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Disclosure to Promote the Right To Information
Whereas the Parliament of India has set out to provide a practical regime of right to
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in order to promote transparency and accountability in the working of every public authority,and whereas the attached publication of the Bureau of Indian Standards is of particular interest
to the public, particularly disadvantaged communities and those engaged in the pursuit of
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timely dissemination of this information in an accurate manner to the public.
!"#$% '(%)
/0)"1 &2 324 #' 5 *)6Jawaharlal Nehru
Step Out From the Old to the New
7"#1&"8+9&"), 7:1&"8+9&")Mazdoor Kisan Shakti Sangathan
The Right to Information, The Right to Live
IS 8900 (1978): Criteria for the rejection of outlying
observations [MSD 3: Statistical Methods for Quality and
Reliability]
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( Reaffirmed 2001 )
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IS :
8900
1978
I ndian St andard
CRITERIA FOR THE REJECTION
OF
OUTLYING OBSERVATIONS
Quality Control and Industrial Statistics Sectional Committee, EC 3
Chairman
DR
P. K. BOSE
Members
Representing
University of Calcutta, Calcutta
SHRI B. ANANTHAKRISHNANAND National Productivity Council, New Delhi
SHRIR. S. GUPTA Alternate )
SHRI M. G. BHADE
Tata Iron and Steel Co Ltd, Jamshedpur
DIRECTOR
Institute of Agricultural Research Statistics ICAR),
New Delhi
DR S. S. PILLAI ( Alternate)
SHRI D. DUTTA The Indian Tube Co Ltd, Jamshedpur
SARI Y. GHOUSEKHAN NGEF, Bangalore
SHRI C. RAJANA ( Alternate )
SHRI S. K. GUPTA
Central Statistical Organization, New Delhi
SHRI A. LAHIRI
Indian Jute Industries Research Association,
Calcutta
SHRI U. DUTTA (
Alternate )
SHRI P. LAKSHMANAN
Indian Statistical Institute, Calcutta
SHRI S. MONDOL
National Test House, Calcutta
SHRI S. K. BANERJEII.Alternate )
DR S. P. MUKHERJEE
Indian Association for Productivity, Quality and
Reliability, Calcutta
SHRI B. H~ATSINGKA ( Alternate )
SHRI Y. P. RAJPUT
Army Statistical Organization ( Ministry of De-
fence), New Delhi
SHRI C. L. VERMA (
Alternate )
SHRI RAMESHSHANKER
Directorate General of Inspection ( Ministry of
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IS : 8988 - 1978
Continuedfrom fiage
1 )
Members
SHRI S. SUBRAMU
SHRI S. N. VOHRA
DR B. N. SINGH,
Director ( Stat )
Rejresenting
Steel Authority of India Ltd, New Delhi
Directorate General of Supplies and Disposals, New
Delhi
Director General, IS1 ( Ex-ofjcio Member )
Secretary
SHRI Y. K. BHAT
Deputy Director (Stat), IS1
Industrial Statistics Subcommittee, EC
3:7
Convener
DR P. K. BOSE
Members
DIRECTOR
DR B. B. P. S. GOEL (
Alternate )
SHRI S. K. GUPTA
SHRI S. B. PANDEY
SHRI Y. P. RAJPUT
SHRI C.
L. VERMA (
Alternate)
DR D. RAY
REPRESENTATIVE
SHRI B. K. SARKAR
SHRI
D. R.
SEN
University of Calcutta,
Institute of Social
Management, Calcutta
Calcutta;
and Indian
Welfare and Business
Institute of Agricultural Research Statistics ( ICAR ),
New Delhi
Central Statistical Organization, New Delhi
Imperial Chemical Industries ( India ) Private Ltd,
Calcutta
Army Statistical Organization ( Ministry of Defence ),
New Delhi
Defence Research and Development Organization
( Ministry of Defence ), New Delhi
Indian Institute of Technology, Kharagpur
Indian Statistical Institute, Calcutta
Delhi Cloth & General Mills Co Ltd, Delhi
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1S t 8900 1978
I ndian St andard
CRITERIA FOR THE REJECTION OF
~OUTLYING OBSERVATIONS
0. FOREWORD
0.1 This Indian Standard was adopted by the Indian Standards Insti-
tution on 25 July 1978, after the draft finalized by the Quality Control
and Industrial Statistics Sectional Committee had been approved by the
Executive Committee.
0.2 An outlying observation or an outlier is one that appears to deviate
markedly from the other observations of the sample in which it occurs.
An outlier may arise merely because of an extreme manifestation of the
random variability inherent in the data or because of the non-random
errors, such as gross deviation from the prescribed experimental procedure,
mistakes in calculations, errors in recording numerical values, other
human errors, loss of the calibration of an instrument, change of measuring
instruments, etc. If it is known that a mistake has occurred, the outlying
observation must be rejected irrespective of its magnitude. If, however,
only a suspicion exists, it may be desirable to determine whether such an
observatien may be rejected or whether it may be accepted as part of the
normal variation expected.
0.3 The procedure consists in testing the statistical significance of the
outlier(s). A null hypothesis ( assumption ) is made that all the obser-
vations including the suspect observations come from thesame population
( or lot ) as the other observations in the sample. A statistical test is then
applied to determine whether this null hypothesis can be rejected at the
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IS : 8900
1978
0.4 When a test method recommends more than one determination for
reporting the average value of the characteristic under consideration, the
precision ( repeatability and reproducibility ) of the test method is found
to be useful in detecting observations which deviate unduly from the rest.
For further guidance in this regard, reference may be made to IS : 5420
( Part I )-1969*.
When the -precision of the test method is not known
quantitatively, one or more of the procedures covered in the standard
may be found helpful. The tests outlined in this standard primarily apply
to the observations in a single random sample or the experimental data
as given by the replicate measures of some property of a given material.
0.5 Although a number of statistical tests based on various considerations
are available to screen the given data for outliers, only those tests which
are simple and efficient have been included in this standard.
0.5.1 In the case of a single outlier ( the smallest or the largest suspect
observation ), two tests are available. One test is based on the standard
deviation whereas the other test is based on the ratio of differences bet-
ween certain order statistics, that is, the observations when they are
arranged in the ascending or descending order of magnitude. The
latter test would be more useful in those cases where the calculation of
standard deviation is to be avoided or a quick judgement is called for.
0.5.2 In the case of two or more outliers at either end of the ordered
sample observations, the test given is based on the ratio of the sample
sum of squares when the doubtful observations are omitted to the sample
sum of squares when the doubtful observations are included, the sample
sum of squares being defined as the sum of the squares of the deviations
of the observations from the corresponding mean.
If simplicity in
calculation is the prime requirement then *the test based on the order
statistics for the case of single outliers may be used by actually omitting
one suspect observation in the sample at a time. However, this test is to
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IS : 8900 - 1978
discard the entire data. However, this guideline is to be applied with
considerable caution in those situations where the data consists of a few
observations only.
0.7 Almost all the statistical criteria for the identification of the outliers
as given in this standard are based on the assumption that the underlying
distribution of the observations is normal. Hence, it is important that
these criteria are not used indiscriminately. In case the assumption of
normality is in doubt, it is advisable to obtain the guidance of a competent
statistician to ascertain the feasibility of the applicability of these test
criteria.
0.8 In reporting the result of a test or analysis, if the final value, observed
or calculated, is to be rounded off, it shall be done in accordance with
IS
:
2-1960*.
1. SCOPE
1.1
This standard lays down the criteria for the detection of outlying
observations in the following three situations:
a) Single outlier ( at either end of the ordered sample observations ),
b) Two or more outliers ( at either end of the ordered sample
observations ), and
c) Two or more outliers ( one at least at each of the two ends of the
ordered sample observations ).
1.2 In situations other than those enumerated in 1.1, it may be more
appropriate to conduct test for non-normality of observations which are
to be covered in a separate standard.
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IS : 8900 - 1978
2.6 Range
The difference between the largest and the smallest obser-
vations in a sample.
2.7
Null Hypothesis -
Hypothesis ( or assumption ) that all the obser-
vations in the sample come from the same parent population, distribution
or lot.
2.8
Level of Significance -
The probability ( or risk ) of rejecting the
null hypothesis when it is true. Conventionally it is taken to be 5 or 1
percent.
2.9 Statistic - A function calculated from the observationsin the sample.
2.10 Critical Value - The value of the appropriate statistic which would
be exceeded by chance with some small probability equivalent to the level
of significance chosen.
2.11 Degrees of Freedom- The number of independent component
values which are necessary to determine a statistic.
3. TESTS FOR SINGLE OUTLIER
3.0 There are many situations when, in a given sample of size
n,
one of
the observations, which is either the largest or the smallest, is suspect. To
detect such outliers, two tests have been given. The first one needs the
calculation of the average and the standard deviation of the sample
observations whereas the second one is based on order statistics and hence
is easier to operate.
3.1 Let x1, x2.. . . . .
x, be the n sample observations arranged in the ascending
order of magnitude so that xl < x2 < ..,......
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IS : 8900 1978
3.1.1 If the largest observation q is doubtful, then the criterion
z-11 =
x, - Z
- shall be calculated.
s
If the calculated, value of In is greater
than or equal to the corresponding critical value in Table 1 for sample size
ta and the specified level of significance, xn would be rejected as an outlier;
not otherwise.
3.1.2
Example I -
The tensile strength ( in megapascals, MPa ) of 10
specimens of brass rods of 10 mm diameter for general engineering purposes
obtained from a lot were obtained as 368,370, 370,370,372, 372, 372, 380,
384 and 397. The doubtful observation is the highest value x10=397 MPa.
The mean ( z )
obtained as:
and the standard deviation ( s ) for the 10 results are
Z e 375.5 MPa, and
s = 9.06 MPa.
Since the calculatedvalue ( 2.373 ) of 3-,, is greater than the tabulated
value 2.176 of In, in Table 1 for the sample size 10 and 5 percent level of
significance, the largest value 357 may be treated as an outlier and rejected.
3.2 The second criterion for the detection of a single outlier is based on the
ratios of differences between relevant ordered observations.
If+ x2 . . . . . Xn
are the ordered set of observations ( arranged in the ascending order of
magnitude ) then depending upon the total number of observations in the
sample, certain ratios of differences shall be calculated as indicated in
Table 2. If these ratios exceed the corresponding critical values at the
chosen level of significance, the existence of an outlier is indicated; not
otherwise.
The ratio to be calculated is indicated as r,, when the sample size is
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4. TEST FOR TWO OR MORE OUTLIERS (
AT EITHER END)
4.0 When two or more observations at the lower end or the upper end of
the ordered set of results deviate unduly from the rest, one way of detecting
the outliers is by the repeated application of the tests recommended in 3 for
single outlier. Although this procedure is simple, it has the disadvantage
that the overall level of significance of the test would not be the same
as in the case of test for a single outlier.
with caution.
Hence this test should be applied
However, separate tests have also been developed when
more than one observation at either end of the ordered set of observations
are suspected to be outliers.
4.1 Let x1, x2.. . . . . . . .xn
be the IZ observations in the sample arranged
according to the ascending order of magnitude. If it is desired to test
whether the
k
largest observations are outliers, the procedure as given below
may be followed:
The mean ( R ) and the sum of the squares ( S2 ) of the deviations of
the observations from their mean are calculated as follows:
n
s* =
2 l~i-Z)~ = n-l )s2
Leaving out the k suspect observations, the mean (A-k ) and the
sum of squares ( S*n_k ) of the remaining ( n-k ) observations are calcula-
ted as:
x1+ x2+ . . . . . . . . .
X&k
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: 8900 -
1978
4.1.1 In case the k smallest observations happen to be suspect, the
procedure followed would be exactly similar to that given in 4.1.
4.1.2
Example 3 -
In the course of preparation of standard reference
sample of bauxite, 13 splits were analyzed by a laboratory for the percentage
content of silicon dioxide ( SiO, ).
The test results were 3.74, 3.76, 3.78,
3.78, 3.78, 3.84, 3.84, 3.85, 3.89, 3.90, 3.90, 3.98 and 4.01.
The two largest observations, namely, 3.98 and 4.01 are suspected to
be outliers.
From all the 13 observations, Zrs = 3.85 and S213 = 0.084
Leaving out the 2 suspect results from the remaining 11 observations,
frr = 3.82 and SalI = 0.034
s21,
0.034
Hence LZ = __ = m =
s**s
0.405
Since the calculated value ( 0.405 ) of
Lz
is greater than the tabulated
value 0.337 of
L,
in Table 3 for the sample size 13 and 5 percent level of
significance, it is inferred that there is not enough evidence to suspect the
two largest observations as outliers.
5.
TESTS FOR TWO OR MORE OUTLIERS m AT LEAST ONE
OUTLIER AT EACH END)
5.0
When there are two or more suspect observations such that they occur
at both ends of the ordered set of sample results, two statistical tests of
significance for their detection have been given. The first test which is
based on the ratio of the sample range to the sample standard deviation is
applicable only when two outliers are present ( one each at the two ends of
the ordered sample results ) . The second test of significance is more broad-
based and is applicable to all situations where two or more suspect obser-
vations
occur in such a manner that at each end of the ordered sample
results there is at least one suspect observation.
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IS : 8900 I 1978
If the smallest and the largest observations, namely, 87.5 and 105.7,
are suspected to be outliers, then from all the 15 observations range (
R )
and standard deviation ( s ) are obtained as:
R = 105.7 - 87.5 = 18.2
I
15
Z (xl--E)s
1
S=
14
= 4.32
18.2
Hence
R/s = 4.y2 =
4.21
Since the tabulated value
ofR/s
in Table 4 corresponding to a sample
size 15 and 5 percent level of significance is 4.17 ( which is less than the
calculated value of 4.21 ), the smallest and largest values are considered as
outliers.
5.2 Test for More Than Two Outliers At Least One Outlier at
Each End ) -
Let x1, x, . . . . . . xn be the sample observations arranged
in the ascending order of magnitude.
If two or more ( say k ) of these
observations are suspect such that there is at least one suspect observation
at each end of the ordered observations in the sample, the procedure as
given below may be followed:
The mean ( z ) of the sample observations is first calculated.
Then
the residuals, that is, the absolute deviations of the observations from
their mean are calculated as:
I *1- 3 I ,
Ixz--zq, Ix,-21
. . . . . .
) I-21
~These n residuals are then arranged in the ascending order of
magnitude and designated as z,, zz . . . . z,,. Thus z1 corresponds to that
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IS : 8988 - 1978
extreme values, as also the sum of squares ( U2n _ k ) are calculated as:
?Z- k
z
G
zn-k =
i - 1
12-
k
f l -
U ,_k = I : k r?l ;,,_k )
i =
1
u2
n-k
The ratio & = u2-
, which is the test criterion, is then compared
with the critical value given in Table 5 for the sample size 12, the value of
k and the desired level of significance.
If the calculated value of & is
found to be smaller than the corresponding tabulated value, then the
k original suspect values are considered to be outliers, not otherwise.
It may be noted that smaller values of & denote significance as in
the case of Lk in Table 3.
5.2.1 Exarr@e 5-
In the data on shearing strength given in Example 4,
if the smallest two observations 875 and 88.7 and the largest observation
105.7 are suspect to be outliers,,then the mean (2) is first calculated as
95.2.
The residuals, that is, the absolute deviations of the sample observa-
tions from their mean are then found out as 7.7, 6.5, 2.3, 1.9, 1*6,0*7, 0.5,
0.2, 0, 0.2, 0.9, 2.0, 3.1, 4.8 and 10.5.
If these residuals (zr ) are arranged in the ascending order of
magnitude, we get 0, 0.2, 0.2, 0.5, 0.7, 0.9, 1.6, 1.9, 2.0, 2.3, 3.1, 4.8, 6.5,
7.7 and 10.5.
Now the 3 residuals corresponding to the 3 original suspected outliers
in the
new
sample are the last three values, namely, 6.5, 7.7 and 10.5
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IS :
8900 1978
Since the calculated value ( 0,159 ) of Es is less than 0,206 which is
the _tabulated value of Ek in Table 5 for the sample size 15,
k -
3 and
5 percent level of significance, all the three observations, namely, 87.5,
88.7 and 105.7, are to be considered as outlietx
TABLE 1 CRITICAL VALUES OF
Tl
OR
Tn
FOR TESTING AN OUTLIER
Clause
3.1 )
SAMPLE SIZE,
fl
SIGNIFICANCE
LEVEL
~_.._---_.-h-_--_
---7
5 Percent
1 Percent
3
1.153 1.155
4 1.463 1.492
5
1.672
I.749
G 1.822 1,944
7 1.938 2.097
8 2032 2221
9
2.110
2.323
10
2.176
2.410
11 2.234
2,485
12 2.285 2.550
13
2.331
2.607
14
2.371 2.659
15
2.409 2.705
16
2.443
2,747
17
2.475
2.785
18 2.504 2.821
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IS :
8900 1978
TABLE 2 CIiITICAL VALUES OF CRITERIA BASED ON ORDER
STATISTICS FOR TESTING AN OUTLIER
: Clause 3.2 j
\VHEN l-ILK OUTLIEll IS
SAMHA
SICNIPrCANCe LEVEL
~_..____~_____
_ ~
SIZE,
r--L-_*__---9
Srrlallrst Valur
Largest Value II
5 Percrnt 1 Percent
lb
Aa2 - ,
.rn -
xn-1 3
0.941 0~988
xn -
Xl
x n - x1 4
ti 765 (1.889
5 0.642 0.780
6 0.560 0698
7 0.507 0.637
11
x2 - A-1
rn -
Xn-1 8 0.554 0.683
xn- 1 - x1 tg - xp 9 0512 0.635
iu 0.477 0.597
1111 xj -
A-1
xu - xn_y
II 0.376 0.679
~--
xn-_l -
x
X - .x:2
12 0.546 0642
I
13 0.521 0.615
lx
*3 - x1
xn -
xn-0
14
___-
xn-P - Xl
xn - x3
15
16
17
18
19
20
03E
0.641
0.525
0616
0.507 0.595
0.490 W577
0.475 0.561
0.462 0547
0.450 0.535
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IS I 8900 - 1978
TABLE 3 CRITXCAL VALUES OF Lx FOR THE DETECTION OF k LARGEST OR SMALLEST i OUTLIERS
( Cluuse 4.1 )
T
_-
--
:
C
E
C
(
:
t
(
i
I
I
(
(
I
T
--
4
--
T
-
I
i-
-I-
3
5
6
a
9
I
5%
--
10
0
3.019
3.033
DO42
>057
I.050
PO66
PO27
PO37
I.072
PO91
3.104
>I18
1.136
YO82
)*I00
PllG
1.130
Pl50
>049
PO64
)_076
1.088
1.104
3204
1.36a
3.321
Y364
1.399
1438
I.222
I.283
l.334
I.378
I.417
P450
l.168
I.229
P282
l.324
I.361
I*400
-
-
--
I.056
I
O46
(
PO58
(
PI12
(
1166
Y220
; ;
I.262
(
.296
1
.336
. .
-
_
1% ~5
--
PO00
PO04
PO21 0.010
bO47 0.032
PO76 0.064
I.112 0.099
t.142 0.129
l.178 0.162
l.208 0,196
I.233 0.224
l.267 0.250
J.294 0.276
I.311 0.300
l.338 0.322
l.358 0.331
I.366 0.354
P387 0377
P488 0.450
l.526 0506
I.574 0.554
1.608 0.588
l.636 0618
I.668 0.646
5%
-i-
7-
__- -
1 0
0 5%
T
-
c
c
:
E
:
I
:.
(
(.
c
:
(
I
c
-
-
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1
_
7-
-_
I
I
I
I
I
I
I
I
I
I
1
1%
10
ro
I.002
PO10
I.028 0022 0.008
I.048 0.045 tiola
1.070 0.070 0032
I.098
0.098
0052
I.120 0.125 0.070
L-147
@I50
0.094
I.172 0.174 0.113
I.194 PI97 0.132
l.219
b.237
k260
I.272
P300
0.151
0.171
0.192
k201
0.231
P377
0.374 0.308
b434 0.434 0.369
k484 @482
0.418
j.522 0,523 0460
l.558 0556 0.498
I.592 0588 0,531
50
10
10
0
PO42
9.060
0.079
PO97
0.115
3.136
Pl54
3.168
3.188
P262
3,327
3376
0.42 1
3.456
3.490
5%
D*O55
0.072
0086
0.099
w115
0.184
0245
0.297
0.342
0.382
0.414
54
.d
JO66
0.126
0183
0235
0.280
0.320
0.356
_- --_
3,030
3044
DO53
D-064
PO78
3.062
I.074
PO88
0144
3154
P196 I.212
3*250 J264
0.292 I.310
3.328
I.350
2 368 3383
-_
(
:
:
(
:
(
:
i
(
(
:
-_
PO01
1.018
PO55
PI06
I.270
j.305
j.337
I.363
I.387
P410
I.427
1.447
l.462
l.484
1.550
l.599
I.642
X.72
l.696
I.722
4
:
7
8
9
10
PO34 0.012
I.054 O-026
PO76 0.038
PO98 0.056
l.122 0.072
PI40 0.090
pi59 pi08
P181 0.126
l.200 @140
I.209 I 0154
P238 t 0.175
I.312 0246
J.376 0.312
I.424 0.364
J.468 0408
P502 0444
I.535 0483
40
:;
14
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1s t
8900 1978
TABLE 4 CRITICAL VALUE OF
R s
FOR THE DETECTION OF
TWO
OUTLIER ( ONE AT EACH END OF ORDERED RESULTS )
( Clnuses5.1 and 5.1.1 )
SAMPLE . ZE,
n
SIGNIFICANCE
LEVXM
I------- -------
_. __--_.---_
5 Percent
1 Percenr
3
2-00
2.00
+ 2.43
2.45
5
2.75
2.80
li
3.01 +i+lU
7
3.22
3.34
8
3-40
3.54
9 3.55 3.72
10
3.68
3.88
11
3.80 4.01
12
3.91 413
13
4-00 424
14 4.09 4.34 ,
15
4.17 443
16
424 451
17
4,.31 4.59
18
4.38 4.66
19
4.43 473
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TABLE 5 CRITICAL VALUES OF
Ek
FOR THE DETECTION OF
k SOME
SMALL AND OTHERS LARGE) OUTLIERS
( Clauses 5.2 nd 5.2.1
)
k
2
-I
3
I 4
T T
-
--
-
-F
I
6 9
1
10
7
8
5
---.-f-----
1%
5
1%
1.004 @OOl
I.016 0.006
I.034
0.014
I.057 0.026
I.083
0.037
0.010
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