Page 166 - Vogel's TEXTBOOK OF QUANTITATIVE CHEMICAL ANALYSIS

4 ERRORS AND STATlSTlCS

If the calculated value of Q exceeds the critical value given in the Q table
(Appendix 14), then the questionable value may be rejected.
In this example Q calculated is 0.727 and Q critical, for a sample size of four,
is 0.831. Hence the result 3.2 pg g- ' should be retained. If, however, in the above
example, three additional measurements were made, with the results:

The value of Q critical for a sample size of seven is 0.570, so rejection of the
value 3.2 pg g - ' is justified.
It should be noted that the value Q has no regard to algebraic sign.

4.1 1 CONFIDENCE INTERVAL
When a small number of observations is made, the value of the standard
deviation s, does not by itself give a measure of how close the sample mean X
might be to the true mean. It is, however, possible to calculate a confidence
interval to estimate the range within which the true mean may be found. The
limits of this confidence interval, known as the confidence limits, are given by
the expression:
ts
Confidence limits of p, for n replicate measurements, p = X + - (1)
&
where t is a parameter that depends upon the number of degrees of freedom
(v) (Section 4.12) and the confidence level required. A table of the values of t
at different confidence levels and degrees of freedom (v) is given in Appendix 12.
Example 3. The mean (2) of four determinations of the copper content of a
sample of an alloy was 8.27 percent with a standard deviation s = 0.17 percent.
Calculate the 95 % confidence limit for the true value.
From the t-tables, the value of t for the 95 per cent confidence level with
(n - l), i.e. three degrees of freedom, is 3.18.
Hence from equation (l), the 95 per cent confidence level,

95 %(C.L.) for p = 8.27 + 3.18 x 0.17
fi

= 8.27 + 0.27 per cent
Thus, there is 95 per cent confidence that the true value of the copper content
of the alloy lies in the range 8.00 to 8.54 per cent.
If the number of determinations in the above example had been 12, then
the reader may wish to confirm that
95 % (C.L.) for p = 8.27 + 2.20 x O. 17
fi
= 8.27 + 0.1 1 per cent