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

s= estimate of the population standard deviation based on the sample data, calculated with n - 1 degrees of freedom as follows:

If x, rather than xn is the doubtful value, the criterion is as follows:

T- = (x - xI)/s

The critical values for either case, for the 1 per cent and 5 per cent levels of significance, are given in Table 1. Table 1 and the following tables give the "one-sided" significance levels. (In a previous ASTM tentative recommended practice (1961), the tables listed values of significance levels double those in the present practice, since it was considered that the experimenter would test either the lowest or the highest observation (or both) for statistical significance. However, to be consistent with actual practice and in an attempt to avoid further misunderstanding, single-sided significance levels are tabulated here so that both

viewpoints can be represented.)

4.2 The hypothesis that we are testing in every case is that all observations in the sample come from the same normal population. Let us adopt, for example,

critical value the 0.05 point of Table 1. On the other hand, if we are interested

Table 1. Suppose, however, that we are interested in outliers occurring on either side, but do not believe that outliers can occur on both sides simultaneously. We might, for example, believe that at some time during the experiment something possibly happened to cause an extraneous variation on the high side or on the low side, but that it was very unlikely that two or more such events could have occurred, one being an extraneous variation on the high side and the other an extraneous variation on the low side. With this point of view we should use the statistic T, = (x, - X)/s or the statistic T1 = (x - xl)/s which ever is larger. If in this instance we use the 0.05 point of Table 1 as our critical value,

the true significance level would be twice 0.05 or 0.10. If we wish a significance level of 0.05 and not 0.10, we must in this case use as a critical value the 0.025

point of Table 1. Similar considerations apply to the other tests given below.

Example 1

As an illustration of the use of T, and Table 1, consider the following ten

observations on breaking strength (in pounds) of 0.104-in. hard-drawn copper wire: 568, 570, 570, 570, 572, 572, 572, 578, 584, 596. The doubtful observation is

6 FRANK E. GRUBBS

with probability less than 0.05. In fact, so large a value would occur by chance not much oftener than 1% of the time. Thus, the weight of the evidence is against the doubtful value having come from the same population as the others (assuming the population is normally distributed). Investigation of the doubtful value is therefore indicated.

4.3 An alternative system, the Dixon criteria, based entirely on ratios of

judgment is called for. For the Dixon test, the sample criterion or statistic changes with sample size. Table 2 gives the appropriate statistic to calculate and also gives the critical values of the statistic for the 1%, 5% and 10% levels

of significance.

Example 2

As an illustration of the use of Dixon's test, consider again the observations on breaking strength given in Example 1, and suppose that a large number of such samples had to be screened quickly for outliers and it was judged too time- consuming to compute s. Table 2 indicates use of

which is a little less than .477, the 5% critical value for n = 10. Under the Dixon criterion, we should therefore not consider this observation as an outlier at the 5% level of significance. This illustrates how border-line cases may be accepted under one test but rejected under another. It should be remembered, however, that the T-statistic discussed above is the best one to use for the single-outlier case, and final statistical judgment should be based on it. See Ferguson, References [6], [7].

Further examination of the sample observations on breaking strength of hard-drawn copper wire indicates that none of the other values needs testing.

(Note 2.)

4.4 A test equivalent to Tn (or TI) based on the sample sum of squared deviations from the mean for all the observations and the sum of squared de-

viations omitting the "outlier" is given by Grubbs in [8]

4.5 The next type of problem to consider is the case where we have the possi- bility of two outlying observations, the least and the greatest observation, in a

Note 2: With experience we may usually just look at the sample values to observe if an outlier is present. However, strictly speaking the statistical test should be applied to all samples to guarantee the significance levels used. Concerning "multiple" tests on a single sample, we comment on this below.

"vertical semi-diameters of Venus made by Lieutenant Herndon in 1846 and given in William Chauvenet's A Manual of Spherical and Practical Astronomy, Vol. II (5th ed., 1876). In the reduction of the observations, Prof. Pierce assumed two unknown quantities and found the following residuals which have been arranged in ascending order of magnitude:

TABLE 3

Critical Values for w/s (Ratio of Range to Sample Standard Deviation)*

* Taken from H. A. David, H. O. Hartley and E. S. Pearson, "The Distribution of the Ratio in a Single Sample of Range to Standard Deviation," Biometrika, Vol. 41 (1954),

pp. 482-493. (Reference[4])

The deviations -1.40 and 1.01 appear to be outliers. Here the suspected observations lie at each end of the sample. Much less work has been accomplished for the case of outliers at both ends of the sample than for the case of one or more outliers at only one end of the sample. This is not necessarily because the "one-sided" case occurs more frequently in practice but because "two-sided" tests are more difficult to deal with. For a high and a low outlier in a single

sample, the procedure below may possess near optimum properties. For optimum procedures when there is at hand an independent estimate, s2 of a2, see "Some Tests for Outliers" by C. P. Quesenberry and H. A. David, Technical Report No. 47, OOR (ARO) project No. 1166, Virginia Polytechnic Institute, Blacks-

burg, Virginia.

4.6 For the observations on the semi-diameters of Venus given above, all the information on the measurement error is contained in the sample of 15 residuals. In cases like this, where no independent estimate of variance is available (i.e. we still have the single sample case), a useful statistic is the ratio of the range of the observations to the sample standard deviation:

If x, is about as far above the mean, x, as xi is below x, and if w/s exceeds some chosen critical value, then one would conclude that boththe doubtful values are outliers. If, however, xl and x, are displaced from the mean by different amounts, some further test would have to be made to decide whether to reject as outlying only the lowest value or only the highest value or both the lowest and highest values.

4.7 For this example the mean of the deviations is x = .018, s = .551, and

the critical values for the 1% and 5% levels, so if the test were being run at the 5% level of significance, we would conclude that this sample contains one or more outliers. The lowest measurement, -1.40", is 1.418" below the sample mean, and the highest measurement, 1.01", is .992" above the mean. Since

Table 1 this value is greater than the critical value for the 5% level, so we reject -1.40. Since we have decided that -1.40 should be rejected, we use the remaining 14 observations and test the upper extreme 1.01, either with the criterion

-

n

8