Материал: [lect] Grubbs - Procedure for Detecting outlying observations in samples (1969)

10 FRANK E. GRUBBS

From Table 1, for n = 14, we find that a value as large as 2.22 would occur by chance more than 5% of the time, so we should retain the value 1.01 in further calculations. We next calculate Dixon's sample criterion:

by Dixon's test, and no further values would be tested in this sample. (Note 3.) 4.8 We next turn to the case where we may have the two largest or the two smallest observations as probable outliers. Here, we employ a test provided by Grubbs [8] which is based on the ratio of the sample sum of squares when the two doubtful values are omitted to the sample sum of squares when the two doubtful values are included. If simplicity in calculation is the prime require- ment, then the Dixon type of test (actually omitting one observation in the sample) might be used for this case. In illustrating the test procedure, we give

the following Examples 4 and 5.

Example 4

In a comparison of strength of various plastic materials, one characteristic studied was the per cent elongation at break. Before comparison of the average elongation of the several materials, it was desirable to isolate for further study any pieces of a given material which gave very small elongation at breakage com- pared with the rest of the pieces in the sample. In this example, one might have primary interest only in outliers to the left of the mean for study, since very high readings indicate exceeding plasticity, a desirable characteristic.

Following are ten measurements of per cent elongation at break made on material No. 23: 3.73, 3.59, 3.94, 4.13, 3.04, 2.22, 3.23, 4.05, 4.11, 2.02. Arranged in ascending order of magnitude, these measurements are: 2.02, 2.22, 3.04, 3.23, 3.59, 3.73, 3.94, 4.05, 4.11, 4.13. The questionable readings are the two lowest, 2.02 and 2.22. We can test these two low readings simultaneously by using the

of Table 4. For the above measurements:

Note 3: It should be noted that in a multiplicity of tests of this kind, the final overall significancelevel will be less than that used in the individualtests, as we areofferingmorethan one chanceof acceptingthe sampleas one producedby a randomoperation.It is not ourpurpose here to cover the theory of multiple tests.

4.9 If simplicity in calculation is very important, or if a large number of samples must be examined individually for outliers, the questionable observations may be tested with the application of Dixon's criteria. Disregarding the lowest range, 4420 we test if the next lowest range 4549 is outlying. With n = 7, we see from Table 2 that r,o is the appropriate statistic. Renumbering the ranges as x, to X7, beginning with 4549, we find

which is only a little less than the 1% critical value, .637, for n = 7. So, if the test is being conducted at any significance level greater than the 1% level, we would conclude that 4549 is an outlier. Since the lowest of the original set of ranges, 4420, is even more outlying than the one we have just tested, it can be classified as an outlier without further testing. We note here, however, that this test did not use all of the sample observations.

4.10 Rejection of Several Outliers. So far we have discussed procedures for detecting one or two outliers in the same sample, but these techniques are not generally recommended for repeated rejection, since if several outliers are present in the sample the detection of one or two spurious values may be "masked" by the presence of other anomalous observations. Outlying observations occur due to a shift in level (or mean), or a change in scale (i.e., change in variance of the observations), or both. Ferguson [6, 7] has studied the power of the various rejection rules relative to changes in level or scale. For several outliers and repeated rejection of observations, Ferguson points out that the sample coefficient of skewness

should be used for "one-sided" tests (change in level of several observations in the same direction), and the sample coefficient of kurtosis

is recommended for "two-sided" tests (change in level to higher and lower values) and also for changes in scale (variance)*. In applying the above tests, the vb, or the b2 , or both, are computed and if their observed values exceed those for significance levels given in the following tables, then the observation farthest from the mean is rejected and the same procedure repeated until no further

5, 10, 15 and 20 (and 25 for b2) were obtained by Ferguson on an IBM 704 Computer using a sampling experiment or "Monte Carlo" procedure. The