors, is described by the well-known distribution of Gauss, and is
also known as the normal distribution.
Half a century after Simpson's work the polymath Johann Carl Friedrich Gauss
(1777-1855) [
12
] systematically investigated the properties of measurement errors and
in so doing set the course of experimental science for the next two centuries. Gauss pos-
tulated that if each observation in a sequence of measurements
Q
1
,
(
q
)
Q
j
,...,
Q
N
was truly independent of all of the others then the deviation from the average value
is a random variable
Q
2
,...,
ξ
j
=
Q
j
−
Q
,
(1.6)
so that
This definition
has the virtue of defining the average error in the measurement process to be zero,
ξ
j
and
ξ
k
are statistically independent of one another if
j
=
k
.
