Page 168 - Vogel's TEXTBOOK OF QUANTITATIVE CHEMICAL ANALYSIS

4 ERRORS AND STATlSTlCS

Had the calculated value for t been less than 1.80 then there would have
been no significance in the results and no apparent bias in the laboratory
procedure, as the tables would have indicated a probability of greater than
1 in 10 of obtaining that value. It should be pointed out that these values refer
to what is known as a double-sided, or two-tailed, distribution because it
concerns probabilities of values both less and greater than the mean. In some
calculations an analyst may only be interested in one of these two cases, and
under these conditions the t-test becomes single-tailed so that the probability
from the tables is halved.
(b) F-test. This is used to compare the precisions of two sets of data,2
for example, the results of two different analytical methods or the results from
two different laboratories. It is calculated from the equation:

N.B. The larger value of s is always used as the numerator so that the value
of F is always greater than unity. The value obtained for F is then checked for
its significance against values in the F-table calculated from an F-distribution
(Appendix 13) corresponding to the numbers of degrees of freedom for the two
sets of data.

Example 5. F-test comparison of precisions.
The standard deviation from one set of 11 determinations was s, = 0.210,
and the standard deviation from another 13 determinations was s, = 0.641. 1s
there any significant difference between the precision of these two sets of results?
From equation (3)

for

The first value (2.28) corresponds to 10 per cent probability, the second value
(2.91) to 5 percent probability and the third value (4.71) to 1 percent probability.
Under these conditions there is less than one chance in 100 that these
precisions are similar. To put it another way, the difference between the two
sets of data is highly significant.
Had the value of F turned out to be less than 2.28 then it would have been
possible to Say that there was no significant difference between the precisions,
at the 10 per cent level.

4.13 COMPARISON OF THE MEANS OF TWO SAMPLES
When a new analytical method is being developed it is usual practice to compare
the values of the mean and precision of the new (test) method with those of an
established (reference) procedure.
The value of t when comparing two sample means 2, and 2, is given by the