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:
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(49.6 4 5.0) (58.9 1 2.43) log C |
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To conclude the discussion about the best-fit line, the following relationship can be shown to |
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exist among Y, Yˆ, and Y : |
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N |
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N |
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N |
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i 1 |
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(Y i Yˆi )2 |
(2.22) |
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(Y i Y |
(Yˆi Y |
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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.
2.3.11Control 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 so that 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 control- |
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analysis data. |
1 |
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To set up a control chart, individual observations might be plotted in sequential order and then |
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compared with control limits established from sufficient past experience. Limits of |
1.96 corre- |
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sponding 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 |
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control limits of |
3.00 . The outer control limits correspond to a confidence level of 99.8%, or a |
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probability of 0.002 that a point will fall outside the limits. One-half of this probability corresponds |
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to a high result and one-half to a low result. However, other confidence limits can be used as well; |
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the choice in each case depends on particular circumstances. |
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Special attention should be paid to one-sided deviation from the control limits, because systematic |
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errors more often cause deviation in one direction than abnormally wide scatter. Two systematic |
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errors of opposite sign would of course cause scatter, but it is unlikely that both would have entered |
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at the same time. It |
is not necessary that the control chart be plotted in a time sequence. In any |
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1 G. Wernimont, Ind. Eng. Chem., Anal. Ed. |
18: 587 (1946); J. A. Mitchell, ibid. 19: 961 (1947). |
2.138 SECTION 2
situation where relatively large numbers of units or small groups are to be compared, the control
chart is a simple means of indicating whether any unit or group is out of line. Thus laboratories,
production machines, test methods, or analysts may be put arbitrarily into a horizontal sequence. |
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Usually it is better to plot the means of small groups of observations on a control chart, rather |
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than individual observations. The random scatter of averages of pairs of observations is 1/(2) |
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1/2 |
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0.71 as great as that of single observations, and the likelihood of two “wild” observations in the |
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same direction is vanishing small. The groups of two to five observations should be chosen in such |
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a way that only change variations operate within the group, whereas assignable causes are sought |
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for variations between groups. If duplicate analyses are performed each day, the pairs form logical |
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groups. |
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Some measure of dispersion of the subgroup data should also be plotted as a parallel control |
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chart. The most reliable measure of scatter is the standard deviation. For small groups, the range |
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becomes increasingly significant as a measure of scatter, and it is usually a simple matter to plot the |
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range as a vertical line and the mean as a point on this line for each group of observations. |
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Bibliography |
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Alder, H. L., and E. B. Roessler, |
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W. H. Freeman, San |
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Youden, W. J., |
Statistical |
Manual |
of the AOAC, |
AOAC, 1111 North 19th St., Arlington, VA, |
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22209. |
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