Report for Analytical Chemists
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CHEMICAL
RESEARCH
Circle No.
34A
·
171 on Reeders'
procedure must be given. It is supposed that this procedure could in principle be realized in endless series of experiments. (There may be practical bounds to do that— e.g., the lifetime of the observer.) It is furthermore supposed that for all the possible results of these experiments, a distinct “probability” p exists (0 ¿ p 5 1) in the sense of measure theory and that the values of this “set function” are solely predetermined by the experimental procedure as such. After having accepted this hypothesis, we may think of probability in the usual intuitive way. But we must never forget that for all applications, we should make sure we have a defined procedure and that the above hypothesis is not flagrantly wrong. This is not as trifling as one may think at first sight: Tossing up a coin will normally yield a sequence with the relative frequency (probability) 1/2 for either side; but if a person is asked to write down a sequence of the two symbols A and B he can do this in a tricky way, so that no probability values can be assigned. To prepare the following conclusions, we shall add the nearly trivial assumption that the probability distribution, either discrete or continuous, behaves decently insofar that not only its own existence is presupposed but also that of its mean X and of its standard deviation . With these hypotheses duly tested and taken as valid, we are entitled to consider the average, x, of a sufficiently large series of experimental results as an estimate for the mean, X, of the whole parent population, and the standard deviation, s, as an estimate for , the standard deviation of the population. The divisor (n 1) in Equation 9b takes care of the fact that the deviations x’ are taken against the averXi age x of the series and not against the mean X of the population. This (To gives a better estimate for . be quite exact: with Equation 9b, s8 will be a better estimate for 2!) So far we have no idea how good these estimates may be, since nothing has been said about any special type of probability function. But from the existence of probability functions (with first and second moments) some useful general rules —
—
SERVICES.
INC.
Service Card
ANALYTICAL CHEMISTRY, VOL. 42, NO. 4, APRIL 1970
for computations follow, which are often erroneously thought to be confined to the “normal” Gaussian distribution. (a) If the result, x, of a measuring procedure is a linear function of several measured quantities zu z2, for which probability funczs tions exist, then with ...
X
+
«
=
+
«2^2
+
+
33 .
·
OLhZh
.
for the means of the respective populations also, the same relation is valid X
-f- otxZi -H
—
+
OÍ3Z3
.
.
.
(10)
cinZh
(b) If the result x is the product of several independent quantities zlt z2,
zs.....then
z2
z3
.
.
X
.
with
x
=
-
also
Zn
Zs. .Zh
Zx Zx
=
(11)
.
is valid.
The random quantities said to be independent, if their common distribution is the product of the individual distribution functions; hence, /(z1; z2 zh) = This means f(z 1) f(z2) f (zj,). that the probability to measure a distinct value of a quantity z* is independent of the outcome of the measurements of all the other quantities from Zi zh. This is a very sharp requirement which is not always fulfilled. For example, sex and size are not independently distributed; or if an analyst has already found A1 in a steel sample, he may expect that he will also find a trace of Ga. (c) If the result, x, of a measuring procedure is a linear function of several measured independent zh, i.e., x quantities z1; z2, z3 ...
are
...
·
.
.
.
.
.
.
.
+
.
—
.
-f- a2z2 -f- a3z3 -faj,Zh the relation between the stan-
«
.
.
.
dard deviations of the respective populations is
2( )
=
2 2( ) + 22 2( 2) + 32 2( 3) 2 2( ) (12) .
.
.
(d) The usefulness and general importance of these rules are easily demonstrated when we use a simple trick: We are free to consider the different random quantities z< (i 1 h), as the variables occurring in h repeated different realizations of the same procedure, for instance —
...
