preface to fifteenth edition

GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS 2.137
The best-fit equation expressed in terms of the confidence intervals for the slope and intercept
is:
E (49.6 5.0) (58.9 2.43) log C
4 1
To conclude the discussion about the best-fit line, the following relationship can be shown to
exist among Y, Ŷ, and : Y
i i i i
N N N
2 2 2
(Y Y) (Yˆ
Y) (Y Y ˆ ) (2.22)
i1 i1 i1
The term on the left-hand side is a constant and depends only on the constituent values provided by
the reference laboratory and does not depend in any way upon the calibration. The two terms on the
right-hand side of the equation show how this constant value is apportioned between the two quantities
that are themselves summations, and are referred to as the sum of squares due to regression
and the sum of squares due to error. The latter will be the smallest possible value that it can possibly
be for the given data.
2.3.11 Control Charts
It is often important in practice to know when a process has changed sufficiently so that steps may
be taken to remedy the situation. Such problems arise in quality control where one must, often
quickly, decide whether observed changes are due to simple chance fluctuations or to actual changes
in the amount of a constituent in successive production lots, mistakes of employees, etc. Control
charts provide a useful and simple method for dealing with such problems.
The chart consists of a central line and two pairs of limit lines or simply of a central line and
one pair of control limits. By plotting a sequence of points in order, a continuous record of the
quality characteristic is made available. Trends in data or sudden lack of precision can be made
evident sothat the causes may be sought.
The control chart is set up to answer the question of whether the data are in statistical control,
that is, whether the data may be retarded as random samples from a single population of data. Because
of this feature of testing for randomness, the control chart may be useful in searching out systematic
sources of error in laboratory research data as well as in evaluating plant-production or controlanalysis
data. 1
To set up a control chart, individual observations might be plotted in sequential order and then
compared with control limits established from sufficient past experience. Limits of 1.96 corresponding
to a confidence level of 95%, might be set for control limits. The probability of a future
observation falling outside these limits, based on chance, is only 1 in 20. A greater proportion of
scatter might indicate a nonrandom distribution (a systematic error). It is common practice with
some users of control charts to set inner control limits, or warning limits, at 1.96 and outer
control limits of 3.00. The outer control limits correspond to a confidence level of 99.8%, or a
probability of 0.002 that a point will fall outside the limits. One-half of this probability corresponds
to a high result and one-half to a low result. However, other confidence limits can be used as well;
the choice in each case depends on particular circumstances.
Special attention should be paid to one-sided deviation from the control limits, because systematic
errors more often cause deviation in one direction than abnormally wide scatter. Two systematic
errors of opposite sign would of course cause scatter, but it is unlikely that both would have entered
at the same time. It is not necessary that the control chart be plotted in a time sequence. In any
1
G. Wernimont, Ind. Eng. Chem., Anal. Ed. 18:587 (1946); J. A. Mitchell, ibid. 19:961 (1947).