Page 167 - Vogel's TEXTBOOK OF QUANTITATIVE CHEMICAL ANALYSIS

COMPARISONS OF RESULTS 4.12

Hence, on increasing the number of replicate determinations both the values of
t and slfidecrease with the result that the confidence interval is smaller. There
is, however, often a limit to the number of replicate analyses that can be sensibly
performed. A method for estimating the optimum number of replicate
determinations is given in Section 4.15.

4.12 COMPARISON OF RESULTS
The comparison of the values obtained from a set of results with either (a) the
true value or (b) other sets of data makes it possible to determine whether the
analytical procedure has been accurate and/or precise, or if it is superior to
another method.
There are two common methods for comparing results: (a) Student's t-test
and (b) the variance ratio test (F-test).
These methods of test require a knowledge of what is known as the number
of degrees of freedom. In statistical terms this is the number of independent
values necessary to determine the statistical quantity. Thus a sample of n values
has n degrees of freedom, whilst the sum C(x - 2)' is considered to have n - 1
degrees of freedom, as for any defined value of X only n - 1 values can be freely
assigned, the nth being automatically defined from the other values.

(a) Student's t-test. This is a test' used for small samples; its purpose
is to compare the mean from a sample with some standard value and to express
some level of confidence in the significance of the comparison. It is also used
to test the difference between the means of two sets of data XI and 2,.
The value of t is obtained from the equation:

where p is the true value.
It is then related to a set of t-tables (Appendix 12) in which the probability
(P) of the t-value falling within certain limits is expressed, either as a percentage
or as a function of unity, relative to the number of degrees of freedom.
Example 4. t-Test when the true mean is known.
If X the mean of the 12 determinations = 8.37, and p the true value = 7.91,
Say whether or not this result is significant if the standard deviation is 0.17.
From equation (2)

From t-tables for eleoen degrees of freedom (one less than those used in the
calculation)
for P = 0.10 ( 10 per cent) 0.05 (5 per cent) 0.01 ( 1 per cent)
t = 1.80 2.20 3.1 1
and as the calculated value for t is 9.4 the result is highly significant. The t-table
tells us that the probability of obtaining the difference of 0.46 between the
experimental and true result is less than 1 in 100. This implies that some
particular bias exists in the laboratory procedure.