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

5.2 The critical values for T[ and T' for the 5% and 1% significance levels are due to David [3] and are given in Table 5. In Table 5 the subscript v = df indicates the total number of degrees of freedom associated with the independent estimate of standard deviation a and n indicates the number of observations

TABLE 5

Critical Values for T When Standard Deviation s, is Independent of Present Sample

The above percentage points are reproduced from H. A. David, "Revised upper percentage

points of the extreme studentized deviate from the sample mean," Biometrika, Vol. 43 (1956), pp. 449-451.

Standardization of Sodium Hydroxide Solutions as Determined by Plant Laboratories Standard Used: Potassium Acid Phthalate (P.A.P)

in the sample under study. We illustrate with an example on interlaboratory testing.

5.3 Example 6-Interlaboratory Testing. In an analysis of interlaboratory test procedures, data representing normalities of sodium hydroxide solutions were determined by twelve different laboratories. In all the standardizations, a tenth normal sodium hydroxide solution was prepared by the Standard Methods Committee using carbon-dioxide-free distilled water, Potassium acid phthalate (P. A. P.), obtained from the National Bureau of Standards, was used as the test standard.

Test data by the twelve laboratories are given in the table below. The P. A. P. readings have been coded to simplify the calculations. The variances between the three readings within all laboratories were found to be homogeneous. A one-way classification in the analysis of variance was first analyzed to determine

if the variation in laboratory results (averages) was statistically significant. This variation was significant, so tests for outliers were then applied to isolate the particular laboratories whose results gave rise to the significant variation. We are indebted to Dr. Grant Wernimont of the Eastman Kodak Co. for the data on Standardization of Sodium Hydroxide Solutions.

The above analysis of variance shows that the variation between laboratories

is highly significant. To test if this (very significant) variation is due to one

(or perhaps two) laboratories that obtained "outlying" results (i.e. perhaps showing non-standard technique), we can test the laboratory averages for outliers. From the analysis of variance, we have an estimate of the variance of

an individual reading as .008793, based on 24 degrees of freedom. The estimated

Since the estimate of within-laboratory variation is independent of any

from the grand average indicates that Laboratory 10 obtained an average reading much lower than the grand average, and that Laboratory 12 obtained a high average compared to the overall average. To first test if Laboratory 10 is an outlier, we compute

This value of T' is obviously significant at a very low level of probability

(P << .01. Refer to Table 5 with n = 12 and v = 24 d.f.). We conclude therefore that the test methods of Laboratory 10 should be investigated.

Excluding Laboratory 10, we compute a new grand average of 1.973 and test if the results of Laboratory 12 are outlying. We have

and this value of T' is significant at P << .01 (Refer to Table 5 with n = 11 and v = 24 d.f.). We conclude that the procedures of Laboratory 12 should also

be investigated.

To verify that the remaining laboratories did indeed obtain homogeneous results, we might repeat the analysis Laboratories 10 and 12. This calculation gives

For this analysis, the variation between labs is not significant at the 5% level and we conclude that all the laboratories except No. 10 and No. 12 exhibit

the same capability in testing procedure.

In conclusion, there should be a systematic investigation of test methods for Laboratories No. 10 and No. 12 to determine why their test precedures are

apparently different from the other ten laboratories.

(For the above example, procedures for ranking means after the initial analysis of variance test could, of course, have been used. For example, Duncan's Multiple Range Test, Scheffe's Test, Tukey's procedure, etc., could have been used. Also, the test of Halperin, Greenhouse and Cornfield [9] could have been

used. We have used David's tables seem tailor-made for one or two specific laboratories.)

6. RECOMMENDEDCRITERIAFORKNOWN

STANDARDDEVIATION

6.1 Frequently the population standard deviation a may be known accurately. In such cases, Table 6 may be used for single outliers and we illustrate with the

following example.

6.2 Example 7 (a known). Passage of the Echo I (Balloon) Satellite was recorded on star-plates when it was visible. Photographs were made by means

of a camera with shutter automatically timed to obtain a series of points for

the Echo path. Since the stars were also photographed at the same times as the Satellite, all the pictures show star-trails and so are called "star-plates."

This table is taken from the paper of Grubbs, Reference [8].

The x- and y-coordinate of each point on the Echo path are read from a photograph, using a stereo-comparator. To eliminate bias of the reader, the

photograph is placed in one position and the coordinates are read; then the photograph is rotated 180? and the coordinates reread. The average of the two

readings is taken as the final reading. Before any further calculations are made, the readings must be "screened" for gross reading or tabulation errors. This is done by examining the difference in the readings taken at the two positions of the photograph.

Recorded below are a sample of six readings made at the two positions and the differences in these readings. On the third reading, the differences are rather

large. Has the operator made an error in positioning the cross-hair on the point? For this example, an independent estimate of aois available since extensive tests on the stero-comparator have shown that the standard deviation in reader's error is about 4 microns. The determination of this standard error was based on such a large sample that we can assume a = 4 microns. The standard