Instant Notes: Analytical Chemistry

Inspection of the data suggests that 0.380 mg dm-^3 is a possible outlier.

Qexptl=0.380 -0.401/(0.410 -0.380) =0.021/0.03 =0.70

Qtab=0.83 for four values at the 95% probability level

As Qexptlis lessthan Qtab, 0.380 mg dm-^3 is not an outlier at the 95% level and

should be retained.

Example 2

If, in Example 1, three additional values of 0.400, 0.413 and 0.411 mg dm-^3 were

included, 0.380 mg dm-^3 is still a possible outlier.

Qexptl=0.380 -0.400/(0.413 -0.380) =0.020/0.033 =0.61

Qtab=0.57 for seven values at the 95% probability level

Now, as Qexptlis greaterthan Qtab, 0.380 mg dm-^3 is an outlier at the 95% level and

should be rejected. Note that because the three additional values are all around

0.4 mg dm-^3 , the suspect value of 0.380 mg dm-^3 appears even more anomalous.

F-test This test is used to compare the precisions of two sets of data which may origi-
nate from two analysts in the same laboratory, two different methods of analysis
for the same analyte or results from two different laboratories. A statistic, F, is
defined as the ratio of the population variances, s 12 /s 22 , or the sample variances,
s 12 /s 22 , of the two sets of data where the larger variance is always placed in the
numerator so that F ≥1.
If the null hypothesis is true, the variances are equal and the value of Fwill be
one or very close to it. As for the Q-test, an experimental value, Fexptl, is calcu-
lated and compared with a tabulated value, Ftab, at a defined probability level,
usually 90% or 95%, and for the number of degrees of freedom, N- 1 , for each
set of data. If Fexptlis lessthan Ftab, then the null hypothesis that there is no
significant differencebetween the two variances and hence between the preci-
sion of the two sets of data, is accepted. However, if Fexptlis greaterthan Ftab,
there is a significant differencebetween the two variances and hence between
the precisions of the two sets of data.
Some values of Ftabat the 95% probability level are given in Table 2. The
columns in the table correspond to the numbers of degrees of freedom for the
numerator set of data, while the rows correspond to the number of degrees of
freedom for the denominator set. Two versions of the table are available,
depending on the exact purpose of the comparison to be made: aone-tailed F-
test will show whether the precision of one set of data is significantly better than
the other, while atwo-tailed F-test will show whether the two precisions are
significantly different.

Table 2. Critical values of F at the 95% (P=0.05) level for a two-tailed test

n 1 579

n 2

5 7.146 6.853 6.681

7 5.285 4.995 4.823

9 4.484 4.197 4.026

n 1 =number of degrees of freedom of the numerator. n 2 =number of degrees of freedom of the

denominator

36 Section B – Assessment of data