1.3. EDA Techniques
1.3.5. Quantitative Techniques
1.3.5.17. |
Grubbs' Test for Outliers |
Detection of Outliers
Grubbs' test detects one outlier at a time. This outlier is expunged from the dataset and the test is iterated until no outliers are detected. However, multiple iterations change the probabilities of detection, and the test should not be used for sample sizes of six or less since it frequently tags most of the points as outliers.
Grubbs' test is also known as the maximum normed residual test.
| H0: | There are no outliers in the data set |
| Ha: | There is at least one outlier in the data set |
| Test Statistic: |
The Grubbs' test statistic is defined as:
 and
are the sample mean and standard deviation. The Grubbs
test statistic is the largest absolute deviation from the
sample mean in units of the sample standard deviation.
|
| Significance Level: |
 .
|
| Critical Region: |
The hypothesis of no outliers is rejected if
![> [(N-1)/SQRT(N)]*SQRT(t(alpha/(2*N),N-2)**2/(N - 2 + t(alpha/(2*N),N-2)**2)](eqns/grubbcr.gif)
 is the
critical value of the
t-distribution
with (N-2) degrees of freedom and a significance level of
/(2N).
In the above formulas for the critical
regions, the Handbook follows the convention that
|
is the upper critical value from the t-distribution
and 
is the lower critical value from the
t-distribution. Note that this is the opposite of
what is used in some texts and software programs. In
particular, Dataplot uses the opposite convention.
*********************
** grubbs test y **
*********************
GRUBBS TEST FOR OUTLIERS
(ASSUMPTION: NORMALITY)
1. STATISTICS:
NUMBER OF OBSERVATIONS = 195
MINIMUM = 9.196848
MEAN = 9.261460
MAXIMUM = 9.327973
STANDARD DEVIATION = 0.2278881E-01
GRUBBS TEST STATISTIC = 2.918673
2. PERCENT POINTS OF THE REFERENCE DISTRIBUTION
FOR GRUBBS TEST STATISTIC
0 % POINT = 0.
50 % POINT = 2.984294
75 % POINT = 3.181226
90 % POINT = 3.424672
95 % POINT = 3.597898
99 % POINT = 3.970215
37.59665 % POINT: 2.918673
3. CONCLUSION (AT THE 5% LEVEL):
THERE ARE NO OUTLIERS.
- The first section prints the sample statistics used in the
computation of the Grubbs' test and the value of the
Grubbs' test statistic.
- The second section prints the upper critical value for the
Grubbs' test statistic distribution corresponding to
various significance levels. The value in the first column,
the confidence level of the test, is equivalent to
100(1-). We reject the
null hypothesis at that significance level if the
value of the Grubbs' test statistic printed in
section one is greater than the critical value printed in the
last column.
- The third section prints the conclusion for a 95% test.
For a different significance level, the appropriate
conclusion can be drawn from the table printed in section
two. For example, for
= 0.10, we look at the row for 90% confidence and compare
the critical value 3.42 to the Grubbs' test statistic 2.92.
Since the test statistic is less than the critical value, we
accept the null hypothesis at the
= 0.10 level.
- Does the data set contain any outliers?
- How many outliers does it contain?
Checking for outliers should be a routine part of any data analysis. Potential outliers should be examined to see if they are possibly erroneous. If the data point is in error, it should be corrected if possible and deleted if it is not possible. If there is no reason to believe that the outlying point is in error, it should not be deleted without careful consideration. However, the use of more robust techniques may be warranted. Robust techniques will often downweight the effect of outlying points without deleting them.
