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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

information for citizens to secure access to information under the control of public authorities,

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

education and knowledge, the attached public safety standard is made available to promote the

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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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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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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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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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 :

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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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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

0021

0037

0.004

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