Grubbs' Test - Finding Outlier
Grubbs' Test - Finding Outlier
Grubbs' Test - Finding Outlier
This test detects outliers from normal distributions. The tested data are the
minimum and maximum values. The result is a probality that indicates that the data
belongs to the core population. If the investigated sample has some other, especially
assymmetric distribution (e.g. lognormal) then these tests give false results!
The test is based on the difference of the mean of the sample and the most
extreme data considering the standard deviation (Grubbs, 1950, 1969; DIN 32645;
DIN 38402).
The test can detect one outlier at a time with different probablities (see table
below) from a data set with assumed normal distribution. If n>25 then the result is
just a coarse approximation.
Tmax =
x n X mean
s
Tmin =
X mean X1
s
where
Xi or Xn = the suspected single outlier (max or min)
s = standard deviation of the whole data set
Xaver = mean
0.1
0.075
0.05
0.025
0.01
0.1
0.075
0.05
0.025
0.01
1.15
1.15
1.15
1.15
1.15
53
2.981
3.151
999
1.42
1.44
1.46
1.48
1.49
54
2.988
3.158
999
1.6
1.64
1.67
1.71
1.75
55
2.995
3.165
999
1.73
1.77
1.82
1.89
1.94
56
3.002
3.172
999
1.83
1.88
1.94
2.02
2.1
57
3.009
3.179
999
1.91
1.96
2.03
2.13
2.22
58
3.016
3.186
999
1.98
2.04
2.11
2.21
2.32
59
3.023
3.193
999
10
2.03
2.1
2.18
2.29
2.41
60
3.03
3.2
999
11
2.09
2.14
2.23
2.36
2.48
61
3.036
3.206
999
12
2.13
2.2
2.29
2.41
2.55
62
3.042
3.212
999
13
2.17
2.24
2.33
2.46
2.61
63
3.048
3.218
999
14
2.21
2.28
2.37
2.51
2.66
64
3.054
3.224
999
15
2.25
2.32
2.41
2.55
2.71
65
3.06
3.23
999
16
2.28
2.35
2.44
2.59
2.75
66
3.066
3.236
999
17
2.31
2.38
2.47
2.62
2.79
67
3.072
3.242
999
18
2.34
2.41
2.5
2.65
2.82
68
3.078
3.248
999
19
2.36
2.44
2.53
2.68
2.85
69
3.084
3.254
999
20
2.38
2.46
2.56
2.71
2.88
70
3.09
3.26
999
21
2.58
2.73
2.91
71
3.095
3.265
999
22
2.6
2.76
2.94
72
3.1
3.27
999
23
2.62
2.78
2.96
73
3.105
3.275
999
24
2.64
2.8
2.99
74
3.11
3.28
999
25
2.66
2.82
3.01
75
3.115
3.285
999
26
2.68
2.84
999
76
3.12
3.29
999
27
2.7
2.86
999
77
3.125
3.295
999
28
2.72
2.88
999
78
3.13
3.3
999
29
2.73
2.9
999
79
3.135
3.305
999
30
2.75
2.91
999
80
3.14
3.31
999
31
2.76
2.93
999
81
3.144
3.314
999
32
2.78
2.95
999
82
3.148
3.318
999
33
2.79
2.96
999
83
3.152
3.322
999
34
2.81
2.97
999
84
3.156
3.326
999
35
2.82
2.98
999
85
3.16
3.33
999
36
2.83
2.992
999
86
3.164
3.334
999
37
2.84
3.004
999
87
3.168
3.338
999
38
2.85
3.016
999
88
3.172
3.342
999
39
2.86
3.028
999
89
3.176
3.346
999
40
2.87
3.04
999
90
3.18
3.35
999
41
2.88
3.05
999
91
3.183
3.353
999
42
2.89
3.06
999
92
3.186
3.356
999
43
2.9
3.07
999
93
3.189
3.359
999
44
2.91
3.08
999
94
3.192
3.362
999
45
2.92
3.09
999
95
3.195
3.365
999
46
2.928
3.098
999
96
3.198
3.368
999
47
2.936
3.106
999
97
3.201
3.371
999
48
2.944
3.114
999
98
3.204
3.374
999
49
2.952
3.122
999
99
3.207
3.377
999
51
2.967
3.137
999
100
3.21
3.38
999
52
2.974
3.144
999