Information Technology Reference
In-Depth Information
Consider the relative number of times
N
j
a measurement error of a given magnitude
occurs in a population of a given (large) size
N
; the relative frequency of occurrence of
any particular error in this population is
N
j
N
.
p
j
=
(1.1)
Here
j
indexes the above measurements into
M
different bins, where typically
N
M
.
The relative number of occurrences provides an estimate of the probability that a
measurement of this size will occur in further experiments. From this ensemble of
N
independent measurements we can define an average value,
M
Q
=
Q
j
p
j
,
(1.2)
j
=
1
with the average of a variable being denoted
by
the overbar and the
N
measurements
put into
M
bins of equal size. The mean value
Q
is often thought to be an adequate char-
acterization of the measure
me
nt and thus an operational definition of the experimental
variable is associated with
Q
Simpson was the first to suggest that the mean value be
accepted as the
best
value for the measured quantity. He further proposed that an isosce-
les triangle be used to represent the theoretical distribution in the measurements around
the mean value. Of course, we now know that using the isosceles triangle as a measure
of variability is wrong, but don't judge Simpson too harshly, after all, he was willing to
put his reputation on the line and speculate on the possible solution to a very difficult
scientific problem in his time and he got the principle right even if he got the equation
wrong.
Subsequently, it was accepted that to be more quantitative one should examine
the degree of variation of the measured value away from its average or “true” value. The
magnitude of this variation is defined not by Simpson's isosceles triangles but by the
standard deviation
.
2
of the measurements,
σ
or the variance
σ
M
Q
j
−
Q
2
p
j
,
2
σ
≡
(1.3)
j
=
1
which, using the definition of the average and the normalization condition for the
probability
M
p
j
=
1
,
(1.4)
j
=
1
reduces to
Q
2
2
σ
=
Q
2
−
.
(1.5)
These equations are probably the most famous in statistics and form the basis of vir-
tually every empirical theory of the physical, social and life sciences that uses discrete
data sets.
