UNIT 16

LATIN SQUARE DESIGN

Structure
16.1 Introduction Steps for Computing Different
Sum of Squares from the Data
Expected Learning Outcomes
The ANOVA Table for Latin
16.2 Definition and Concept of
Square Design
Latin Square Design
16.5 Advantages and
Defining the Latin Square Design
Disadvantages of Latin
A Brief History of Latin Square Square Design
Design
16.6 Efficiency Comparison with
Concept Behind the Latin Square Randomised Block Design
Design and Completely
16.3 Layout of Latin Square Randomised Design
Design 16.7 Latin Square Design with
16.4 Statistical Analysis of Latin One Missing Observation
Square Design 16.8 Summary
Linear Model for the Analysis of 16.9 Terminal Questions
Latin Square Design
16.10 Answers / Solutions
Hypotheses to be Tested

Estimation of Model Parameters

16.1 INTRODUCTION
After studying the Completely Randomised Design (CRD) and Randomised
Block Design (RBD) in the previous two units, we shall now explain and make
detailed study of the third type of design in this unit, which is “Latin Square
Design” (LSD). In the first two units we mention the “Soil of the Fertility” in
agricultural experiments which was observed to play very important role in the
making the ‘Blocks’ as well as the experimental units, that is, ‘Plots’ from
which the ultimate data are measured or recorded for its analysis. We
mentioned the effect of sizes and shapes of blocks and plots on the results of
the experiment and, accordingly, mentioned certain rules of constructing these
in any type of experiment; be it agricultural, educational, psychological,
economical or other experiments. While dealing with the Randomised Block
Design experiment in the Unit 15, we defined and explained the term “Blocking
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 Block 4 Design of Experiments

Technique” and consequently, associated the construction of blocks and plots

with the direction of “Fertility Gradient” of the agricultural field (area) in case of
agricultural experiments or the experimental area in experiments other than
agricultural ones.
When mentioning about some advantages of Randomised Block Design over
Completely Randomised Design, the former one was said to be an
advancement over the latter one, in the sense that it follows the third principle
of Design of Experiments also, namely, “Local Control”, in addition to the first
two principles which Completely Randomised Design follows. The principle of
local control was seen to be governed by the Blocking Technique. However, in
Randomised Block Design, during the process of using Blocking Technique,
the idea of direction of fertility gradient, which is a very crucial factor for the
experiment was not known to the experimenter. If this knowledge would have
been to the experimenter, he/she could use it for efficiently deciding the
direction to which the blocks are to be made. This fact has been discussed in
Sub-section 15.2.1 of unit 15 and accordingly three guidelines for blocking
method have been mentioned. The third guideline mentioned there is as
follows:
“When the fertility gradient occurs in two directions with both gradients
equally strong and perpendicular to each other, use blocks that are as
square as possible or choose some other designs, like, “Latin Square
Design” (LSD)”.
Thus, the purpose for considering Latin Square Design in experiments,
particularly in field experiments, is clear. However, this is not the only reason
for using Latin Square Design; there are also some other reasons which we
shall discuss afterwards. Obviously, you can say that Latin Square Design is
an advancement of Randomised Block Design.
In this unit, we shall exclusively discuss Latin Square Design and present its
concept, layout with practical examples and the method of applying statistical
techniques for the analysis of the data obtained through a Latin Square
Design. Section 16.2 shall first present the concept of Latin Squares, used by
Mathematicians and its resemblance with Latin Square Design, definition of
Latin Square Design and reasons of attempting sometimes Latin Square
Design instead of other designs. This section also presents the concept of the
design with a brief note on its history. Section 16.3 will discuss its layout with
examples. Section 16.4 will be devoted to the statistical analysis of a Latin
Square Design with the description of those components of the analytical part
which lead the study towards the problem of testing of some hypotheses with
the help of Analysis of Variance (ANOVA) technique. This section will also
mention about the hypotheses which could be tested for the data generated
from a Latin Square Design, the method of partitioning the total variability into
a number of its components and the method of constructing the ANOVA Table
which could be used for computing F-ratios for different causes of variations.
Section 16.5 will present some advantages and limitations of Latin Square
Design, whereas Section 16.6 presents the efficiency comparison of it with
respect to Completely Randomised Design and Randomised Block Design.
Section 16.7 will discuss the problem of missing observations in Latin Square
Design and will suggest how to resolve the problem.
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Unit 16 Latin Square Design

Expected Learning Outcomes

After studying this unit, you should be able to:
❖ define a Latin Square Design (LSD) and explain that in what sense it is
considered to be an advancement over Randomised Block Design (RBD);
❖ describe the layout of Latin Square Design and explain how the
treatments should be assigned to experimental units under this design;
❖ discuss that how the technique of Analysis of Variance can be utilised in
order to analyse the data coming from this design, in addition to the model
to be used, different hypotheses to be tested and computation of Sum of
Squares which are needed for completing the Analysis of Variance
process so that conclusions of the analysis could be obtained;
❖ describe the application of Latin Square design (LSD) is advantageous or
disadvantageous for analysing the data as compared to Randomised
Block Design;
❖ explain how the efficiency of Latin Square design (LSD) can be computed
with respect to Randomised Block Design; and
❖ discuss the problem of missing observation(s) and how it can be tackled
in Latin Square design.

16.2 DEFINITION AND CONCEPT OF LATIN

SQUARE DESIGN
In fact, the term “Latin Square” is borrowed directly from Mathematics.
Before proceeding to the discussion of Latin Square Design, let us describe
what a “Latin Square” means in Mathematics. According to the
mathematicians,
“A Latin Square (may be abbreviated as “LS”) is a (n x n) Square Table filled
with n different symbols (say, Greek alphabets α, β, γ, δ, etc.), in such a way
that each symbol occurs exactly once in each row and exactly once in each
column.”
In this sense, the study of Latin Squares is a matter of Mathematics in general
and of Combinatorics, in particular. We shall present here a brief description of
Latin Square only in reference to Latin Square Design.
With any given number of letters, we can see that more than one number of
Latin Squares can be formed. As an example, with 4 Greek alphabets α, β, γ,
δ, we can construct the following Latin squares:
LS-1 LS-2 LS-3

C1 C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4
R1 α β γ δ R1 γ δ α β R1 α δ γ β
R2 β γ δ α R2 β γ δ α R2 β α δ γ
R3 γ δ α β R3 α β γ δ R3 γ β α δ
R4 δ α β γ R4 δ α β γ R4 δ γ β α

where R1, R2, R3, R4 stands for the row numbers and C1, C2, C3, C4 for the
column numbers. 167
 Block 4 Design of Experiments

In all the above three Latin Squares, it can be seen that the tables are
squares, each having four rows and four columns, and the alphabets are so
arranged within each of them in such a way that each alphabet occurs once in
a row and once in a column. However, the above three Latin Squares are not
the only squares we can construct from 4 alphabets. It is a matter of
Combinatorics and Mathematics to list all the possible squares with p numbers
(or, symbols). In fact, given a set of symbols, we can construct other Latin
squares by simply permuting rows, permuting columns and permuting the
symbols. As for example, you can see that LS-2 is obtained from LS-1 by
changing R1 and R3 and LS-3 is obtained from LS-1 by changing C2 and C4.
Transformed Set and Standard Squares: The totality of Latin Square
obtained from a single Latin Square by permuting the rows, columns and
letters is called a “Transformed Set”. A pp Latin Square with p letters in the
natural order occurring in the first row and in the first column is called a
“Standard Square”. Thus, LS-1 is a standard square whereas LS-2 and LS-
3 are ‘Transformed Sets’ from LS-1.
It can be proved that the number of squares that can be generated from a
given standard square of order p  p, by permutation of rows, columns and
letters is ( p!) . All these squares are not necessarily different. Also, we obtain
p! (p − 1) ! different Latin Square by permuting all the p columns and the (p − 1)
rows except the first row. Thus, with p = 3, we can generate 12 Latin Square;
with p = 4, we get 144 Latin Square and so on.
Remark 16.1: Although, the study of Latin Square is a simple matter to
mathematicians, it is multifaceted to a statistician also, particularly to an
experimental designer who used to be engaged in designing different kinds of
experiments in an efficient manner. The name “Latin Square Design” (LSD)
is, in fact, borrowed from the concept of Latin squares due to the reason that
layout of a Latin Square Design generally resembles with a Latin Square. We
shall see in Section 16.3 that for deciding the layout of a Latin Square Design
how the concept of Latin Square Design is useful. Actually, Latin Square
Design is considered to be an example of Latin Square.
We shall now discuss in the next Sub-section how an Latin Square Design can
be given a precise definition, what is the concept behind Latin Square and in
what sense Latin Square Design has an upper hand over the previously
discussed designs.
16.2.1 Defining the Latin Square Design
Keeping in mind the structure of a Latin Square, as defined above; it is quite
easy to put forward the precise definition of an Latin Square Design. We have
the following definition of Latin Square Design:
A “Latin Square Design” (LSD) is a method of placing some treatments
(say, t treatments) in a balanced fashion within a square field or experimental
area, each one repeated t times in such a way that each treatment appears at
random exactly one time in each row and each column in the design.
As mentioned in the Remark 16.1, the Latin Square Design gets its name from
the fact that we can write it as a Latin Square with Latin letters to correspond
168 to the treatments. The treatment factor levels are the Latin letters in Latin
Unit 16 Latin Square Design

Square Design. The number of rows and columns has to correspond to the
number of treatment levels. So, if we have four treatments then we would
need to have four rows and four columns in order to create a Latin square.
This gives us a design where we have each of the treatments in each row and
in each column only once. We shall show afterwards that this kind of design is
used to reduce systematic error occurred not only due to rows (treatments)
but also due to columns of the square. This indicates towards the fact that
while in Randomised Block Design, only one blocking variable is removed;
the Latin Square Design designs are carefully constructed to allow the
removal of two blocking variables simultaneously.
Another important fact about Latin Square Design is that while the process of
removal of two blocking factors is possible simultaneously; this process is
accomplished with the process of reducing the number of experimental units
also needed to conduct the experiment. For illustrating this fact, let us
consider the Latin Square Design with 4 treatments A, B, C and D. We
observe that if we use a simple random design, it will require 4  4  4 = 64
experimental units, while Latin Square Design needs only 4  4 = 16
experimental units; which is a reduction of 75% in the number of required
experimental units.
16.2.2 A Brief History of Latin Square Design
As pointed out earlier, the Latin Square Design (LSD) borrows its name from
the famous works on Latin Squares by a number of mathematicians, which
have a long history. The concept probably originated with problems
concerning the movement and disposition of pieces on a chess board.
According to a work of Preece (1983), the history of Latin Square dates back
to 1624. However, Euler (1782) was attributed for the systematic development
and study of Latin Squares and their combinatorial properties which was
carried on by Cayley (1877 – 1890).
In the year 1925, R. A. Fisher, at Rothamsted Experimental Station in
Harpenden, recommended that the concept of Latin Squares can be applied
for agricultural crop experiments. In the same year, Ronald Aylmer (1925) was
also of the same opinion and hence, introduced the Latin Square Designs in
Statistics. At about the same time, Jerzy Neyman developed the same idea
during his doctoral study at the University of Warsaw. However, there is
evidence of their much earlier use in experiments.
16.2.3 Concept Behind the Latin Square Design
Now we shall discuss under what situations Latin Square Design are useful for
reducing the Experimental errors, so as to increase the efficiency of the design.
We know that in Randomised Block Design, the blocking technique is used to
reduce the experimental error by eliminating the contribution of known sources
of variation among the experimental units, which is the idea of one of the
principles of Design of Experiments (DOE), namely, the “Local Control”. This
is done by dividing the experimental area into Blocks, each Block consisting of
some experimental units (that is, plots), such that variability within each block is
minimised and variability among blocks is maximised. One such known source
of variation in field experiments which highly affect the outputs of experimental
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units is the “Fertility of Soil” or “Fertility Gradient”. You know that Randomised
Block Design considers only the Unidirectional Fertility Gradient in order to apply
the principle of local control and accordingly, the shape and direction of blocks
are decided to control (or, eliminate) only one source of nuisance variability, that
is, Unidirectional Fertility Gradient. This fact is also illustrated with the help of
Examples 1 and 2 under Sub-section 15.2.2 of Unit 15.
Sometimes, we may come across with a number of experiments where there
might be two sources of nuisance variability. For instance, in case a farmer
has a field; whose soil fertility might change simultaneously in two
perpendicular directions; say, from North to South direction and from East to
West direction due to many of the other reasons. As for instance, there might
be a Humidity Gradient in the field from north to south due to a slop in the level
of the field in this direction and at the same time there might be a Sunshine
Gradient from East to West direction due to more exposure of sunlight to the
western part of the field as compared to the eastern part. Both of these might
be causes of variation in the fertility of the soil. Let for the concerned field,
output (yield) is expected to vary from West to East due to fertility gradient
(obviously, in horizontal direction or, in rows) causing due to sunshine
exposures and from North to South (that is, in vertical direction or, in columns)
causing due to humidity changes. This means that due to any of the reasons,
there would be two-directional variation in the whole field. So, there are two
sources of nuisance variability, due to which the output of the field may vary
both in rows and columns. Therefore, in order to remove the variation for these
two sources simultaneously from the experimental error variation, both rows
and columns can be used as blocking factors. Latin Square Design is the
design which uses the concept of two blocking factors. Whenever we have
situations where there arises the need for more than one blocking factor, a
Latin Square Design allows us to remove all these sources of variations using
blocking technique simultaneously for all the factors. In fact, in Latin Square
Design (LSD), a field is blocked into columns and rows, that is, each row is a
level of the Row Factor, and each column is a level of the Column Factor. In
other words, we can remove the variation from our measured response in both
directions if we consider both rows and columns as Factors in our design.
We know that in experimental designs, some treatments are applied randomly
with replications on plots which are grouped into different blocks. In Latin
Square Design, treatments are assigned at random within rows and columns,
with each treatment appearing once per row and once per column. Therefore,
Latin Square Design consists of equal number of rows, columns and
treatments, that is, Latin Square Design is actually a Latin Square or it is an
example of Latin Square.
The Latin Square Design (LSD), perhaps, represents the most popular
alternative design when two blocking factors need to be controlled
simultaneously. It is useful where the experimenter desires to control variation
in two different directions. It is to be mentioned here that Latin Square Design
is not only useful in field experiments but also equally useful in industrial
experimentation as well as other experiments.
We can explain the concept of Latin Square Design with the help of the
following example:
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Unit 16 Latin Square Design

Assuming that a factory produces an item with six technicians and the same
number of machines. Also assume that columns represent the technicians,
and the rows represent the machines. Then, obviously we have a 6  6
squares with six rows and six columns and we can randomly assign the
specific technicians to a row and the specific machines to a column. Let the six
treatments are six different protocols for producing the item. The interest of the
concerned manufacturer in this experimentation is the average time needed to
produce each item. If technicians and machines both have an effect on the
time required to produce the item, which is a very common phenomenon in
manufacturing system, then by using a Latin Square Design, this variation due
to technicians or machines will be effectively removed from the analysis.
Now, you may like to answer the following Self-Assessment Question:

SAQ 1
Define a Latin Square Design (LSD). Explain how the name Latin Square
Design is derived.

16.3 LAYOUT OF LATIN SQUARE DESIGN

In Sub-section 16.2.1, we have shown that from a given Standard Latin
Square for a given number of letters; it is possible to construct more than one
Latin Square. Since, Latin Square Design is virtually based upon a Latin
Square in the sense that allotment of treatments to different rows and columns
is made following the rule of a Latin Square; this means that given a Latin
Square Design with some specific number of treatments, it is possible to
construct different Latin Square Designs all of which satisfy the given criterion.
Therefore, any of them can be chosen to represent the layout of the
concerned Latin Square Design.
Let us clarify and illustrate the above points with the help of the following
example:
Let in an experiment, to be conducted by an international company producing
different types of nutritional diets for young children, the objective of the
company be to compare effects of 4 diets say, A, B, C and D on the growth
rate of children in India during a certain period of time as the variate under
study. Since the children to be selected for the experiment may come from
different socio-economic groups (or, with different social status) and from
different ages groups; it was intended to eliminate the variation due to social
status and ages of the children. Therefore, social status and age are the two
factors whose variabilities are to be eliminated from the experimental error. Let
us denote these two factors respectively by U and V.
In order to represent this experiment in the form of an appropriate design, let
us use a Latin Square Design. Therefore, depending upon the number of
diets, there must be 16 children as experimental units, each of which should
be selected from a different social status and age-group combination. It is,
therefore, necessary that there should be four children belonging to each
social status group and each of these four children should come from a
different age group. In the language of an agricultural experiment then, we can
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 Block 4 Design of Experiments

say that there are p = 4 treatments and accordingly, the field is a perfect
square field having 4 rows and 4 columns. Let us denote the factor ‘Social
Status’ by U, consisting of four levels arranged in 4 rows R1, R2, R3 and R4
and the factor ‘Age Group’ by V consisting of four levels arranged in four
columns C1, C2, C3 and C4. As mentioned above, the aim behind using the
Latin Square Design for the analysis is to control the two nuisance variability
caused due to Factors U and V.
Thus, the experiment should be a 4  4 Latin Square Design with 4
treatments. Let us, therefore, consider the following standard Latin Square
Design for deciding the layout:
LSD-1

C1 C2 C3 C4
R1 A B C D
R2 B C D A
R3 C D A B
R4 D A B C

It may be considered as one of the layouts of the design. In this, we decide to

allocate the first Treatment A randomly to the first levels of both social status
and age groups (that is, of the factors U and V); Treatment B to first level of U
but second level of V; Treatment C to first level of factor U and third level of V
and Treatment D to the first level of U but fourth level of V. Thus, the first
horizontal block is completely allocated with a particular treatment to each plot.
As soon as the first row is finished, all other rows can automatically be
allocated following a specific rule because it is a standard Latin Square
Design.
The question then arises whether this is the only way to allocate treatments to
cells in Latin Square Design. The answer is ‘No’. There are other ways for this
allocation, since we may use other transformed sets for the purpose. For
example, we may use the layouts given by LS-2 or LS-3 of Section 16.2 which
are as follows:
LSD-2 LSD-3
C1 C2 C3 C4 C1 C2 C3 C4
R1 C D A B R1 A D C B
R2 B C D A R2 B A D C
R3 A B C D R3 C B A D
R4 D A B C R4 D C B A

or, the following layouts:

LSD-4 LSD-5

C1 C2 C3 C4 C1 C2 C3 C4
R1 B C D A R1 A C B D
R2 D B A C R2 B A D C

R3 A D C B R3 D B C A
R4 C A B D R4 C D A B

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Unit 16 Latin Square Design

Obviously, the randomisation of treatments should be done either row-wise or

column-wise with a precaution that each treatment should occur once in a row
and once in a column.
Now, you may try to answer the following Self-Assessment Question:

SAQ 2
Mention the rule of allocating different treatments to different rows and
columns in case the design chosen is Latin Square Design. If there be 5
treatments denoted by μ, θ, φ, α and β ; show how would you assign them to
rows and columns in order to prepare the layout of a Latin Square Design,
given that the treatments assigned to first column are in the sequence
μ, α, φ, θ and β from top to bottom.

16.4 STATISTICAL ANALYSIS OF LATIN

SQUARE DESIGN
Till now what we have discussed in the previous sections, it is clear that in
Latin Square Design, there are three factors which affect the output of the
experiment; namely, effect due to the (i) rows which show the levels of one of
the nuisance variable (factor); (ii) columns which show the levels of the
another nuisance variable (factor) and the (iii) treatments which are applied to
the experimental units; all with p levels. In this sense, the analysis of the data
collected from a Latin Square Design should be analysed as a Three-way
classified data having p3 observations. But, due to special type of allocation of
treatments to each cell, there is only one observation per cell instead of k
observations per cell as it would be in the usual Three-way classified data.
Therefore, for the analysis of a Latin Square Design, we are supposed to
obtain only the Sum of Squares due to each of the three factors and Error
Sum of Square as usual. Let us denote the row effect, column effect and
treatment effect, respectively, by the letters U, V and T. In order to obtain the
Row and Column Sum of Squares, we can apply the method as used for this
purpose in Randomised Block Design. However, we have an additional sum of
square which is ‘Treatment Sum of Square’. For the computation of this Sum
of Squares, we first prepare a separate table collecting the observations
treatment-wise from each row or column of the layout. As soon as these three
Sum of Squares are computed, the Error Sum of Squares (ESS) can be
computed by subtracting all these three Sum of Squares from the Total Sum of
Squares.
16.4.1 Linear Model to be Used for the Analysis of
Latin Square Design
Since there is one observation per cell, we cannot get any interaction Sum of
Squares between the three factors. Hence, the fixed effect non-interaction
linear model will be used, which is given by:
xijk = μ + αi + β j + τk + eijk ; … (16.1)

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 Block 4 Design of Experiments

where xijk stands for the observation coming from the ith row, jth column and
under the kth treatment for i, j, k = 1, 2, …, p; μ, α i , β j and τ k being
respectively, the Fixed General Effect, Effect due to the ith level of the Factor U
(Row Effects), Effect due to the jth level of the Factor V (Column Effects) and
Effect due to the kth level of the Factor T (Treatment Effects). The p2 random
variables eijk is the error component, assumed to be independently and
normally distributed with mean zero and variance σ 2 , that is, eijk s are i.i.d.
random variables where eijk ~ N 0, σ2 . ( )
16.4.2 Hypotheses to be Tested
In Latin Square Design (LSD), since we come across with the analysis of
effects of three different factors over the output (yield), namely, row, column
and treatment effects, we have the following three corresponding null
hypotheses along with corresponding alternative hypotheses for testing their
significance: The hypotheses are:
(i) H0R : α1 = α2 =  = αp = 0;

Against the alternative hypothesis

H1R : α1  α2    αp or equivalently,

H1R : At least one αi is not zero.

(ii) H0C : β1 = β2 =  = βp = 0;

Against the alternative hypothesis

H1C : β1  β2    βp or equivalently,

H1C : At least one β j is not zero.

(iii) H0Tr : τ1 = τ2 =  = τp = 0

Against the alternative hypothesis

H0Tr : τ1  τ2    τp or equivalently

H1Tr : At least one τ k is not zero.

Using the notations similar to described in the Unit 12, we can see that
αi = (μi00 − μ) for all i; where, μioo stands for the mean of the ith level of the
( )
Factor U, βj = μojo − μ for all j; where, μojo denotes the mean of the jth level of
the Factor V and τk = (μook − μ) for all k; where, μook denoting the mean of the
kth level of the Factor T.
Using the results of Unit 12, further, we know that

 α =  i=1 (μioo − μ) = 0 ,
p p
i=1 i

 β =  j=1 (μojo − μ) = 0.
p p
j=1 j

Similarly, it can be shown that

 τ = k =1 (μook − μ) = 0.
p p

174 k =1 k
Unit 16 Latin Square Design

Now, you may try to answer the following Self-Assessment Question:

SAQ 3
State which ANOVA model would be most suitable for the analysis of data
obtained in a Latin Square Design. Explain the parameters which would be
used in the model with an explanation of all of these parameters.

16.4.3 Estimation of Model Parameters

While discussing the Two-way Analysis of Variance in the Unit 12, we
presented the method of estimating the parameters μ, αi and β j . Following
similar method as described in Unit 12 and notations used above in this unit,
we can show that the least square estimates of the parameters μ, αi , β j and
τ k can be obtained as

μ̂ = xooo ; αˆ i = xioo − xooo ; βˆ j = x ojo − x ooo and τˆ k = x ook − x ooo ;

where xooo , xioo , xojo and xook are, respectively, the Grand Mean of all the
observations, Mean of the ith level of the Factor U, Mean of the jth level of the
Factor V and Mean of the kth level of the Factor T as obtained on the basis of
the data.
Partition of the Total Sum of Squares:
The model (16.1), after substitution of estimates of the parameters become
x ijk = μˆ + αˆ i + βˆ j + τ̂k + eˆ ijk ;

= xooo + ( xioo − xooo ) + ( xojo − xooo ) + ( xook − xooo )

+ ( xijk − xioo − xojo − xook + 2xooo ) .

Therefore, Total Sum of Square (TSS) will be

   (x − xooo ) = p2 i=1 ( xioo − xooo ) + p2  j=1 ( xojo − xooo )

p p p 2 p 2 p 2

i=1 j=1 k =1 ijk

(
+p2 k =1 ( xook − xooo ) + i=1 j=1k =1 xijk − xioo − xojo − xook + 2xooo )
p 2 p p p 2
.

As per definition of each Sum of Squares, given in the Units 11 and 12, we
see that the above expression is equivalent to be written as
TSS = Sum of Squares due to Row (SSR) + Sum of Squares due to
Column (SSC) + Sum of Squares due to Treatment (SSTr) +
Sum of Squares due to Error (ESS);
which shows that how the Total variability of the data could be partitioned into
a number of variations.
Associated Degrees of Freedom (df):
The Degrees of Freedom (df) associated with TSS, SSR, SSC, SSTr and ESS
( ) ( )
will be p2 − 1 , (p − 1) , (p − 1) , (p − 1) and p2 − 1 − ( 3p − 3 ) = (p − 1)(p − 2) ,
respectively.

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 Block 4 Design of Experiments

Expectation of Sum of Squares:

Following the steps mentioned in Sub-sections 11.5.4 and 12.5.4, respectively
in the Units 11 and 12, we can show that
 TSS 
= σ ;
2
(i) E 2
 p − 1

 SSR  p
 = σ + p − 1  i=1αi ;
2 p 2
(ii) E
 p − 1 

 SSC  p

p
(iii) E  = σ2 + β2 ;
 p − 1  p − 1 j =1 j

 SSTr  p

p
(iv) E  = σ2 + τ 2 and
 p − 1  p − 1 k =1 k

 ESS 
= σ .
2
(v) E
 ( p − 1)( p − 2 ) 
These results indicate that while Total Mean Sum of Squares (MSST) and
Error Mean Sum of Squares (MSSE) are unconditionally unbiased for the total
variability, σ 2 ; the other Mean Sum of Squares, namely, Mean Sum of
Squares due to Rows (MSSR), Mean Sum of Squares due to Columns (MSSC)
and Mean Sum of Squares due to Error (MSSE) are unbiases for σ 2 , only
when the respective conditions αi = 0, βi = 0, τ i = 0 for all i = 1, 2, 3, …, p.

16.4.4 Steps for Computing Different Sum of Squares

We shall now describe the method of computing different Sum of Squares in a
Latin Square Design. Let us consider a Latin Square Design with p (> 4)
treatments; T1, T2, …, Tp whose layout is given along with observations xijk
and totals of rows and columns as follows:
Table 16.1: Layout of p  p Latin Square Design

Column
C1 C2 C3 … Cp Total

R1 T1 T2 T3 Tp
… R1
x111 x122 x133 x 1pp

R2 T2 T3 T4 T1
… R2
x 212 x 223 x 234 x 2p1

Row T3 T4 T5 T2
R3
… R3
x 313 x 324 x 335 x 3p2

. . . . … . .
. . . . … . .
. . . . … . .
Tp T1 T2 T p −1
Rp
… Rp
x p1p x p 21 x p32 xpp(p −1)

Total C1 C2 C3 … Cp G

176
Unit 16 Latin Square Design

Where in the table

Ri denotes the Sum of the ith row (i = 1, 2, …, p);
Cj denotes the Sum of the jth column (j = 1, 2, …, p);

G denotes the Grand Total, that is, G = i=1Ri =  j=1Cj .

p p

Let us, further denote by Tk, the sum of those observations which come from
kth treatment from each of the rows or from each of the column (k = 1, 2, …, p).
Steps:
1. Calculate the Row Totals R1, R2, …, Rp.
2. Calculate the Column Totals C1, C2, …, Cp.
3. Calculate the Treatment Totals T1, T2, …, Tp.

Calculate the Grand Total G = i=1Ri =  j=1Cj .

p p
4.

  
p p p 2
5. Calculate the Raw Sum of Squares given by i=1 j=1 k =1 ijk
x .

Ri2
 i=1 p .
p
6. Calculate the Sum

C2j

p
7. Calculate the Sum j =1
.
p

Tk2
 k =1 p .
p
8. Calculate the Sum

G2
9. Calculate the Correction Factor (C.F.), given by .
p2

G2
i=1 j=1k =1xijk2 −
p p p
10. Calculate the Total Sum of Squares (TSS) as .
p2

Ri2 G2

p
11. Calculate the Sum of Squares due to Rows (SSR) as i =1
− 2.
p p

C2j G2

p
12. Calculate the Sum of Squares due to Columns (SSC) as j =1
− .
p p2

Tk2 G2
 k =1 p − p2 .
p
13. Calculate the Sum of Squares due to Treatments (SSTr) as

14. Calculate the Error Sum of Squares (ESS) as

ESS = TSS – SSR – SSC – SSTr.

16.4.5 The ANOVA Table for Latin Square Design

On the basis of different sources of variation, different Sum of Squares
calculated as above, corresponding degree of freedom, the ANOVA Table for
Latin Square Design is given by

177
 Block 4 Design of Experiments

Table 16.2: ANOVA Table for Latin Square Design

Degrees of Mean Sum of
Source Sum of Squares
Freedom Squares F- Ratio
of Variation (SS)
(df) (MSS)
FR =
p i =1 ( xioo − x ooo )
p 2
RSS
Due to Row p −1 = MSSR MSSR
= SSR p −1
MSSE
FC =
p j=1(xojo − xooo )
p
Due to
2
CSS MSS C
p −1 = MSSC
Column p −1 MSSE
= SSC

FT =
p k =1(x ook − xooo )2
p
Due to SSTr MSSTr
p −1 = MSSTr
Treatment
= SSTr p −1 MSSE

  
p p p
i =1 j =1 k =1
ESS
(p − 1)(p − 2) = MSSE
Error
(x − xioo − x ojo − x ook + 2x ooo ) (p − 1)(p − 2) ----
2
ijk

= ESS

   (x − xooo )
p p p 2

Total p2 − 1 i =1 j =1 k =1 ijk
---- ----
= TSS

From the ANOVA table, the conclusions can be drawn for the null hypotheses
(I) H0R : α1 = α2 =  = αp = 0; (II) H0C : β1 = β2 =  = βp = 0 and
(III) H0Tr : τ1 = τ2 =  = τp = 0 as follows:

1. Using the F distribution and the computed F Ratio FR, as obtained in the
ANOVA Table, we test the null hypothesis H0 R of equality of all the row
effects (p number of effects) and
(i) Reject the Null Hypothesis H0 R which states that all the row
effects are equal; at the given level of significance α if the
computed ratio,
MSSR
FR = > Fα; (p −1),(p−1)(p− 2) , the Tabulated value;
MSSE
where Fα; (p −1), (p −1) (p −2) is the upper α point of the F distribution with
df (p – 1), (p –1) ( p –2) to be observed from the F-table in
Appendix given at the end of this Volume 2;
(ii) Otherwise, accept or do not reject the Null Hypothesis, H0 R
implying that all the row effects are equally effective.
2. Using the F distribution and the computed F Ratio, FC, we test the null
hypothesis H0C :β1 = β2 =  = βp = 0 , which states that the effects of all
the p columns are equal and
(i) Reject the Null Hypothesis H0C , at the given level of significance
α if the computed ratio,
MSSC
FC = > Fα; (p –1),(p –1)(p –2) , the Tabulated value;
MSSE
Where Fα; (p −1), (p −1) (p −2) is the upper α − point of the F distribution with
df (p – 1), (p –1)(p – 2), to be observed from the tables of F
178 distribution given in Appendix given at the end of this Volume 2;
Unit 16 Latin Square Design

(ii) Otherwise, accept or do not reject the null hypothesis H0C , or

reject the alternative hypothesis H1C .

3. Using the F distribution and the computed F Ratio, i.e., FT, as obtained
in the ANOVA table, we test the null hypothesis H0Tr stating the equality
of all the treatment effects (p effects) and
(i) Reject the Null Hypothesis H0Tr which states that all the
treatment effects are equal; at the given level of significance α if
the computed ratio
MSSTr
FT = > Fα; (p –1), (p –1) (p –2) , the Tabulated value;
MSSE

where Fα; (p −1), (p −1) (p −2) is the upper α − point of the F distribution
with df (p – 1), (p –1)( p –2) to be observed from the F-table in
Appendix given at the end of this Volume 2;
(ii) Otherwise, accept or do not reject the null hypothesis H0Tr ,
implying that all the treatment effects are equally effective.
Let us illustrate the entire computational procedure in Latin Square Design
through a real problem, which is given below:
Example 1: In an agricultural experiment, with the aim of testing the effect of
five types of spacing methods between the brinjal plants on the yield of it, a
5x5 Latin Square Design was applied. The field layout and yields under the
Latin Square Design are shown below. Here spacings are the treatments
which are denoted by letters A, B, C, D and E.
Column
Row
1 2 3 4 5

1 B 260 E 259 A 338 C 195 D 255

2 D 245 A 280 E 279 B 182 C 250
3 E 287 B 203 C 334 D 202 A 266

4 A 271 C 227 D 295 E 182 B 270

5 C 266 D 230 B 325 A 220 E 210

Analyze the data and give your conclusions.

Solution: Here the given Latin Square Design is of dimension 5  5. So, we
have p = 5. The steps of computations are:
1. Calculation of Row Totals: 1307, 1236, 1292, 1245, 1251.
2. Calculation of Column Totals; 1329, 1199, 1571, 981, 1251.
3. Calculation of Treatment Totals:
A= 1375; B = 1240; C = 1272; D = 1227; E = 1217.
4. Calculation of the Grand Total G : G = 6331.

  
p p p
5. Calculation of the Raw Sum of Squares: i=1 j=1
x = 1650359.
2
k =1 ijk

179
 Block 4 Design of Experiments

Ri2 Ri2 8020235

 i=1 p :  i=1 p = 5 = 1604047.
p p
6. Calculation of the Sum

C2j C2j 8199245

 
p p
7. Calculation of the Sum j =1
: j =1
= = 1639849.
p p 5

Tk2 Tk2 8032827

k =1 p : k =1 p = 5 = 1606565.40
p p
8. Calculation of the Sum

G2
9. Calculation of the Correction Factor (C.F.) :
p2

G2 40081561
= = 1603262.44.
p2 25
10. Calculation of Total Sum of Squares (TSS):
G2
i=1 j=1k =1 ijk p2 = 1650359 − 1603262.44 = 47096.56.
p p p
x 2
−

11. Calculation of Sum of Squares due to Rows (SSR):

2
p Ri G2
 i=1 p p2 = 1604047 − 1603262.44 = 784.56.
−

12. Calculation of Sum of Squares due to Columns (SSC):

2
p Cj G2
 j=1 p − p2 = 1639849 − 1603262.44 = 36586.56.
13. Calculation of Sum of Squares due to Treatments (SSTr):
2
p Tk G2
k =1 p p2 = 1606565.40 − 1603262.44 = 3302.96.
−

14. Calculation of Error Sum of Squares (ESS):

ESS = TSS – SSR – SSC - SSTr
= 47096.56 – 784.56 – 36586.56 – 3302.96
= 6420.08.
Therefore, the ANOVA table will be
Sum of Mean Sum of
Source Degrees of Squares Squares F Ratio
of Variation Freedom (df)
(SS) (MSS)
Due to
5 −1= 4 784.56 = SSR 196.14 = MSSR FR = 0.366
Rows
Due to
5 −1= 4 36586.56 = SSC 9146.64 = MSSC FC = 7.096
Columns
Due to
5 −1= 4 3302.96 =SSTr 825.74 = MSSTr FT = 1.543
Treatments

Error ( 5 − 1)( 5 − 2) =12 6420.08 = ESS 535.01 = MSSE ----

Total 25 − 1 = 24 47096.56 = TSS ---- ----

Conclusions:
1. Since, FR=0.366 at df (4,12) ˂ F (Tabulated) =3.26 at 5% level of
significance observed from the F-table in Appendix given at the end of
180
Unit 16 Latin Square Design

this Volume 2, the hypothesis H0R is not rejected, that is, all the row
effects on the average yield are equal.
2. Since, FC = 17.096 at df (4, 12) > F (Tabulated) = 3.26 at 5% level of
significance observed from the F-table in Appendix given at the end of
this Volume 2, the hypothesis H0C is rejected. Therefore, we conclude
that all the column effects on the average yield are not the same.
3. We see that FT = 1.543 at df (4, 12) ˂ F (Tabulated) =3.26 at 5% level of
significance observed from the F-table in Appendix given at the end of
this Volume 2, therefore, the hypothesis H0Tr is not rejected, therefore, it
indicates that all the treatment effects on the average yield are same.
Now, you may try to answer the following Self-Assessment Question:

SAQ 4
Perform an Analysis of Variance for the following Latin Square Design with six
treatments A, B, C, D, E and F in order to test the hypothesis that all the
treatments are equally effective:
B 92 F 80 E 120 D 84 A 99 C 82
D 78 A 72 B 90 E 122 C 110 F 98
E 118 C 100 F 110 A 50 B 94 D 74
A 80 D 98 C 98 B 66 F 82 E 90
C 90 B 70 D 66 F 90 E 98 A 66
F 90 E 112 A 78 C 82 D 98 B 94

16.5 ADVANTAGES AND DISADVANTAGES OF

LATIN SQUARE DESIGN
(a) Advantages:
(i) The Latin Square Design has been described as an advancement
over Randomised Block Design in the sense that blocking technique
in it is used for two perpendicular directions instead of one direction
only as is the case with Randomised Block Design. The
simultaneous two-directional blocking or grouping of the units, in
fact, eliminates those two major sources of variation, namely,
variation among rows and among columns from the experimental
error which are not much relevant to the comparisons among the
treatments which is the major concern of a Latin Square Design.
This itself means that Latin Square Design is more efficient than
Randomised Block Design and Completely Randomised Desgin. In
the words of Professor R. A. Fisher “If experimentation were only
concerned with the comparison of four to eight treatments or
varieties, Latin Square Design would not only be the principal but
almost the universal design employed”.
(ii) Latin Square Design is considered to be an incomplete three-way
design because of the reason that all the three factors, namely,
Rows, Columns and Treatments are at the same number of levels
(say, p); and in a complete three-way layout with all the factors at p 181
 Block 4 Design of Experiments

levels, therefore, we require (p x p x p) = p3 experimental units,

whereas in Latin Square Design we take observations on only (p x
p) = p2 experimental units according to the plan of the design. In this
sense, Latin Square Design is an economical design over complete
three-way design because of the reduced number of experimental
units.
(iii) The layout of Latin Square Design and its analysis remains relatively
simple.
(b) Disadvantages:
(i) Whereas Completely Randomised Design and Randomised Block
Design both can be accommodated in any shape of the field; Latin
Square Design requires preferably a square shape field but nothing
less than rectangular field.
(ii) We have seen that Randomised Block Design is available for a wide
range of treatments without any restriction upon the number of
replications, whereas in LSD, the number of treatments is to remain
same as the number of rows and columns. This is a serious
disadvantage of Latin Square Design. This actually restricts upon
the dimension of Latin Square Design. In fact, Latin Square Design
with dimension greater than 12  12 are seldom used because then
we need a field so large that squares do not remain homogeneous.
(iii) On the other hand, Latin Square Design with small dimension is
also not suitable because then the degrees of freedom associated
with error is too less which is not desirable as then the results
might not be reliable. Generally, Latin Square Design with
dimension 4  4 to 8  8 are found most appropriate. We can see
that for a 2  2 Latin Square Design, the degrees of freedom of
error becomes zero, which makes the MSSE infinite.
(iv) The fundamental assumption that there is no interaction between
different factors may not be true in general.

16.6 EFFICIENCY COMPARISON

(a) Latin Square Design Versus Randomised Block Design
We have seen that in Randomised Block Design, blocks are either
arranged row-wise or column-wise and hence the efficiency of Latin
Square Design over Randomised Block Design can be obtained in two
ways; (i) when blocks are given row-wise and (ii) when blocks are given
column-wise.
(i) When Rows are taken as Blocks:
In such situation, the Relative Efficiency (RE) of Latin Square
Design over Randomised Block Design is given by
MSSC + ( p − 1) MSSE
Re lativeEfficiency (Column) = .
pMSSE

It is called “Column Efficiency”.

(ii) When Columns are taken as Blocks:
In this case, the Relative Efficiency (RE) of Latin Square Design
182 over Randomised Block Design is given by
Unit 16 Latin Square Design

MSSR + (p − 1) MSSE
Relative Efficiency (Row) = .
pMSSE

It is known as “Row Efficiency”. In both the expressions MSSC,

MSSR and MSSE are those Mean Sum of Square which are obtained
in the analysis of Latin Square Design.
(b) Latin Square Design Versus Completely Randomised Design
Relative efficiency of Latin Square Design over Completely Randomised
Design is given by
MSSR + MSSC + (p − 1) MSSE
Relative Efficiency = .
(p + 1) MSSE
All these expressions of Relative Efficiency are derived by Prof. O.
Kempthorne.

16.7 LATIN SQUARE DESIGN WITH ONE

MISSING OBSERVATION
The situation of missing observation in experimental design is a very common
feature. Due to some unavoidable reasons, one or more observations might
be missing while recording the output of the experiment. The reasons might be
damages made by animals or pets in an agricultural experiment; the death of
an animal or animals in an experiment of measuring the effect of some diets
on them and others.
Similar to Completely Randomised Design (CRD) and Randomised Block
Design (RBD), for simplicity of mathematical treatment, we consider the case
of Latin Square Design when only one observation is missing.
As we assumed in the case of Completely Randomised Design and
Randomised Block Design, here also we assume without loss of generality
that in a p  p Latin Square Design, the observation which is missing belongs
to the first row, first column and this plot receives the first treatment, denoted
by T1, that is the observation x111 is missing.

We know that the “Missing Plot Technique”, which we have used in

Completely Randomised Design and Randomised Block Design, is based
upon the “Least Square Principle” under which the Error Sum of Squares is
minimised with respect to the missing observation in order to get least square
estimate of the observation. After estimating the observation, this estimated
value is used in the mathematical treatment of the design concerned for final
results. We shall use the same technique here also.
Let us denote the missing value as X1. Let us denote the sum of the
observations in the first row, first column and for the first treatment T1, without
including the missing value X1 respectively by R1* , C1* and T1* . Let G* denote
the sum of all the observations, that is, Grand Total, excluding the missing
value. Then the TSS, SSR, SSC and SSTr as defined in the steps under the
Sub-section 16.4.4 will be

183
 Block 4 Design of Experiments

(G )
2
*
+ X1
TSS =  i=1 j=1 x + X − where (i, j, k )  (1,1,1) ;
p p p 2 2
k =1 ijk 1
p2

(R ) (G )
2 2
*
1 + X1 + R 22 +  + Rp2 *
+ X1
S SR = − ;
p p2

(C ) (G )
2 2
*
1 + X1 + C22 +  + Cp2 *
+ X1
S SC = − ;
p p2

(T ) (G )
2 2
1
*
+ X1 + T22 +  + Tp2 *
+ X1
SSTr = − .
p p2

Thus, since,
ESS = TSS – SSR – SSC – SSTr,
due to the above expressions, we have
ESS =
(G ) − (R ) + (G ) − (C ) + (G ) − (T )
2 2 2 2 2 2
*
+ X1 *
1 + X1 *
+ X1 *
1 + X1 *
+ X1 1
*
+ X1
X −
2
1
p2 p p2 p p2 p

(G )
2
*
+ X1
+ + terms independent of X1.
p2

(R ) (C ) − (T )
2 2 2
*
1 + X1 *
1 + X1 1
*
+ X1
=X 2
1 − −
p p p
(G )
2
*
+ X1
+2 + terms independent of X1.
p2

In order to minimize the Error Sum of Squares (ESS), we differentiate ESS

with respect to X1 and equate the resultant expression to zero, we get
ESS R* + X1 C* + X1 T * + X1 G* + X1
= 2X1 − 2 1 −2 1 −2 1 +4 = 0.
X1 p p p p2

This expression yields the estimated value of X1 as

ˆ =
X
(
p R1* + C1* + T1* − 2G* ) .
1
(p − 1)(p − 2 )
Thus, the value of the missing observation is easily computed. After getting
this value of X1, it is substituted for the missing observation in the table of
observations and the usual method of finding different Sum of Squares is
used.

Obviously, if instead of first row, first column and the first treatment, the
missing observation corresponds to ith row, column and treatment, the
estimated value of the missing observation will be

ˆ =
X
(
p Ri* + Ci* + Ti* − 2G* ) for i = 1, 2, …, p.
i
(p − 1)(p − 2 )
184
Unit 16 Latin Square Design

Now, you may try to answer the following Self-Assessment Question:

SAQ 5
Compare the Latin Square Design with Randomised Block Design for their
advantages and disadvantages.

16.8 SUMMARY
In this unit, we have discussed:
• The concept of Latin Squares, as developed by mathematicians a long
back, with examples since, the Latin Square Design (LSD) derives its
name from it and the method of allocating treatments to different plots in
Latin Square Design is borrowed from the definition of Latin Squares.
• The precise definition of Latin Square Design along with a brief history of
the development of it.
• The concept of two directional blocking techniques as used in Latin
Square Design in detail and its usefulness in order to reduce the
experimental error as an advancement of the unidirectional blocking
technique, used in Randomised Block Design.
• The layout of a Latin Square Design with the help of some illustrations.
• The statistical analysis of the data obtained in a Latin Square Design
along with the description of appropriate ANOVA model to be used,
hypotheses to be tested using ANOVA technique, estimation of the
parameters involved in the model on the basis of the data observed,
steps for computation of different sum of squares, preparation of the
ANOVA table and its use for the test procedures.
• The various advantages and disadvantages of Latin Square Design in
respect of Randomised Block Design.
• The efficiency of Latin Square Design over Randomised Block Design
and Completely Randomised Design and the expressions of relative
efficiencies.
• The case of missing observations in Latin Square Design. The
estimation of the missing observation using the missing plot technique,
that is, finding the least square estimate by minimizing the Error Sum of
Squares along with the expression of the estimate.

16.9 TERMINAL QUESTIONS

1. In the context of blocking of plots in an experimental field, explain in
what sense Latin Square Design is considered to be an advancement
over RBD. What are the blocking factors in a Latin Square Design?
2. With the six treatments, A, B, C, D, E and F provide the layout of a Latin
Square Design which is in the standard square form. Also provide a
transformed Latin Square Design with the same treatments.

185
 Block 4 Design of Experiments

3. Show how would you present the data along with the levels of the three
factors, namely, treatments, rows and columns in a tabular form for an
Latin Square Design with h factors of each factor.
4. Give the format of the ANOVA table to be used in a Latin Square Design
and explain how the null hypotheses can be tested on the basis of this
table.
5. An industrial experimenter wishes to compare the effects of five types of
electronic circuits A, B, C, D and E on the laptops. It was found that two
other factors, namely, voltage levels and size of mother boards might
highly affect the performance of the laptop. The experimenter designed
the experiment in the form of a Latin Square Design for comparing the
effectiveness of the 5 electronic circuits. The layout of the design is
depicted below, where numerical figures are some coded measures of
the performance of laptops:
Mother Board Size
1 2 3 4 5
1 E 52.0 D 69.0 C 55.0 A 53.0 B 65.0
2 C 60.0 B 60.0 A 59.0 E 54.0 D 62.0
Voltage 3 A 51.0 E 59.0 D 50.0 B 68.0 C 55.0
4 D 66.0 A 60.0 B 68.0 C 63.0 E 67.0
5 B 62.0 C 62.0 E 66.0 D 54.0 A 59.0
Perform an ANOVA and test for the main factors: type of electronic
circuits, voltage levels and size of mother boards.
6. What is the problem of missing plots in Latin Square Design? How would
you estimate the missing observation which occurs for the observation
x111, where 111 in the suffix stands for the first level of all the three
factors?

16.10 ANSWERS / SOLUTIONS

Self Assessment Questions (SAQs)
1. Hint: See Sub-sections 16.2.2 and 16.2.1 for your answer.
2. See the example in Section 16.3 for the rule of allotment of treatments to
different rows and columns. Given that in a 5x5 Latin Square Design, the
treatments are μ, θ, φ, α and β and these are allotted in the sequence
μ, α, φ, θ and β from top to bottom in the first column. Then following the
rule of allotting one treatment only once in a row and in a column, we have
the following arrangement of the treatments:
μ α φ θ β

α φ θ β μ
φ θ β μ α

θ β μ α φ

β μ α φ θ

3. Hint: See Sub-section 16.4.1 for providing your answer.

186
4. The data are reproduced below:
Unit 16 Latin Square Design

B 92 F 80 E 120 D 84 A 99 C 82
D 78 A 72 B 90 E 122 C 110 F 98
E 118 C 100 F 110 A 50 B 94 D 74
A 80 D 98 C 98 B 66 F 82 E 90
C 90 B 70 D 66 F 90 E 98 A 66
F 90 E 112 A 78 C 82 D 98 B 94

For the given data, we have following values:

Calculation of Row Totals: 557, 570, 546, 514, 480, 554.
Calculation of Column Totals: 548, 532, 562, 494, 581, 504.
Calculation of Treatment Totals: 445, 506, 562, 498, 660, 550.
Calculation of the Grand Total G : G = 3221.

  
p p p
Calculation of the Raw Sum of Squares: i=1 j=1
x = 297733.
2
k =1 ijk

Ri2 Ri2 1734777

i=1 p : i=1 p = 6 = 289129.5.
p p
Calculation of the Sum

C2j C2j 1734785

 
p p
Calculation of the Sum j =1
: j =1
= = 289130.83.
p p 6

Tk2 Tk2 1756009

k =1 p :  k =1 p = 6 = 292668.17.
p p
Calculation of the Sum

G2
Calculation of the Correction Factor (C.F.), :
p2
G2 10374841
= = 288190.03.
p2 36

Calculation of Total Sum of Squares (TSS):

G2
i=1 j=1k =1xijk2 −
p p p
= 297733 − 288190.03 = 9542.97.
p2

Calculation of Sum of Squares due to Rows (SSR):

Ri2 G2
i=1 p − p2 = 289129.5 − 288190.03 = 939.47.
p

Calculation of Sum of Squares due to Columns (SSC):

C2j G2

p
j =1
− = 289130.83 − 288190.03 = 940.80.
p p2

Calculation of Sum of Squares due to Treatments (SSTr):

Tk2 G2
k =1 p − p2 = 292668.17 − 288190.03 = 4478.14.
p

Calculation of Error Sum of Squares (ESS):

ESS = TSS – SSR – SSC - SSTr
= 9542.97 − 939.47 − 940.80 − 4478.14 = 3184.56. 187
 Block 4 Design of Experiments

Therefore, the ANOVA Table would be

Source Degrees of Sum of Mean Sum of F Ratio
of Variation Freedom (df) Squares (SS) Squares (MSS)

Due to 6 −1 = 5 939.47= SSR 187.894 = MSSR FR = 1.180

Rows
Due to 6 −1 = 5 940.80 = SSC 188.16 = MSSC FC = 1.182
Columns
Due to 6 −1 = 5 4478.14 = SSTr 895.628 = MSSTr FT = 5.625
Treatments
Error ( 6 − 1)( 6 − 2) = 20 3184.56 = ESS 159.228 = MSSE ----

Total 36 - 1 = 35 9542.97 = TSS ---- ----

Conclusions:
1. Since, FR = 1.180 at df (5, 20 ) ˂ F (Tabulated) =2.71 at (5, 20) df
and 5% level of significance to be observed from the F-table in
Appendix given at the end of this Volume 2, the hypothesis H0R is
not rejected, that is, the effect of all the rows are same on all the
treatments.
2. Since, FC = 1.182 at df (5, 20) < F (Tabulated) = 2.71 at (5, 20) df
and at 5% level of significance to be observed from the F-table in
Appendix given at the end of this Volume 2, the hypothesis H0C is
not rejected. Therefore, we conclude that the effects of all the
columns are not significantly different from each other.
3. We see that FT = 5.625 at df (5, 20) > F (Tabulated) = 2.71 at 5%
level of significance to be observed from the F-table in Appendix
given at the end of this Volume 2, therefore, the hypothesis H0Tr is
rejected. It indicates that all the treatment effects do not differ
significantly from each other.
5. Hint: Consult Sub-section 16.4.1 and 16.5.2 for your answer.
Terminal Questions (TQs)
1. Hint: See Sub-section 16.2.4 for your answer.
2. The layout of the 6x6 Latin Square Design in the Standard Latin Square
form will be
A B C D E F
B C D E F A
C D E F A B
D E F A B C
E F A B C D
F A B C D E
Another Latin Square Design in the transformed form with the same
treatments will be
C D A F E B
B C D A F E
E B C D A F
F E B C D A
A F E B C D

188 D A F E B C
Unit 16 Latin Square Design

3. Hint: See Sub-section 16.4.4 and Table 16.1 for your answer.
4. Hint: See Sub-section 16.4.5 and Table 16.2 for the answer.
5. The data given in the exercise are presented below:
Mother Board Size
1 2 3 4 5
1 E 52.0 D 69.0 C 55.0 A 53.0 B 65.0
2 C 60.0 B 60.0 A 59.0 E 54.0 D 62.0
Voltage 3 A 51.0 E 59.0 D 50.0 B 68.0 C 55.0
4 D 66.0 A 60.0 B 68.0 C 63.0 E 67.0
5 B 62.0 C 62.0 E 66.0 D 54.0 A 59.0

Following the lines of calculations, as presented in Example 1, we see

that here Rows are the different levels of the factor “Voltage” and Columns
are the different levels of the factor “Mother Board Size”. Treatments are
denoted by letters A, B, C, D and E. Therefore, we do the computations
accordingly as follows:
1. Calculation of Row Totals: 294, 295, 283, 324, 303.
2. Calculation of Column Totals: 291, 310, 298, 292, 308.
3. Calculation of Treatment Totals:
A= 282; B = 323; C = 295; D = 301; E = 298.
4. Calculation of the Grand Total G : G = 1499.

  
p p p
5. Calculation of the Raw Sum of Squares: i=1 j=1
x = 90675.
2
k =1 ijk

Ri2 Ri2 450335

 i=1 p :  i=1 p = 5 = 90067.
p p
6. Calculation of the Sum

C2j C2j 449713

 
p p
7. Calculation of the Sum j =1
: j =1
= = 89942.6.
p p 5

Tk2 Tk2 450283

 k =1 p : k =1 p = 5 = 90056.6.
p p
8. Calculation of the Sum

G2
9. Calculation of the Correction Factor (C.F.), :
p2

G2 2247001
= = 89880.04.
p2 25

10. Calculation of Total Sum of Squares (TSS):

G2
i=1 j=1k =1xijk2 −
p p p
= 90675 − 89880.04 = 794.96.
p2

11. Calculation of Sum of Squares due to Rows (SSR):

2
p R G2
i=1 pi − p2 = 90067 − 89880.04 = 186.96.

189
 Block 4 Design of Experiments

12. Calculation of Sum of Squares due to Columns (SSC):

2
p Cj G2
 j=1 p p2 = 89942..6 − 89880.04 = 62.56.
−

13. Calculation of Sum of Squares due to Treatments (SSTr):

2
p Tk G2
k =1 p p2 = 90056.6 − 89880.04 = 176.56.
−

14. Calculation of Error Sum of Squares (ESS):

ESS = TSS – SSR – SSC – SSTr
= 794.96 – 186.96 – 62.56 − 176.56 = 368.88.
Therefore, the ANOVA Table would be
Source Degrees of Sum of Mean Sum of F Ratio
of Variation Freedom (df) Squares Squares
(SS) (MSS)
Due to Rows 5 −1= 4 186.96 = SSR 46.74 = MSSR FR =1.52

Due to 5 −1= 4 62.56 = SSC 15.64 = MSSC FC = 0.51

Columns
Due to 5 −1= 4 176.56 = SSTr 44.14 = MSSTr FTr = 1.44
Treatments
Error ( 5 − 1)( 5 − 2) =12 368.88 = ESS 30.74 = MSSE ----

Total 25 − 1 =24 794.96 = TSS ---- ----

Conclusions:
1. Since, FC = 1.52 at df (4, 12) ˂ F (Tabulated) =3.26 at 5% level of
significance to be observed from the F-table in Appendix given at
the end of this Volume 2, the hypothesis H0R is not rejected, that
is, all levels of voltage effects are same on all the treatments.
2. Since, FC = 0.51 at df (4, 12) < F (Tabulated) = 3.26 at 5% level of
significance to be observed from the F-table given in Appendix
given at the end of this Volume 2, the hypothesis H0C is not
rejected. Therefore, we conclude that the effects of size of the
mother board do not significantly affect the performance of the
laptops.
3. We see that FT = 1.44 at df (4, 12) ˂ F (Tabulated) =3.26 at 5%
level of significance to be observed from the F-table given in
Appendix given at the end of this Volume 2, therefore, the
hypothesis H0Tr is not rejected, therefore, it indicates that all the
treatment effects on the average provide same performance of
laptops.
6. Hint: See Section 16.7 for your answer.

190
 Appendix

Table I Student’s t Distribution (t table)

The first column of this table indicates the degrees of freedom and first row a
specified upper tail area (α). The entry represents the value of t-statistic such that the
area under the curve of t-distribution to its upper tail is equal to α.

One- tail α = 0.10 0.05 0.025 0.01 0.005

ν =1 3.078 6.314 12.706 31.821 63.657
2 1.886 2.920 4.303 6.965 9.925
3 1.638 2.353 3.182 4.541 5.841
4 1.533 2.132 2.776 3.747 4.604
5 1.476 2.015 2.571 3.365 4.032
6 1.440 1.943 2.447 3.143 3.707
7 1.415 1.895 2.365 2.998 3.499
8 1.397 1.860 2.306 2.896 3.355
9 1.383 1.833 2.262 2.821 3.250
10 1.372 1.812 2.228 2.764 3.169
11 1.363 1.796 2.201 2.718 3.106
12 1.356 1.782 2.179 2.681 3.055
13 1.350 1.771 2.160 2.650 3.012
14 1.345 1.761 2.145 2.624 2.977
15 1.341 1.753 2.131 2.602 2.947
16 1.337 1.746 2.120 2.583 2.921
17 1.333 1.740 2.110 2.567 2.898
18 1.330 1.734 2.101 2.552 2.878
19 1.328 1.729 2.093 2.539 2.861
20 1.325 1.725 2.086 2.528 2.845
21 1.323 1.721 2.080 2.518 2.831
22 1.321 1.717 2.074 2.508 2.819
23 1.319 1.714 2.069 2.500 2.807
24 1.318 1.711 2.064 2.492 2.797
25 1.316 1.708 2.060 2.485 2.787
26 1.315 1.706 2.056 2.479 2.779
27 1.314 1.703 2.052 2.473 2.771
28 1.313 1.701 2.048 2.467 2.763
29 1.311 1.699 2.045 2.462 2.756
30 1.310 1.697 2.042 2.457 2.750
40 1.303 1.684 2.021 2.423 2.704
60 1.296 1.671 2.000 2.390 2.660
120 1.289 1.658 1.980 2.358 2.617
∞ 1.282 1.645 1.960 2.326 2.576

191
 Table II F Distribution (F table)
F-table contains the values of F-statistic for different set of degrees of freedom (1 , 2 ) of numerator and
denominator such that the area under the curve of F-distribution to its right (upper tail) is equal to α.
F values for α = 0.1
Degrees of Degrees of Freedom for Numerator(ν1)
Freedom for
Denominator 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 24 30 40 60 120 ∞
(ν2)
1 39.86 49.50 53.60 55.83 57.23 58.21 58.91 59.44 59.86 60.20 60.47 60.70 60.91 61.07 61.22 61.35 61.47 61.57 61.66 61.74 62.00 62.26 62.53 62.79 63.06 63.33

2 8.53 9.00 9.16 9.24 9.29 9.33 9.35 9.37 9.38 9.39 9.40 9.41 9.41 9.42 9.42 9.43 9.43 9.44 9.44 9.44 9.45 9.46 9.47 9.47 9.48 9.49
3 5.54 5.46 5.39 5.34 5.31 5.28 5.27 5.25 5.24 5.23 5.22 5.22 5.21 5.20 5.20 5.20 5.19 5.19 5.19 5.18 5.18 5.17 5.16 5.15 5.14 5.13
4 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.96 3.94 3.92 3.91 3.90 3.89 3.88 3.87 3.86 3.86 3.85 3.85 3.84 3.83 3.82 3.80 3.79 3.78 3.76
5 4.06 3.78 3.62 3.52 3.45 3.40 3.37 3.34 3.32 3.30 3.28 3.27 3.26 3.25 3.24 3.23 3.22 3.22 3.21 3.21 3.19 3.17 3.16 3.14 3.12 3.11
6 3.78 3.46 3.29 3.18 3.11 3.05 3.01 2.98 2.96 2.94 2.92 2.90 2.89 2.88 2.87 2.86 2.86 2.85 2.84 2.84 2.82 2.80 2.78 2.76 2.74 2.72
7 3.59 3.26 3.07 2.96 2.88 2.83 2.79 2.75 2.72 2.70 2.68 2.67 2.65 2.64 2.63 2.62 2.61 2.61 2.60 2.59 2.58 2.56 2.54 2.51 2.49 2.47
8 3.46 3.11 2.92 2.81 2.73 2.67 2.62 2.59 2.56 2.54 2.52 2.50 2.49 2.48 2.46 2.45 2.45 2.44 2.43 2.42 2.40 2.38 2.36 2.34 2.32 2.29
9 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.44 2.42 2.40 2.38 2.36 2.35 2.34 2.33 2.32 2.31 2.31 2.30 2.28 2.25 2.23 2.21 2.18 2.16
10 3.29 2.92 2.73 2.61 2.52 2.46 2.41 2.38 2.35 2.32 2.30 2.28 2.27 2.26 2.24 2.23 2.22 2.22 2.21 2.20 2.18 2.16 2.13 2.11 2.08 2.06
11 3.23 2.86 2.66 2.54 2.45 2.39 2.34 2.30 2.27 2.25 2.23 2.21 2.19 2.18 2.17 2.16 2.15 2.14 2.13 2.12 2.10 2.08 2.05 2.03 2.00 1.97
12 3.18 2.81 2.61 2.48 2.39 2.33 2.28 2.24 2.21 2.19 2.17 2.15 2.13 2.12 2.10 2.09 2.08 2.08 2.07 2.06 2.04 2.01 1.99 1.96 1.93 1.90
13 3.14 2.76 2.56 2.43 2.35 2.28 2.23 2.20 2.16 2.14 2.12 2.10 2.08 2.07 2.05 2.04 2.03 2.02 2.01 2.01 1.98 1.96 1.93 1.90 1.88 1.85
14 3.10 2.73 2.52 2.39 2.31 2.24 2.19 2.15 2.12 2.10 2.07 2.05 2.04 2.02 2.01 2.00 1.99 1.98 1.97 1.96 1.94 1.91 1.89 1.86 1.83 1.80
15 3.07 2.70 2.49 2.36 2.27 2.21 2.16 2.12 2.09 2.06 2.04 2.02 2.00 1.99 1.97 1.96 1.95 1.94 1.93 1.92 1.90 1.87 1.85 1.82 1.79 1.76
16 3.05 2.67 2.46 2.33 2.24 2.18 2.13 2.09 2.06 2.03 2.01 1.99 1.97 1.95 1.94 1.93 1.92 1.91 1.90 1.89 1.87 1.84 1.81 1.78 1.75 1.72
17 3.03 2.64 2.44 2.31 2.22 2.15 2.10 2.06 2.03 2.00 1.98 1.96 1.94 1.93 1.91 1.90 1.89 1.88 1.87 1.86 1.84 1.81 1.78 1.75 1.72 1.69
18 3.01 2.62 2.42 2.29 2.20 2.13 2.08 2.04 2.00 1.98 1.95 1.93 1.92 1.90 1.89 1.87 1.86 1.85 1.85 1.84 1.81 1.78 1.75 1.72 1.69 1.66
19 2.99 2.61 2.40 2.27 2.18 2.11 2.06 2.02 1.98 1.96 1.93 1.91 1.89 1.88 1.86 1.85 1.84 1.83 1.82 1.81 1.79 1.76 1.73 1.70 1.67 1.63
20 2.97 2.59 2.38 2.25 2.16 2.09 2.04 2.00 1.96 1.94 1.91 1.89 1.87 1.86 1.84 1.83 1.82 1.81 1.80 1.79 1.77 1.74 1.71 1.68 1.64 1.61
21 2.96 2.57 2.36 2.23 2.14 2.08 2.02 1.98 1.95 1.92 1.90 1.88 1.86 1.84 1.83 1.81 1.80 1.79 1.78 1.78 1.75 1.72 1.69 1.66 1.62 1.59
22 2.95 2.56 2.35 2.22 2.13 2.06 2.01 1.97 1.93 1.90 1.88 1.86 1.84 1.83 1.81 1.80 1.79 1.78 1.77 1.76 1.73 1.70 1.67 1.64 1.60 1.57
23 2.94 2.55 2.34 2.21 2.11 2.05 1.99 1.95 1.92 1.89 1.87 1.85 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.74 1.72 1.69 1.66 1.62 1.59 1.55
24 2.93 2.54 2.33 2.19 2.10 2.04 1.98 1.94 1.91 1.88 1.85 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.74 1.73 1.70 1.67 1.64 1.61 1.57 1.53
25 2.92 2.53 2.32 2.18 2.09 2.02 1.97 1.93 1.89 1.87 1.84 1.82 1.80 1.79 1.77 1.76 1.75 1.74 1.73 1.72 1.69 1.66 1.63 1.59 1.56 1.52
26 2.91 2.52 2.31 2.17 2.08 2.01 1.96 1.92 1.88 1.86 1.83 1.81 1.79 1.77 1.76 1.75 1.73 1.72 1.71 1.71 1.68 1.65 1.61 1.58 1.54 1.50
27 2.90 2.51 2.30 2.17 2.07 2.00 1.95 1.91 1.87 1.85 1.82 1.80 1.78 1.76 1.75 1.74 1.72 1.71 1.70 1.70 1.67 1.64 1.60 1.57 1.53 1.49
28 2.89 2.50 2.29 2.16 2.06 2.00 1.94 1.90 1.87 1.84 1.81 1.79 1.77 1.75 1.74 1.73 1.71 1.70 1.69 1.69 1.66 1.63 1.59 1.56 1.52 1.48
29 2.89 2.50 2.28 2.15 2.06 1.99 1.93 1.89 1.86 1.83 1.80 1.78 1.76 1.75 1.73 1.72 1.71 1.69 1.68 1.68 1.65 1.62 1.58 1.55 1.51 1.47
30 2.88 2.49 2.28 2.14 2.05 1.98 1.93 1.88 1.85 1.82 1.79 1.77 1.75 1.74 1.72 1.71 1.70 1.69 1.68 1.67 1.64 1.61 1.57 1.54 1.50 1.46

Appendix
40 2.84 2.44 2.23 2.09 2.00 1.93 1.87 1.83 1.79 1.76 1.74 1.71 1.70 1.68 1.66 1.65 1.64 1.62 1.61 1.61 1.57 1.54 1.51 1.47 1.42 1.38
60 2.79 2.39 2.18 2.04 1.95 1.87 1.82 1.77 1.74 1.71 1.68 1.66 1.64 1.62 1.60 1.59 1.58 1.56 1.55 1.54 1.51 1.48 1.44 1.40 1.35 1.29
120 2.75 2.35 2.13 1.99 1.90 1.82 1.77 1.72 1.68 1.65 1.63 1.60 1.58 1.56 1.55 1.53 1.52 1.50 1.49 1.48 1.45 1.41 1.37 1.32 1.26 1.19
192

∞ 2.71 2.30 2.08 1.94 1.85 1.77 1.72 1.67 1.63 1.60 1.57 1.55 1.52 1.51 1.49 1.47 1.46 1.44 1.43 1.42 1.38 1.34 1.30 1.24 1.17 1.00

192
 Appendix
F values for α = 0.05
Degrees of
Freedom for Degrees of freedom for numerator(ν1)
193

Denominator (ν2)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 24 30 40 60 120 ∞
1 161 199 216 225 230 234 237 239 240 242 243 244 245 245 246 246 247 247 248 248 249 250 251 252 253 254
2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.39 19.40 19.40 19.41 19.42 19.43 19.43 19.43 19.44 19.44 19.44 19.45 19.45 19.46 19.47 19.48 19.49 19.50
3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.76 8.74 8.73 8.72 8.70 8.69 8.68 8.67 8.67 8.66 8.64 8.62 8.59 8.57 8.55 8.53
4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.94 5.91 5.89 5.87 5.86 5.84 5.83 5.82 5.81 5.80 5.77 5.75 5.72 5.69 5.66 5.63
5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.70 4.68 4.66 4.64 4.62 4.60 4.59 4.58 4.57 4.56 4.53 4.50 4.46 4.43 4.40 4.37
6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.03 4.00 3.98 3.96 3.94 3.92 3.91 3.90 3.88 3.87 3.84 3.81 3.77 3.74 3.70 3.67
7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.60 3.57 3.55 3.53 3.51 3.49 3.48 3.47 3.46 3.44 3.41 3.38 3.34 3.30 3.27 3.23
8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.31 3.28 3.26 3.24 3.22 3.20 3.19 3.17 3.16 3.15 3.12 3.08 3.04 3.01 2.97 2.93
9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.10 3.07 3.05 3.03 3.01 2.99 2.97 2.96 2.95 2.94 2.90 2.86 2.83 2.79 2.75 2.71
10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.94 2.91 2.89 2.86 2.85 2.83 2.81 2.80 2.79 2.77 2.74 2.70 2.66 2.62 2.58 2.54
11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.82 2.79 2.76 2.74 2.72 2.70 2.69 2.67 2.66 2.65 2.61 2.57 2.53 2.49 2.45 2.40
12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.72 2.69 2.66 2.64 2.62 2.60 2.58 2.57 2.56 2.54 2.51 2.47 2.43 2.38 2.34 2.30
13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 2.63 2.60 2.58 2.55 2.53 2.51 2.50 2.48 2.47 2.46 2.42 2.38 2.34 2.30 2.25 2.21
14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 2.57 2.53 2.51 2.48 2.46 2.44 2.43 2.41 2.40 2.39 2.35 2.31 2.27 2.22 2.18 2.13
15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.51 2.48 2.45 2.42 2.40 2.38 2.37 2.35 2.34 2.33 2.29 2.25 2.20 2.16 2.11 2.07
16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.46 2.42 2.40 2.37 2.35 2.33 2.32 2.30 2.29 2.28 2.24 2.19 2.15 2.11 2.06 2.01
17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 2.41 2.38 2.35 2.33 2.31 2.29 2.27 2.26 2.24 2.23 2.19 2.15 2.10 2.06 2.01 1.96
18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.37 2.34 2.31 2.29 2.27 2.25 2.23 2.22 2.20 2.19 2.15 2.11 2.06 2.02 1.97 1.92
19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.34 2.31 2.28 2.26 2.23 2.21 2.20 2.18 2.17 2.16 2.11 2.07 2.03 1.98 1.93 1.88
20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.31 2.28 2.25 2.22 2.20 2.18 2.17 2.15 2.14 2.12 2.08 2.04 1.99 1.95 1.90 1.84
21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.28 2.25 2.22 2.20 2.18 2.16 2.14 2.12 2.11 2.10 2.05 2.01 1.96 1.92 1.87 1.81
22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30 2.26 2.23 2.20 2.17 2.15 2.13 2.11 2.10 2.08 2.07 2.03 1.98 1.94 1.89 1.84 1.78
23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 2.24 2.20 2.18 2.15 2.13 2.11 2.09 2.08 2.06 2.05 2.01 1.96 1.91 1.86 1.81 1.76
24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 2.22 2.18 2.15 2.13 2.11 2.09 2.07 2.05 2.04 2.03 1.98 1.94 1.89 1.84 1.79 1.73
25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.20 2.16 2.14 2.11 2.09 2.07 2.05 2.04 2.02 2.01 1.96 1.92 1.87 1.82 1.77 1.71
26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 2.18 2.15 2.12 2.09 2.07 2.05 2.03 2.02 2.00 1.99 1.95 1.90 1.85 1.80 1.75 1.69
27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25 2.20 2.17 2.13 2.10 2.08 2.06 2.04 2.02 2.00 1.99 1.97 1.93 1.88 1.84 1.79 1.73 1.67
28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19 2.15 2.12 2.09 2.06 2.04 2.02 2.00 1.99 1.97 1.96 1.91 1.87 1.82 1.77 1.71 1.65
29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22 2.18 2.14 2.10 2.08 2.05 2.03 2.01 1.99 1.97 1.96 1.94 1.90 1.85 1.81 1.75 1.70 1.64
30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.13 2.09 2.06 2.04 2.01 1.99 1.98 1.96 1.95 1.93 1.89 1.84 1.79 1.74 1.68 1.62
40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.04 2.00 1.97 1.95 1.92 1.90 1.89 1.87 1.85 1.84 1.79 1.74 1.69 1.64 1.58 1.51
60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.95 1.92 1.89 1.86 1.84 1.82 1.80 1.78 1.76 1.75 1.70 1.65 1.59 1.53 1.47 1.39
120 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.96 1.91 1.87 1.83 1.80 1.78 1.75 1.73 1.71 1.69 1.67 1.66 1.61 1.55 1.50 1.43 1.35 1.25
∞ 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.79 1.75 1.72 1.69 1.67 1.64 1.62 1.60 1.59 1.57 1.52 1.46 1.39 1.32 1.22 1.00
 F values for α = 0.025
Degrees of
Freedom for Degrees of freedom for numerator(ν1)
Denominator
(ν2)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 24 30 40 60 120 ∞

1 648 800 864 900 922 937 948 957 963 969 973 977 980 983 985 987 989 990 992 993 997 1001 1006 1010 1014 1018
2 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39 39.40 39.41 39.41 39.42 39.43 39.43 39.44 39.44 39.44 39.45 39.45 39.46 39.47 39.47 39.48 39.49 39.50
3 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47 14.42 14.37 14.34 14.30 14.28 14.25 14.23 14.21 14.20 14.18 14.17 14.12 14.08 14.04 13.99 13.95 13.90
4 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90 8.84 8.79 8.75 8.72 8.68 8.66 8.63 8.61 8.59 8.58 8.56 8.51 8.46 8.41 8.36 8.31 8.26
5 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68 6.62 6.57 6.52 6.49 6.46 6.43 6.40 6.38 6.36 6.34 6.33 6.28 6.23 6.18 6.12 6.07 6.02
6 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52 5.46 5.41 5.37 5.33 5.30 5.27 5.24 5.22 5.20 5.18 5.17 5.12 5.07 5.01 4.96 4.90 4.85
7 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82 4.76 4.71 4.67 4.63 4.60 4.57 4.54 4.52 4.50 4.48 4.47 4.42 4.36 4.31 4.25 4.20 4.14
8 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36 4.30 4.24 4.20 4.16 4.13 4.10 4.08 4.05 4.03 4.02 4.00 3.95 3.89 3.84 3.78 3.73 3.67
9 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03 3.96 3.91 3.87 3.83 3.80 3.77 3.74 3.72 3.70 3.68 3.67 3.61 3.56 3.51 3.45 3.39 3.33
10 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78 3.72 3.66 3.62 3.58 3.55 3.52 3.50 3.47 3.45 3.44 3.42 3.37 3.31 3.26 3.20 3.14 3.08
11 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59 3.53 3.47 3.43 3.39 3.36 3.33 3.30 3.28 3.26 3.24 3.23 3.17 3.12 3.06 3.00 2.94 2.88
12 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44 3.37 3.32 3.28 3.24 3.21 3.18 3.15 3.13 3.11 3.09 3.07 3.02 2.96 2.91 2.85 2.79 2.73
13 6.41 4.97 4.35 4.00 3.77 3.60 3.48 3.39 3.31 3.25 3.20 3.15 3.12 3.08 3.05 3.03 3.00 2.98 2.96 2.95 2.89 2.84 2.78 2.72 2.66 2.60
14 6.30 4.86 4.24 3.89 3.66 3.50 3.38 3.29 3.21 3.15 3.09 3.05 3.01 2.98 2.95 2.92 2.90 2.88 2.86 2.84 2.79 2.73 2.67 2.61 2.55 2.49
15 6.20 4.77 4.15 3.80 3.58 3.41 3.29 3.20 3.12 3.06 3.01 2.96 2.92 2.89 2.86 2.84 2.81 2.79 2.77 2.76 2.70 2.64 2.59 2.52 2.46 2.40
16 6.12 4.69 4.08 3.73 3.50 3.34 3.22 3.12 3.05 2.99 2.93 2.89 2.85 2.82 2.79 2.76 2.74 2.72 2.70 2.68 2.63 2.57 2.51 2.45 2.38 2.32
17 6.04 4.62 4.01 3.66 3.44 3.28 3.16 3.06 2.98 2.92 2.87 2.82 2.79 2.75 2.72 2.70 2.67 2.65 2.63 2.62 2.56 2.50 2.44 2.38 2.32 2.25
18 5.98 4.56 3.95 3.61 3.38 3.22 3.10 3.01 2.93 2.87 2.81 2.77 2.73 2.70 2.67 2.64 2.62 2.60 2.58 2.56 2.50 2.45 2.38 2.32 2.26 2.19
19 5.92 4.51 3.90 3.56 3.33 3.17 3.05 2.96 2.88 2.82 2.76 2.72 2.68 2.65 2.62 2.59 2.57 2.55 2.53 2.51 2.45 2.39 2.33 2.27 2.20 2.13
20 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.84 2.77 2.72 2.68 2.64 2.60 2.57 2.55 2.52 2.50 2.48 2.46 2.41 2.35 2.29 2.22 2.16 2.09
21 5.83 4.42 3.82 3.48 3.25 3.09 2.97 2.87 2.80 2.73 2.68 2.64 2.60 2.56 2.53 2.51 2.48 2.46 2.44 2.42 2.37 2.31 2.25 2.18 2.11 2.04
22 5.79 4.38 3.78 3.44 3.22 3.05 2.93 2.84 2.76 2.70 2.65 2.60 2.56 2.53 2.50 2.47 2.45 2.43 2.41 2.39 2.33 2.27 2.21 2.15 2.08 2.00
23 5.75 4.35 3.75 3.41 3.18 3.02 2.90 2.81 2.73 2.67 2.62 2.57 2.53 2.50 2.47 2.44 2.42 2.39 2.37 2.36 2.30 2.24 2.18 2.11 2.04 1.97
24 5.72 4.32 3.72 3.38 3.15 2.99 2.87 2.78 2.70 2.64 2.59 2.54 2.50 2.47 2.44 2.41 2.39 2.36 2.35 2.33 2.27 2.21 2.15 2.08 2.01 1.94
25 5.69 4.29 3.69 3.35 3.13 2.97 2.85 2.75 2.68 2.61 2.56 2.51 2.48 2.44 2.41 2.38 2.36 2.34 2.32 2.30 2.24 2.18 2.12 2.05 1.98 1.91
26 5.66 4.27 3.67 3.33 3.10 2.94 2.82 2.73 2.65 2.59 2.54 2.49 2.45 2.42 2.39 2.36 2.34 2.31 2.29 2.28 2.22 2.16 2.09 2.03 1.95 1.88
27 5.63 4.24 3.65 3.31 3.08 2.92 2.80 2.71 2.63 2.57 2.51 2.47 2.43 2.39 2.36 2.34 2.31 2.29 2.27 2.25 2.19 2.13 2.07 2.00 1.93 1.85
28 5.61 4.22 3.63 3.29 3.06 2.90 2.78 2.69 2.61 2.55 2.49 2.45 2.41 2.37 2.34 2.32 2.29 2.27 2.25 2.23 2.17 2.11 2.05 1.98 1.91 1.83
29 5.59 4.20 3.61 3.27 3.04 2.88 2.76 2.67 2.59 2.53 2.46 2.48 2.43 2.39 2.36 2.30 2.27 2.25 2.23 2.21 2.15 2.09 2.03 1.96 1.89 1.81
30 5.57 4.18 3.59 3.25 3.03 2.87 2.75 2.65 2.57 2.51 2.33 2.41 2.37 2.34 2.31 2.28 2.26 2.23 2.21 2.20 2.14 2.07 2.01 1.94 1.87 1.79
40 5.42 4.05 3.46 3.13 2.90 2.74 2.62 2.53 2.45 2.39 2.26 2.29 2.25 2.21 2.18 2.15 2.13 2.11 2.09 2.07 2.01 1.94 1.88 1.80 1.72 1.64

Appendix
60 5.29 3.93 3.34 3.01 2.79 2.63 2.51 2.41 2.33 2.27 2.22 2.17 2.13 2.09 2.06 2.03 2.01 1.98 1.96 1.94 1.88 1.82 1.74 1.67 1.58 1.48
120 5.15 3.80 3.23 2.89 2.67 2.52 2.39 2.30 2.22 2.16 2.10 2.05 2.01 1.98 1.94 1.92 1.89 1.87 1.84 1.82 1.76 1.69 1.61 1.53 1.43 1.31
∞
194

5.02 3.69 3.12 2.79 2.57 2.41 2.29 2.19 2.11 2.05 1.99 1.94 1.90 1.87 1.83 1.80 1.78 1.75 1.73 1.71 1.64 1.57 1.48 1.39 1.27 1.00
 F values for α = 0.01

Appendix
Degrees of
Freedom Degrees of freedom for numerator (ν1)
195

for
Denominator
(ν2) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 24 30 40 60 120 ∞
1 4063 4992 5404 5637 5760 5890 5890 6025 6025 6025 6025 6167 6167 6167 6167 6167 6167 6167 6167 6167 6235 6261 6287 6313 6339 6366
2 98.50 99.00 99.15 99.27 99.30 99.34 99.34 99.38 99.38 99.38 99.42 99.42 99.42 99.42 99.42 99.42 99.46 99.46 99.46 99.46 99.46 99.47 99.47 99.48 99.49 99.50
3 34.11 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.34 27.23 27.13 27.05 26.98 26.92 26.87 26.83 26.79 26.75 26.72 26.69 26.60 26.51 26.41 26.32 26.22 26.13
4 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.55 14.45 14.37 14.31 14.25 14.20 14.15 14.11 14.08 14.05 14.02 13.93 13.84 13.75 13.65 13.56 13.46
5 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05 9.96 9.89 9.83 9.77 9.72 9.68 9.64 9.61 9.58 9.55 9.47 9.38 9.29 9.20 9.11 9.02
6 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87 7.79 7.72 7.66 7.60 7.56 7.52 7.48 7.45 7.42 7.40 7.31 7.23 7.14 7.06 6.97 6.88
7 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62 6.54 6.47 6.41 6.36 6.31 6.28 6.24 6.21 6.18 6.16 6.07 5.99 5.91 5.82 5.74 5.65
8 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81 5.73 5.67 5.61 5.56 5.52 5.48 5.44 5.41 5.38 5.36 5.28 5.20 5.12 5.03 4.95 4.86
9 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5.26 5.18 5.11 5.05 5.01 4.96 4.92 4.89 4.86 4.83 4.81 4.73 4.65 4.57 4.48 4.40 4.31
10 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85 4.77 4.71 4.65 4.60 4.56 4.52 4.49 4.46 4.43 4.41 4.33 4.25 4.17 4.08 4.00 3.91
11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 4.54 4.46 4.40 4.34 4.29 4.25 4.21 4.18 4.15 4.12 4.10 4.02 3.94 3.86 3.78 3.69 3.60
12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30 4.22 4.16 4.10 4.05 4.01 3.97 3.94 3.91 3.88 3.86 3.78 3.70 3.62 3.54 3.45 3.36
13 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19 4.10 4.02 3.96 3.91 3.86 3.82 3.78 3.75 3.72 3.69 3.66 3.59 3.51 3.43 3.34 3.26 3.17
14 8.86 6.51 5.56 5.04 4.70 4.46 4.28 4.14 4.03 3.94 3.86 3.80 3.75 3.70 3.66 3.62 3.59 3.56 3.53 3.51 3.43 3.35 3.27 3.18 3.09 3.00
15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 3.73 3.67 3.61 3.56 3.52 3.49 3.45 3.42 3.40 3.37 3.29 3.21 3.13 3.05 2.96 2.87
16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3.69 3.62 3.55 3.50 3.45 3.41 3.37 3.34 3.31 3.28 3.26 3.18 3.10 3.02 2.93 2.85 2.75
17 8.40 6.11 5.19 4.67 4.34 4.10 3.93 3.79 3.68 3.59 3.52 3.46 3.40 3.35 3.31 3.27 3.24 3.21 3.19 3.16 3.08 3.00 2.92 2.84 2.75 2.65
18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 3.51 3.43 3.37 3.32 3.27 3.23 3.19 3.16 3.13 3.10 3.08 3.00 2.92 2.84 2.75 2.66 2.57
19 8.19 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52 3.43 3.36 3.30 3.24 3.19 3.15 3.12 3.08 3.05 3.03 3.00 2.93 2.84 2.76 2.67 2.58 2.49
20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37 3.29 3.23 3.18 3.13 3.09 3.05 3.02 2.99 2.96 2.94 2.86 2.78 2.70 2.61 2.52 2.42
21 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40 3.31 3.24 3.17 3.12 3.07 3.03 2.99 2.96 2.93 2.90 2.88 2.80 2.72 2.64 2.55 2.46 2.36
22 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35 3.26 3.18 3.12 3.07 3.02 2.98 2.94 2.91 2.88 2.85 2.83 2.75 2.67 2.58 2.50 2.40 2.31
23 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30 3.21 3.14 3.07 3.02 2.97 2.93 2.89 2.86 2.83 2.80 2.78 2.70 2.62 2.54 2.45 2.35 2.26
24 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26 3.17 3.09 3.03 2.98 2.93 2.89 2.85 2.82 2.79 2.76 2.74 2.66 2.58 2.49 2.40 2.31 2.21
25 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22 3.13 3.06 2.99 2.94 2.89 2.85 2.81 2.78 2.75 2.72 2.70 2.62 2.54 2.45 2.36 2.27 2.17
26 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18 3.09 3.02 2.96 2.90 2.86 2.82 2.78 2.75 2.72 2.69 2.66 2.59 2.50 2.42 2.33 2.23 2.13
27 7.68 5.49 4.60 4.11 3.78 3.56 3.39 3.26 3.15 3.06 2.99 2.93 2.87 2.82 2.78 2.75 2.71 2.68 2.66 2.63 2.55 2.47 2.38 2.29 2.20 2.10
28 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12 3.03 2.96 2.90 2.84 2.79 2.75 2.72 2.68 2.65 2.63 2.60 2.52 2.44 2.35 2.26 2.17 2.06
29 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.09 3.00 2.93 2.87 2.81 2.77 2.73 2.69 2.66 2.63 2.60 2.57 2.50 2.41 2.33 2.23 2.14 2.03
30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 2.98 2.91 2.84 2.79 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.47 2.39 2.30 2.21 2.11 2.01
40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 2.80 2.73 2.66 2.61 2.56 2.52 2.48 2.45 2.42 2.39 2.37 2.29 2.20 2.11 2.02 1.92 1.81
60 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72 2.63 2.56 2.50 2.44 2.39 2.35 2.31 2.28 2.25 2.22 2.20 2.12 2.03 1.94 1.84 1.73 1.60
120 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56 2.47 2.40 2.34 2.28 2.23 2.19 2.15 2.12 2.09 2.06 2.03 1.95 1.86 1.76 1.66 1.53 1.38
∞ 6.64 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41 2.32 2.25 2.19 2.13 2.08 2.04 2.00 1.97 1.94 1.91 1.88 1.79 1.70 1.59 1.47 1.33 1.00
 F values for α = 0.005
Degrees of
Freedom Degrees of freedom for numerator (ν1)
for
Denominator
(ν2) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 24 30 40 60 120 ∞
1 16211 19999 21615 22500 23056 23437 23715 23925 24091 24224 24334 24426 24505 24572 24630 24681 24727 24767 24803 24836 24940 25044 25148 25253 25359 25464
2 198.50 199.00 199.17 199.25 199.30 199.33 199.36 199.37 199.39 199.40 199.41 199.42 199.42 199.43 199.43 199.44 199.44 199.44 199.45 199.45 199.46 199.47 199.47 199.48 199.49 199.50
3 55.55 49.80 47.47 46.19 45.39 44.84 44.43 44.13 43.88 43.69 43.52 43.39 43.27 43.17 43.08 43.01 42.94 42.88 42.83 42.78 42.62 42.47 42.31 42.15 41.99 41.83
4 31.33 26.28 24.26 23.15 22.46 21.97 21.62 21.35 21.14 20.97 20.82 20.70 20.60 20.51 20.44 20.37 20.31 20.26 20.21 20.17 20.03 19.89 19.75 19.61 19.47 19.32
5 22.78 18.31 16.53 15.56 14.94 14.51 14.20 13.96 13.77 13.62 13.49 13.38 13.29 13.21 13.15 13.09 13.03 12.98 12.94 12.90 12.78 12.66 12.53 12.40 12.27 12.14
6 18.63 14.54 12.92 12.03 11.46 11.07 10.79 10.57 10.39 10.25 10.13 10.03 9.95 9.88 9.81 9.76 9.71 9.66 9.62 9.59 9.47 9.36 9.24 9.12 9.00 8.88
7 16.24 12.40 10.88 10.05 9.52 9.16 8.89 8.68 8.51 8.38 8.27 8.18 8.10 8.03 7.97 7.91 7.87 7.83 7.79 7.75 7.64 7.53 7.42 7.31 7.19 7.08
8 14.69 11.04 9.60 8.81 8.30 7.95 7.69 7.50 7.34 7.21 7.10 7.01 6.94 6.87 6.81 6.76 6.72 6.68 6.64 6.61 6.50 6.40 6.29 6.18 6.06 5.95
9 13.61 10.11 8.72 7.96 7.47 7.13 6.88 6.69 6.54 6.42 6.31 6.23 6.15 6.09 6.03 5.98 5.94 5.90 5.86 5.83 5.73 5.62 5.52 5.41 5.30 5.19
10 12.83 9.43 8.08 7.34 6.87 6.54 6.30 6.12 5.97 5.85 5.75 5.66 5.59 5.53 5.47 5.42 5.38 5.34 5.31 5.27 5.17 5.07 4.97 4.86 4.75 4.64
11 12.23 8.91 7.60 6.88 6.42 6.10 5.86 5.68 5.54 5.42 5.32 5.24 5.16 5.10 5.05 5.00 4.96 4.92 4.89 4.86 4.76 4.65 4.55 4.45 4.34 4.23
12 11.75 8.51 7.23 6.52 6.07 5.76 5.52 5.35 5.20 5.09 4.99 4.91 4.84 4.77 4.72 4.67 4.63 4.59 4.56 4.53 4.43 4.33 4.23 4.12 4.01 3.90
13 11.37 8.19 6.93 6.23 5.79 5.48 5.25 5.08 4.94 4.82 4.72 4.64 4.57 4.51 4.46 4.41 4.37 4.33 4.30 4.27 4.17 4.07 3.97 3.87 3.76 3.65
14 11.06 7.92 6.68 6.00 5.56 5.26 5.03 4.86 4.72 4.60 4.51 4.43 4.36 4.30 4.25 4.20 4.16 4.12 4.09 4.06 3.96 3.86 3.76 3.66 3.55 3.44
15 10.80 7.70 6.48 5.80 5.37 5.07 4.85 4.67 4.54 4.42 4.33 4.25 4.18 4.12 4.07 4.02 3.98 3.95 3.91 3.88 3.79 3.69 3.58 3.48 3.37 3.26
16 10.58 7.51 6.30 5.64 5.21 4.91 4.69 4.52 4.38 4.27 4.18 4.10 4.03 3.97 3.92 3.87 3.83 3.80 3.76 3.73 3.64 3.54 3.44 3.33 3.22 3.11
17 10.38 7.35 6.16 5.50 5.07 4.78 4.56 4.39 4.25 4.14 4.05 3.97 3.90 3.84 3.79 3.75 3.71 3.67 3.64 3.61 3.51 3.41 3.31 3.21 3.10 2.98
18 10.22 7.21 6.03 5.37 4.96 4.66 4.44 4.28 4.14 4.03 3.94 3.86 3.79 3.73 3.68 3.64 3.60 3.56 3.53 3.50 3.40 3.30 3.20 3.10 2.99 2.87
19 10.07 7.09 5.92 5.27 4.85 4.56 4.34 4.18 4.04 3.93 3.84 3.76 3.70 3.64 3.59 3.54 3.50 3.46 3.43 3.40 3.31 3.21 3.11 3.00 2.89 2.78
20 9.94 6.99 5.82 5.17 4.76 4.47 4.26 4.09 3.96 3.85 3.76 3.68 3.61 3.55 3.50 3.46 3.42 3.38 3.35 3.32 3.22 3.12 3.02 2.92 2.81 2.69
21 9.83 6.89 5.73 5.09 4.68 4.39 4.18 4.01 3.88 3.77 3.68 3.60 3.54 3.48 3.43 3.38 3.34 3.31 3.27 3.24 3.15 3.05 2.95 2.84 2.73 2.61
22 9.73 6.81 5.65 5.02 4.61 4.32 4.11 3.94 3.81 3.70 3.61 3.54 3.47 3.41 3.36 3.31 3.27 3.24 3.21 3.18 3.08 2.98 2.88 2.77 2.66 2.55
23 9.63 6.73 5.58 4.95 4.54 4.26 4.05 3.88 3.75 3.64 3.55 3.47 3.41 3.35 3.30 3.25 3.21 3.18 3.15 3.12 3.02 2.92 2.82 2.71 2.60 2.48
24 9.55 6.66 5.52 4.89 4.49 4.20 3.99 3.83 3.69 3.59 3.50 3.42 3.35 3.30 3.25 3.20 3.16 3.12 3.09 3.06 2.97 2.87 2.77 2.66 2.55 2.43
25 9.48 6.60 5.46 4.84 4.43 4.15 3.94 3.78 3.64 3.54 3.45 3.37 3.30 3.25 3.20 3.15 3.11 3.08 3.04 3.01 2.92 2.82 2.72 2.61 2.50 2.38
26 9.41 6.54 5.41 4.79 4.38 4.10 3.89 3.73 3.60 3.49 3.40 3.33 3.26 3.20 3.15 3.11 3.07 3.03 3.00 2.97 2.87 2.77 2.67 2.56 2.45 2.33
27 9.34 6.49 5.36 4.74 4.34 4.06 3.85 3.69 3.56 3.45 3.36 3.28 3.22 3.16 3.11 3.07 3.03 2.99 2.96 2.93 2.83 2.73 2.63 2.52 2.41 2.29
28 9.28 6.44 5.32 4.70 4.30 4.02 3.81 3.65 3.52 3.41 3.32 3.25 3.18 3.12 3.07 3.03 2.99 2.95 2.92 2.89 2.79 2.69 2.59 2.48 2.37 2.25
29 9.23 6.40 5.28 4.66 4.26 3.98 3.77 3.61 3.48 3.38 3.29 3.21 3.15 3.09 3.04 2.99 2.95 2.92 2.88 2.86 2.76 2.66 2.56 2.45 2.33 2.21
30 9.18 6.35 5.24 4.62 4.23 3.95 3.74 3.58 3.45 3.34 3.25 3.18 3.11 3.06 3.01 2.96 2.92 2.89 2.85 2.82 2.73 2.63 2.52 2.42 2.30 2.18
40 8.83 6.07 4.98 4.37 3.99 3.71 3.51 3.35 3.22 3.12 3.03 2.95 2.89 2.83 2.78 2.74 2.70 2.66 2.63 2.60 2.50 2.40 2.30 2.18 2.06 1.93
60 8.49 5.79 4.73 4.14 3.76 3.49 3.29 3.13 3.01 2.90 2.82 2.74 2.68 2.62 2.57 2.53 2.49 2.45 2.42 2.39 2.29 2.19 2.08 1.96 1.83 1.69
120 8.18 5.54 4.50 3.92 3.55 3.28 3.09 2.93 2.81 2.71 2.62 2.54 2.48 2.42 2.37 2.33 2.29 2.25 2.22 2.19 2.09 1.98 1.87 1.75 1.61 1.43

Appendix
∞ 7.88 5.30 4.28 3.72 3.35 3.09 2.90 2.74 2.62 2.52 2.43 2.36 2.29 2.24 2.19 2.14 2.10 2.06 2.03 2.00 1.90 1.79 1.67 1.53 1.36 1.36
196