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IS 10427-1 (1982): Designs for industrial experimentation, Part 1: Standard designsDisclosure 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.

“! $ ' +-” Satyanarayan Gangaram Pitroda

“Invent a New India Using Knowledge”

“01 ' 5 ” Jawaharlal Nehru

“Step Out From the Old to the New”

“1 +, 1 +” Mazdoor Kisan Shakti Sangathan

“The Right to Information, The Right to Live”

“! > 0 B ” Bharthari—Ntiatakam

“Knowledge is such a treasure which cannot be stolen”

“Invent a New India Using Knowledge”

””

IS 10427-1 (1982): Designs for industrial experimentation, Part 1: Standard designs [MSD 3: Statistical Methods for Quality and Reliability]

© BIS 2004

B U R E A U O F I N D I A N S T A N D A R D S MANAK BHAVAN, 9 BAHADUR SHAH ZAFAR MARG

NEW DELHI 110002

Edition 1.1 (1992-02)

Price Group 7

(Incorporating Amendment No. 1)

© BIS 2004

BUREAU OF INDIAN STANDARDS

This publication is protected under the Indian Copyright Act (XIV of 1957) and reproduction in whole or in part by any means except with written permission of the publisher shall be deemed to be an infringement of copyright under the said Act.

Indian Standard DESIGNS FOR INDUSTRIAL

EXPERIMENTATION PART 1 STANDARD DESIGNS

Quality Control and Industrial Statistics Sectional Committee, EC 3

Chairman Representing

DR P. K. BOSE Indian Association for Productivity, Quality and Reliability (IAPQR), Calcutta

Members

SHRI B. HIMATSINGKA ( Alternate to Dr P. K. Bose )

SHRI B. ANANTH KRISHNANAND National Productivity Council, New Delhi SHRI R. S. GUPTA ( Alternate )

SHRI M. G. BHADE The Tata Iron and Steel Co Ltd, Jamshedpur DR S. K. CHATTERJEE National Institute of Occupational Health,

Ahmadabad SHRI P. K. KULKARNI ( Alternate )

SHRI C. A. CRASTO The Cotton Textiles Export Promotion Council, Bombay

DIRECTOR Indian Agricultural Statistics Research Institute (ICAR), New Delhi

DR S. S. PILLAI ( Alternate ) SHRI Y. GHOUSE KHAN NGEF Ltd, Bangalore

SHRI C. RAJANNA ( Alternate ) SHRI A. LAHIRI Indian Jute Industries’ Research Association,

Calcutta SHRI U. DUTTA ( Alternate )

SHRI R. N. MONDAL Tea Board, Calcutta DR S. P. MUKHERJEE Calcutta University, Calcutta SHRI A. N. NANKANA Indian Statistical Institute, Calcutta SHRI M. K. V. NARAYAN The Indian Tube Co Ltd, Jamshedpur

SHRI P. N. GUPTA ( Alternate ) SHRI T. R. PURI Army Statistical Organization, Ministry of Defence,

New Delhi SHRI R. D. AGRAWAL ( Alternate )

SHRI E. K. RAMACHANDARAN National Test House, Calcutta SHRI S. MONDAL ( Alternate )

SHRI RAMESH SHANKER Directorate General of Inspection, Ministry of Defence, New Delhi

SHRI N. S. SENGAR ( Alternate ) ( Continued on page 2 )

IS : 10427 (Part 1) - 1982

2

Members Representing

SHRI T. V. RATNAM The South India Textile Research Association, Coimbatore

DR D. RAY Defence Research and Development Organization, Ministry of Defence, New Delhi

SHRI G. SAMSON Indian Telephone Industries Ltd, Bangalore SHRI P. V. RAO ( Alternate )

SHRI A. SANJEEVA RAO Shriram Fibres Limited, Madras SHRI R. SETHURAMAN Electronics Trade and Technology Development

Corporation Ltd, New Delhi SHRI N. SRINIVASAN ( Alternate )

SHRI S. SHIVARAM The State Trading Corporation of India Ltd, New Delhi

DR S. S. SRIVASTAVA Central Statistical Organization, New Delhi SHRI S. SUBRAMU Steel Authority of India Ltd, New Delhi SHRI Y. K. BHAT,

Director (Stat) Director General, ISI ( Ex-officio Member )

Secretary

Industrial Statistics Subcommittee, EC 3 : 7

Convener

DR P. K. BOSE Indian Association for Productivity, Quality and Reliability (IAPQR), Calcutta

Members

SHRI B. K. CHOWDHURY ( Alternate to Dr P. K. Bose )

DIRECTOR Indian Agricultural Statistics Research Institute (ICAR), New Delhi

DR B. B. P. S. GOEL ( Alternate ) DR S. P. MUKHERJEE Calcutta University, Calcutta DR M. PADMANABHAN National Sample Survey Organisation, New Delhi SHRI T. R. PURI Army Statistical Organization, Ministry of Defence,

New Delhi SHRI R. D. AGRAWAL ( Alternate )

SHRI C. RAJANNA NGEF Ltd, Bangalore SHRI Y. GHOUSE KHAN ( Alternate )

DR D. RAY Defence Research and Development Organization, Ministry of Defence, New Delhi

SHRI B. K. SARKAR Indian Statistical Institute, Calcutta SHRI D. R. SEN Delhi Cloth Mills (DCM), Delhi DR S. S. SRIVASTAVA Central Statistical Organization, New Delhi

IS : 10427 (Part 1) - 1982

3

Indian Standard DESIGNS FOR INDUSTRIAL

EXPERIMENTATION PART 1 STANDARD DESIGNS

0. F O R E W O R D 0.1 This Indian Standard was adopted by the Indian Standards Institution on 15 December 1982, after the draft finalized by the Quality Control and Industrial Statistics Sectional Committee had been approved by the Executive Committee. 0.2 Industrial organizations are constantly faced with the problem of decision making regarding product/process design, process specifications, quality improvement, dominant factors affecting quality, cost reduction, import substitution, etc. In all such problems, one is confronted with several alternatives and one has to choose that alternative which satisfies the requirements at minimum cost. For taking a right decision in all such cases, an experiment may have to be carried out either to discover something about a particular process or to compare the effect of several conditions on the phenomenon under study. 0.3 The effectiveness of an experiment depends to a large extent on the manner in which the data is collected. The method of data collection may adversely affect the conclusions that can be drawn from the experiment. If, therefore, proper designing of an experiment is not made, it may happen that no inferences may be drawn or if drawn, may not answer the questions to which the experimenter is seeking an answer. The designing of an experiment is essentially the determination of the pattern of observations to be collected. A good experimental design is one that answers efficiently and unambiguously those questions which are to be resolved and furnishes the required information with a minimum of experimental effort. For this purpose the experiments may be statistically designed. 0.4 The main advantage of designing an experiment statistically is to obtain unambiguous results at a minimum cost. The statistically designed experiments also enable the experimenter to :

a) isolate the effects of known extraneous factors, b) evaluate the inter-relationship or interaction between the factors, c) evaluate the experimental error, d) determine beforehand the size of the experiment for the specified

precision in the results, e) extract maximum information from the data, and f) remove uncertainty from the conclusions.

IS : 10427 (Part 1) - 1982

4

0.5 This edition 1.1 incorporates Amendment No. 1 (February 1992). Side bar indicates modification of the text as the result of incorporation of the amendment. 0.6 In reporting the results of tests or analyses, 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 provides methods of planning and conducting experiments under various conditions. It also describes the procedures for analysing data recorded from such experiments. The various designs described in the standard are completely randomised design, randomised block design, latin square design, balanced incomplete block design and factorial designs.

2. TERMINOLOGY

2.0 For the purpose of this standard the following definitions shall apply. 2.1 Experimental Design — The arrangement in which an experimental programme is to be conducted and the selection of the levels of one or more factors or factor combinations to be included in the experiment. 2.2 Randomisation — The procedure of allocating treatments at random to experimental units. 2.3 Replication — The repetition of treatments over experimental units under similar conditions. 2.4 Experimental Error — The experimental error is that part of the total variation which cannot be assigned to any given cause or which is not associated with any deliberate variation in the experimental conditions. 2.5 Blocks — A set of homogeneous experimental units which give rise to observations, among which the error variation is expected to be less than the whole set of observations. 2.6 Degrees of Freedom — The number of independent component values which are necessary to determine a statistic.

3. COMPLETELY RANDOMISED DESIGN

3.1 This is the simplest pattern of collecting experimental data when the experimenter is interested in testing the effect of a set of treatments. In this design the experimental units are allotted at random to the treatments, so that every experimental material unit

*Rules for rounding off numerical values ( revised ).

IS : 10427 (Part 1) - 1982

5

gets the same chance of receiving every treatment. In addition the units should be processed in random order at all subsequent stages in the experiment where this order is likely to affect the results. Let the total experiment material consist of n homogeneous experimental units and there be a set of t treatments to be tested. The treatments are then allocated randomly to all the experimental units. For this purpose a table of random numbers may be used to assign the units to the treatments. For instance, let the ith treatment be replicted ri times

so that = n. The treatments may be numbered arbitrarily from

1 to t and the experimental units from 1 to n. ri units selected at random from the n units, using a table of random numbers, may be allotted to the first treatment, r2 units selected randomly from the remaining units to the second treatment, and so on. 3.2 The observations obtained from the n experimental units are to be recorded as shown in Table 1.

Grand total = T1 + T2 + ...... + Tt = G

3.3 From Table 1, the following sums of squares are calculated :

TABLE 1 DISTRIBUTION OF TREATMENTS IN EXPERIMENTAL UNITS TREATMENTS

1 2 3 t

Total T1 T2 T3 ............... Tt

a) Correction factor for mean (CF) : Square the grand total G and divide by n

b) Total sum of squares (Stot) : Square each yij-value and add. From this sum subtract CF,

Stot = – CF

c) Sum of squares between treatments (ST) : Square the treatment totals (Ti) and divide them by the number of observations (ri) on which each total is based and add. Subtract from this CF.

t Σ

y11 y21 y31 yt1 y12 y_2 y32 yt2 y13 y23 y33 yt3 y1r1

y2r2 y3r3

6

d) Sum of squares due to experimental error (SE) :

3.4 The different sum of squares can be entered in the analysis of variance table as shown in Table 2.

3.5 The mean square column in Table 2 is obtained by dividing sum of square component by its degrees of freedom.

3.6 The interest of the experimenter lies in testing the null hypothesis that all treatments effects are equal. For this, the statistic F = MST/ MSE is computed. This value of F is then compared with the tabulated F-value (variance ratio table) at desired level of significance (say 5 percent) for [ ( t-1 ), ( n-t ) ] degrees of freedom. If the calculated value of F exceeds the tabulated value, the null hypothesis is rejected, meaning thereby that there is sufficient reason to believe that all treatment effects are not equal.

3.7 If the null hypothesis of all treatment effects being equal is rejected, proceed to calculate the critical difference between two

TABLE 2 ANALYSIS OF VARIANCE TABLE FOR COMPLETELY RANDOMISED DESIGN

SOURCE OF VARIATION

Between treatments ( t-1 )

Experimental error ( n-t )

ij

2 ij Σ

7

treatments. The critical difference between any two treatments, say ti and tj is given as follows :

Critical difference (CD) =

where tα is the α% value of t-variate with ( n-t ) degrees of freedom, α being the level of significance.

3.7.1 The mean for each treatment is then calculated as yi = Ti/ri.

3.7.2 The absolute difference between any two treatment means is compared with its critical difference. If the difference between the two treatment means is less than the critical difference, the two treatments are not considered to be significantly different. If the absolute difference between two treatment means is greater than the critical difference, it is considered that these are significantly different.

3.8 Merits and Demerits

3.8.1 Merits

3.8.2 Demerits — As the experimental units are heterogeneous, the units that receive one treatment may not be similar to those which receive another treatment and so the entire variation among the units enters into the experimental error. So the experimental error is higher as compared to other designs.

3.9 Complete randomisation is therefore only appropriate, where a) the experimental units are homogeneous, and b) an appreciable fraction of the units is likely to be destroyed or fail

to respond.

3.10 Example 1 — Six hourly samples were taken from each of the three different types of mixes during grinding of final cement for determination of rapid calcium oxide content. The test results of the samples are given in Table 3. It has to be tested whether the different types of mixes produce the same calcium oxide content.

a) It allows complete flexibility as any number of treatments and replicates may be used. The number of replicates, if desired, can be varied from treatment to treatment;

b) The statistical analysis is easy even if the number of replicates are not the same for all treatments or if the experimental errors differ from treatment to treatment; and

c) The relative loss of information due to missing observations is smaller than with any other design and they do not pose any problem in the analysis of data.

1 ri ---- 1

8

3.10.1 The various calculations are as given below:

3.10.2 The analysis of variance table for above experiment is given in Table 4.

3.10.3 The tabulated value of F at 5 percent level of significance, for (2,15) degrees of freedom is 3.68, which is less than the calculated value. Hence the three different types of mixes are significantly different, as far as calcium oxide content is concerned.

TABLE 3 CALCIUM OXIDE CONTENT (IN PERCENT)

( Clause 3.10 )

(1) (2) (3)

43.9 45.0 44.2 44.0 45.1 44.4 43.9 45.1 44.1 44.0 45.2 44.0 43.9 45.0 44.2 44.1 45.0 44.1

Total 263.8 270.4 265.0, G = 799.2

a) Correction factor (CF) = = 35 484.48

b) Total sum of squares

= 4.28 c) Sum of squares between mixes

d) Sum of squares due to error = 4.28 – 4.12 = 0.16

TABLE 4 ANALYSIS OF VARIANCE TABLE

SOURCE OF VARIATION

DEGREES OF FREEDOM

SUM OF SQUARES

Between mixes 2 4.12 2.06 187

Experimental error 15 0.16 0.011

Total 17 4.28

263.8( )2

6 ---------------------- 270.4( )

9

3.10.4 The critical difference for different pairs is calculated next. For instance, the critical difference value at 5 percent level, for mixes I and III is given by:

Critical difference (CD)

= 0.13 3.10.4.1 The average for mix I (y1) = 43.97 and for mix III (y3) = 44.17. Since | | (= 0.20) is greater than the critical difference value, mixes I and III are significantly different at 5 percent significance level.

4. RANDOMISED BLOCK DESIGN

4.1 In this case, the total experimental material is divided into 6 blocks or groups of homogeneous material and then, each block is sub-divided into t experimental units, where t denotes the number of treatments to be tested. 4.2 The t treatments are then allocated to t experimental units within each block at random. Separate random allocation is made for each block. For random allocation, first t treatments are given random numbers from 1 to t. Then, a random number between 1 and t is chosen and whatever number comes, the treatment having that number is allocated to the first experimental unit in a block. The procedure is continued till all the treatments are allocated within a block. The procedure is then repeated with the other blocks. 4.3 The observations obtained from the bt experimental units are to be recorded as given in Table 5.

TABLE 5 DISTRIBUTION OF TREATMENTS IN BLOCKS

BLOCK 1 2 3 . . . . b TOTAL

TREATMENT

1 . . . .

2 . . . .

B1 B2 B3 Bb

10

4.4 From the above table, the following sums of squares are calculated:

4.5 The different sums of squares may now be entered in the analysis of variance table as shown in Table 6.

a) Correction factor (CF) =

c) Sum of squares between treatments (ST)

d) Sum of squares between blocks (SB)

e) Sum of squares due to experimental error (SE) = Stot – ST – SB

TABLE 6 ANALYSIS OF VARIANCE TABLE FOR RANDOMISED BLOCK DESIGN

SOURCE OF VARIATION

DEGREES OF FREEDOM

(1) (2) (3) (4) (5)

Between blocks b-1 MSB = SB/(b-1)

Between treatments t-1 MST = ST/(t-1)

Experimental error ( b-1 ) ( t-1 ) SE = Stot – SB – ST MSE = SE/(b-1)(t-1)

Total bt – 1

2 ) CF–+ + +=

1 b ---

t Σ

b Σ

j 1=

Bj 2

t -------- CF–=–=

2 CF–=

11

4.6 Since the interest of the experimental lies in testing the hypothesis whether all treatment effects are identical, the following statistic is computed :

F = MST/MSE

4.6.1 The calculated value of F is then compared with the tabulated F-value (variance ratio table) at desired level of significance (say 5 percent) for [ ( t-1 ), ( b-1 ) ( t-1 ) ] degrees of freedom. If the calculated value of F is less than the tabulated value, the null hypothesis is not rejected meaning thereby that there is no significant difference among treatments. If the calculated value of F exceeds the tabulated value, reject the null hypothesis meaning thereby that there is sufficient reason to believe that all treatment effects are not equal.

4.7 Merits and Demerits

4.7.1 Merits

4.7.2 Demerits a) The effect of only one factor is investigated, and b) Single grouping of only one factor yields higher experimental

error as compared to more complicated designs.

4.8 Example — An experiment was conducted for ash content of instant tea. For this purpose five different varieties of instant tea were tested in three laboratories. The test results are given in Table 7. It has to be determined whether there is significant difference among varieties of instant tea.

4.8.1 The various calculations are obtained as follows:

a) Correction factor (CF) = = 5 992.00

b) Total sum of squares =

[(20.1)2 + (20.1)2 + ... + (20.5)2] – CF = 9.86

a) By means of grouping, the experimental error is reduced and the treatment comparisons are made more sensitive than with completely randomised designs;

b) Any number of treatments and replicates may be included. However, if the number of treatments is large (20 or more), the efficiency of error control decreases; and

c) The statistical analysis is straightforward. When data from some individual units are missing, the ‘missing plot’ technique enables the available results to be fully utilized.

299.8( )2

12

4.8.2 The analysis of variance table for the above data is given in Table 8.

TABLE 7 ASH CONTENT OF INSTANT TEA

( Clause 4.8 )

VARIETY

c) Sum of squares between varieties of tea

= [ (60.5)2 +…

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.

“! $ ' +-” Satyanarayan Gangaram Pitroda

“Invent a New India Using Knowledge”

“01 ' 5 ” Jawaharlal Nehru

“Step Out From the Old to the New”

“1 +, 1 +” Mazdoor Kisan Shakti Sangathan

“The Right to Information, The Right to Live”

“! > 0 B ” Bharthari—Ntiatakam

“Knowledge is such a treasure which cannot be stolen”

“Invent a New India Using Knowledge”

””

IS 10427-1 (1982): Designs for industrial experimentation, Part 1: Standard designs [MSD 3: Statistical Methods for Quality and Reliability]

© BIS 2004

B U R E A U O F I N D I A N S T A N D A R D S MANAK BHAVAN, 9 BAHADUR SHAH ZAFAR MARG

NEW DELHI 110002

Edition 1.1 (1992-02)

Price Group 7

(Incorporating Amendment No. 1)

© BIS 2004

BUREAU OF INDIAN STANDARDS

This publication is protected under the Indian Copyright Act (XIV of 1957) and reproduction in whole or in part by any means except with written permission of the publisher shall be deemed to be an infringement of copyright under the said Act.

Indian Standard DESIGNS FOR INDUSTRIAL

EXPERIMENTATION PART 1 STANDARD DESIGNS

Quality Control and Industrial Statistics Sectional Committee, EC 3

Chairman Representing

DR P. K. BOSE Indian Association for Productivity, Quality and Reliability (IAPQR), Calcutta

Members

SHRI B. HIMATSINGKA ( Alternate to Dr P. K. Bose )

SHRI B. ANANTH KRISHNANAND National Productivity Council, New Delhi SHRI R. S. GUPTA ( Alternate )

SHRI M. G. BHADE The Tata Iron and Steel Co Ltd, Jamshedpur DR S. K. CHATTERJEE National Institute of Occupational Health,

Ahmadabad SHRI P. K. KULKARNI ( Alternate )

SHRI C. A. CRASTO The Cotton Textiles Export Promotion Council, Bombay

DIRECTOR Indian Agricultural Statistics Research Institute (ICAR), New Delhi

DR S. S. PILLAI ( Alternate ) SHRI Y. GHOUSE KHAN NGEF Ltd, Bangalore

SHRI C. RAJANNA ( Alternate ) SHRI A. LAHIRI Indian Jute Industries’ Research Association,

Calcutta SHRI U. DUTTA ( Alternate )

SHRI R. N. MONDAL Tea Board, Calcutta DR S. P. MUKHERJEE Calcutta University, Calcutta SHRI A. N. NANKANA Indian Statistical Institute, Calcutta SHRI M. K. V. NARAYAN The Indian Tube Co Ltd, Jamshedpur

SHRI P. N. GUPTA ( Alternate ) SHRI T. R. PURI Army Statistical Organization, Ministry of Defence,

New Delhi SHRI R. D. AGRAWAL ( Alternate )

SHRI E. K. RAMACHANDARAN National Test House, Calcutta SHRI S. MONDAL ( Alternate )

SHRI RAMESH SHANKER Directorate General of Inspection, Ministry of Defence, New Delhi

SHRI N. S. SENGAR ( Alternate ) ( Continued on page 2 )

IS : 10427 (Part 1) - 1982

2

Members Representing

SHRI T. V. RATNAM The South India Textile Research Association, Coimbatore

DR D. RAY Defence Research and Development Organization, Ministry of Defence, New Delhi

SHRI G. SAMSON Indian Telephone Industries Ltd, Bangalore SHRI P. V. RAO ( Alternate )

SHRI A. SANJEEVA RAO Shriram Fibres Limited, Madras SHRI R. SETHURAMAN Electronics Trade and Technology Development

Corporation Ltd, New Delhi SHRI N. SRINIVASAN ( Alternate )

SHRI S. SHIVARAM The State Trading Corporation of India Ltd, New Delhi

DR S. S. SRIVASTAVA Central Statistical Organization, New Delhi SHRI S. SUBRAMU Steel Authority of India Ltd, New Delhi SHRI Y. K. BHAT,

Director (Stat) Director General, ISI ( Ex-officio Member )

Secretary

Industrial Statistics Subcommittee, EC 3 : 7

Convener

DR P. K. BOSE Indian Association for Productivity, Quality and Reliability (IAPQR), Calcutta

Members

SHRI B. K. CHOWDHURY ( Alternate to Dr P. K. Bose )

DIRECTOR Indian Agricultural Statistics Research Institute (ICAR), New Delhi

DR B. B. P. S. GOEL ( Alternate ) DR S. P. MUKHERJEE Calcutta University, Calcutta DR M. PADMANABHAN National Sample Survey Organisation, New Delhi SHRI T. R. PURI Army Statistical Organization, Ministry of Defence,

New Delhi SHRI R. D. AGRAWAL ( Alternate )

SHRI C. RAJANNA NGEF Ltd, Bangalore SHRI Y. GHOUSE KHAN ( Alternate )

DR D. RAY Defence Research and Development Organization, Ministry of Defence, New Delhi

SHRI B. K. SARKAR Indian Statistical Institute, Calcutta SHRI D. R. SEN Delhi Cloth Mills (DCM), Delhi DR S. S. SRIVASTAVA Central Statistical Organization, New Delhi

IS : 10427 (Part 1) - 1982

3

Indian Standard DESIGNS FOR INDUSTRIAL

EXPERIMENTATION PART 1 STANDARD DESIGNS

0. F O R E W O R D 0.1 This Indian Standard was adopted by the Indian Standards Institution on 15 December 1982, after the draft finalized by the Quality Control and Industrial Statistics Sectional Committee had been approved by the Executive Committee. 0.2 Industrial organizations are constantly faced with the problem of decision making regarding product/process design, process specifications, quality improvement, dominant factors affecting quality, cost reduction, import substitution, etc. In all such problems, one is confronted with several alternatives and one has to choose that alternative which satisfies the requirements at minimum cost. For taking a right decision in all such cases, an experiment may have to be carried out either to discover something about a particular process or to compare the effect of several conditions on the phenomenon under study. 0.3 The effectiveness of an experiment depends to a large extent on the manner in which the data is collected. The method of data collection may adversely affect the conclusions that can be drawn from the experiment. If, therefore, proper designing of an experiment is not made, it may happen that no inferences may be drawn or if drawn, may not answer the questions to which the experimenter is seeking an answer. The designing of an experiment is essentially the determination of the pattern of observations to be collected. A good experimental design is one that answers efficiently and unambiguously those questions which are to be resolved and furnishes the required information with a minimum of experimental effort. For this purpose the experiments may be statistically designed. 0.4 The main advantage of designing an experiment statistically is to obtain unambiguous results at a minimum cost. The statistically designed experiments also enable the experimenter to :

a) isolate the effects of known extraneous factors, b) evaluate the inter-relationship or interaction between the factors, c) evaluate the experimental error, d) determine beforehand the size of the experiment for the specified

precision in the results, e) extract maximum information from the data, and f) remove uncertainty from the conclusions.

IS : 10427 (Part 1) - 1982

4

0.5 This edition 1.1 incorporates Amendment No. 1 (February 1992). Side bar indicates modification of the text as the result of incorporation of the amendment. 0.6 In reporting the results of tests or analyses, 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 provides methods of planning and conducting experiments under various conditions. It also describes the procedures for analysing data recorded from such experiments. The various designs described in the standard are completely randomised design, randomised block design, latin square design, balanced incomplete block design and factorial designs.

2. TERMINOLOGY

2.0 For the purpose of this standard the following definitions shall apply. 2.1 Experimental Design — The arrangement in which an experimental programme is to be conducted and the selection of the levels of one or more factors or factor combinations to be included in the experiment. 2.2 Randomisation — The procedure of allocating treatments at random to experimental units. 2.3 Replication — The repetition of treatments over experimental units under similar conditions. 2.4 Experimental Error — The experimental error is that part of the total variation which cannot be assigned to any given cause or which is not associated with any deliberate variation in the experimental conditions. 2.5 Blocks — A set of homogeneous experimental units which give rise to observations, among which the error variation is expected to be less than the whole set of observations. 2.6 Degrees of Freedom — The number of independent component values which are necessary to determine a statistic.

3. COMPLETELY RANDOMISED DESIGN

3.1 This is the simplest pattern of collecting experimental data when the experimenter is interested in testing the effect of a set of treatments. In this design the experimental units are allotted at random to the treatments, so that every experimental material unit

*Rules for rounding off numerical values ( revised ).

IS : 10427 (Part 1) - 1982

5

gets the same chance of receiving every treatment. In addition the units should be processed in random order at all subsequent stages in the experiment where this order is likely to affect the results. Let the total experiment material consist of n homogeneous experimental units and there be a set of t treatments to be tested. The treatments are then allocated randomly to all the experimental units. For this purpose a table of random numbers may be used to assign the units to the treatments. For instance, let the ith treatment be replicted ri times

so that = n. The treatments may be numbered arbitrarily from

1 to t and the experimental units from 1 to n. ri units selected at random from the n units, using a table of random numbers, may be allotted to the first treatment, r2 units selected randomly from the remaining units to the second treatment, and so on. 3.2 The observations obtained from the n experimental units are to be recorded as shown in Table 1.

Grand total = T1 + T2 + ...... + Tt = G

3.3 From Table 1, the following sums of squares are calculated :

TABLE 1 DISTRIBUTION OF TREATMENTS IN EXPERIMENTAL UNITS TREATMENTS

1 2 3 t

Total T1 T2 T3 ............... Tt

a) Correction factor for mean (CF) : Square the grand total G and divide by n

b) Total sum of squares (Stot) : Square each yij-value and add. From this sum subtract CF,

Stot = – CF

c) Sum of squares between treatments (ST) : Square the treatment totals (Ti) and divide them by the number of observations (ri) on which each total is based and add. Subtract from this CF.

t Σ

y11 y21 y31 yt1 y12 y_2 y32 yt2 y13 y23 y33 yt3 y1r1

y2r2 y3r3

6

d) Sum of squares due to experimental error (SE) :

3.4 The different sum of squares can be entered in the analysis of variance table as shown in Table 2.

3.5 The mean square column in Table 2 is obtained by dividing sum of square component by its degrees of freedom.

3.6 The interest of the experimenter lies in testing the null hypothesis that all treatments effects are equal. For this, the statistic F = MST/ MSE is computed. This value of F is then compared with the tabulated F-value (variance ratio table) at desired level of significance (say 5 percent) for [ ( t-1 ), ( n-t ) ] degrees of freedom. If the calculated value of F exceeds the tabulated value, the null hypothesis is rejected, meaning thereby that there is sufficient reason to believe that all treatment effects are not equal.

3.7 If the null hypothesis of all treatment effects being equal is rejected, proceed to calculate the critical difference between two

TABLE 2 ANALYSIS OF VARIANCE TABLE FOR COMPLETELY RANDOMISED DESIGN

SOURCE OF VARIATION

Between treatments ( t-1 )

Experimental error ( n-t )

ij

2 ij Σ

7

treatments. The critical difference between any two treatments, say ti and tj is given as follows :

Critical difference (CD) =

where tα is the α% value of t-variate with ( n-t ) degrees of freedom, α being the level of significance.

3.7.1 The mean for each treatment is then calculated as yi = Ti/ri.

3.7.2 The absolute difference between any two treatment means is compared with its critical difference. If the difference between the two treatment means is less than the critical difference, the two treatments are not considered to be significantly different. If the absolute difference between two treatment means is greater than the critical difference, it is considered that these are significantly different.

3.8 Merits and Demerits

3.8.1 Merits

3.8.2 Demerits — As the experimental units are heterogeneous, the units that receive one treatment may not be similar to those which receive another treatment and so the entire variation among the units enters into the experimental error. So the experimental error is higher as compared to other designs.

3.9 Complete randomisation is therefore only appropriate, where a) the experimental units are homogeneous, and b) an appreciable fraction of the units is likely to be destroyed or fail

to respond.

3.10 Example 1 — Six hourly samples were taken from each of the three different types of mixes during grinding of final cement for determination of rapid calcium oxide content. The test results of the samples are given in Table 3. It has to be tested whether the different types of mixes produce the same calcium oxide content.

a) It allows complete flexibility as any number of treatments and replicates may be used. The number of replicates, if desired, can be varied from treatment to treatment;

b) The statistical analysis is easy even if the number of replicates are not the same for all treatments or if the experimental errors differ from treatment to treatment; and

c) The relative loss of information due to missing observations is smaller than with any other design and they do not pose any problem in the analysis of data.

1 ri ---- 1

8

3.10.1 The various calculations are as given below:

3.10.2 The analysis of variance table for above experiment is given in Table 4.

3.10.3 The tabulated value of F at 5 percent level of significance, for (2,15) degrees of freedom is 3.68, which is less than the calculated value. Hence the three different types of mixes are significantly different, as far as calcium oxide content is concerned.

TABLE 3 CALCIUM OXIDE CONTENT (IN PERCENT)

( Clause 3.10 )

(1) (2) (3)

43.9 45.0 44.2 44.0 45.1 44.4 43.9 45.1 44.1 44.0 45.2 44.0 43.9 45.0 44.2 44.1 45.0 44.1

Total 263.8 270.4 265.0, G = 799.2

a) Correction factor (CF) = = 35 484.48

b) Total sum of squares

= 4.28 c) Sum of squares between mixes

d) Sum of squares due to error = 4.28 – 4.12 = 0.16

TABLE 4 ANALYSIS OF VARIANCE TABLE

SOURCE OF VARIATION

DEGREES OF FREEDOM

SUM OF SQUARES

Between mixes 2 4.12 2.06 187

Experimental error 15 0.16 0.011

Total 17 4.28

263.8( )2

6 ---------------------- 270.4( )

9

3.10.4 The critical difference for different pairs is calculated next. For instance, the critical difference value at 5 percent level, for mixes I and III is given by:

Critical difference (CD)

= 0.13 3.10.4.1 The average for mix I (y1) = 43.97 and for mix III (y3) = 44.17. Since | | (= 0.20) is greater than the critical difference value, mixes I and III are significantly different at 5 percent significance level.

4. RANDOMISED BLOCK DESIGN

4.1 In this case, the total experimental material is divided into 6 blocks or groups of homogeneous material and then, each block is sub-divided into t experimental units, where t denotes the number of treatments to be tested. 4.2 The t treatments are then allocated to t experimental units within each block at random. Separate random allocation is made for each block. For random allocation, first t treatments are given random numbers from 1 to t. Then, a random number between 1 and t is chosen and whatever number comes, the treatment having that number is allocated to the first experimental unit in a block. The procedure is continued till all the treatments are allocated within a block. The procedure is then repeated with the other blocks. 4.3 The observations obtained from the bt experimental units are to be recorded as given in Table 5.

TABLE 5 DISTRIBUTION OF TREATMENTS IN BLOCKS

BLOCK 1 2 3 . . . . b TOTAL

TREATMENT

1 . . . .

2 . . . .

B1 B2 B3 Bb

10

4.4 From the above table, the following sums of squares are calculated:

4.5 The different sums of squares may now be entered in the analysis of variance table as shown in Table 6.

a) Correction factor (CF) =

c) Sum of squares between treatments (ST)

d) Sum of squares between blocks (SB)

e) Sum of squares due to experimental error (SE) = Stot – ST – SB

TABLE 6 ANALYSIS OF VARIANCE TABLE FOR RANDOMISED BLOCK DESIGN

SOURCE OF VARIATION

DEGREES OF FREEDOM

(1) (2) (3) (4) (5)

Between blocks b-1 MSB = SB/(b-1)

Between treatments t-1 MST = ST/(t-1)

Experimental error ( b-1 ) ( t-1 ) SE = Stot – SB – ST MSE = SE/(b-1)(t-1)

Total bt – 1

2 ) CF–+ + +=

1 b ---

t Σ

b Σ

j 1=

Bj 2

t -------- CF–=–=

2 CF–=

11

4.6 Since the interest of the experimental lies in testing the hypothesis whether all treatment effects are identical, the following statistic is computed :

F = MST/MSE

4.6.1 The calculated value of F is then compared with the tabulated F-value (variance ratio table) at desired level of significance (say 5 percent) for [ ( t-1 ), ( b-1 ) ( t-1 ) ] degrees of freedom. If the calculated value of F is less than the tabulated value, the null hypothesis is not rejected meaning thereby that there is no significant difference among treatments. If the calculated value of F exceeds the tabulated value, reject the null hypothesis meaning thereby that there is sufficient reason to believe that all treatment effects are not equal.

4.7 Merits and Demerits

4.7.1 Merits

4.7.2 Demerits a) The effect of only one factor is investigated, and b) Single grouping of only one factor yields higher experimental

error as compared to more complicated designs.

4.8 Example — An experiment was conducted for ash content of instant tea. For this purpose five different varieties of instant tea were tested in three laboratories. The test results are given in Table 7. It has to be determined whether there is significant difference among varieties of instant tea.

4.8.1 The various calculations are obtained as follows:

a) Correction factor (CF) = = 5 992.00

b) Total sum of squares =

[(20.1)2 + (20.1)2 + ... + (20.5)2] – CF = 9.86

a) By means of grouping, the experimental error is reduced and the treatment comparisons are made more sensitive than with completely randomised designs;

b) Any number of treatments and replicates may be included. However, if the number of treatments is large (20 or more), the efficiency of error control decreases; and

c) The statistical analysis is straightforward. When data from some individual units are missing, the ‘missing plot’ technique enables the available results to be fully utilized.

299.8( )2

12

4.8.2 The analysis of variance table for the above data is given in Table 8.

TABLE 7 ASH CONTENT OF INSTANT TEA

( Clause 4.8 )

VARIETY

c) Sum of squares between varieties of tea

= [ (60.5)2 +…