6.3.4. Outlier Tests
The three outlier tests supported here will detect whether a minimum or a maximum value is an outlier and test its significance by comparing the test statistic with a critical value. The critical table values are generated by the program for any alpha level, not just for 1%, 5%, 10%. For Dixon and Grubbs tests, if the onetail option is selected, a separate onetailed test is performed for the minimum and maximum values of data. If a twotailed test is selected the larger of the minimum and maximum test values is tested against a twotailed table value. ESD test is twotailed only.
If significant outlier(s) are found, their values are displayed in a separate table.
6.3.4.1. Dixon
Dixon’s test does not assume normal distribution of data and is useful in detecting outliers in small sized groups.
The data is sorted in ascending order and two separate Q values are obtained for the lowest and highest values in data. Separate tests are performed for minimum and maximum values according to formulas given by Dixon (1953).

Minimum 
Maximum 
3 ≤ n ≤ 7 


8 ≤ n ≤ 10 


11 ≤ n ≤ 13 


14 ≤ n ≤ 1000 


If a twotailed test is selected, the larger of the Q_{Min} and Q_{Max} is tested against the table value computed for alpha / 2. Note that the onetail table Q is very accurate but the twotailed Q is an approximation.
Example
Sachs, L. (1984), p. 278 gives the following four observations 157, 326, 177, 176 and finds that 326 is an outlier at 5% level (onetailed test).
Dixon’s Outlier Test
Alpha = 0.05
Onetailed tests

Dixon’s Q 
Table Q 
Pass/Fail 
C1 Q(Min) 
0.1124 
0.7655 
Pass 
Q(Max) 
0.8817 
0.7655 
**Fail** 
N = 4, Q(Min)=(X(2)X(1))/(X(N)X(1)), Q(Max)=(X(N)X(N1))/(X(N)X(1))

Outlier Value 
C1 
326 
On the other hand, although in the sequence 1, 2, 3, 4, 5, 9 the value 9 looks like an outlier, it is not significantly different from the rest of the numbers at 5% level.
Dixon’s Outlier Test
Alpha = 0.05
Onetailed tests

Dixon’s Q 
Table Q 
Pass/Fail 
C1 Q(Min) 
0.1250 
0.5624 
Pass 
Q(Max) 
0.5000 
0.5624 
Pass 
N = 6, Q(Min)=(X(2)X(1))/(X(N)X(1)), Q(Max)=(X(N)X(N1))/(X(N)X(1))
6.3.4.2. Grubbs
Grubbs test requires that the data is approximately normally distributed and thus it is accurate for larger values of n. For onetailed tests the test statistic G is calculated for minimum and maximum observations separately as follows:
For a twotailed test greater of the G_{Min} and G_{Max} values is tested against the twotailed table value for the given alpha. Table G is accurate for both one and twotail tests.
Example
Tietjen and Moore (1972) test the following sequence of numbers 199.31, 199.53, 200.19, 200.82, 201.92, 201.95, 202.18, 245.57 for outliers.
Grubbs’ Outlier Test
Alpha = 0.05
Onetailed tests

Grubbs’ G 
Table G 
Pass/Fail 
C1 G(Min) 
0.4494 
2.0317 
Pass 
G(Max) 
2.4688 
2.0317 
**Fail** 
G = Maximum deviation from mean / Standard Deviation

Outlier Value 
C1 
245.57 
6.3.4.3. ESD (Generalised Extreme Studentised Deviate)
While Dixon’s and Grubbs’ tests can detect one outlier at a time, the Generalized Extreme Studentised Deviate (ESD) test can be used to test several outliers simultaneously. It is basically a Grubbs’ test run several times on the same sample, each time testing and omitting the most extreme observation and reducing the degrees of freedom by one. The user specifies the maximum number of extreme values to be tested.
The ESD test is twotailed.
Example
In this example Rosner (1983) searches for 10 outliers in a series of 54 numbers. Open GOODFIT, select Statistics 1 → Goodness of Fit Tests → Outlier Tests and select VitE (C8) as [Variable]. On the Output Options Dialogue enter 10 into the Number of outliers to test (ESD) box, check only the ESD test and click [Finish] to obtain the following results:
ESD Outlier Test
Alpha = 0.05
Twotailed test
Number of outliers to check (ESD) = 10

ESD Ri 
Table Ri 
Pass/Fail 
C1 (Max) 1 
3.1189 
3.1588 
Pass 
C1 (Max) 2 
2.9430 
3.1514 
Pass 
C1 (Max) 3 
3.1794 
3.1439 
**Fail** 
C1 (Max) 4 
2.8102 
3.1362 
Pass 
C1 (Min) 5 
2.8156 
3.1282 
Pass 
C1 (Max) 6 
2.8482 
3.1201 
Pass 
C1 (Max) 7 
2.2793 
3.1118 
Pass 
C1 (Max) 8 
2.3104 
3.1032 
Pass 
C1 (Min) 9 
2.1016 
3.0945 
Pass 
C1 (Max) 10 
2.0672 
3.0854 
Pass 
Ri = Generalised Extreme Studentised Deviate

Outlier Value 
C1 
5.34 