statistical procedure and their uses in plant pathology

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EXPERIMENTAL DESIGN
“It is the plan used in experimentation”
Experimental design means how the observations or measurements or data should be taken to
answer a particular problem under investigation in a valid, efficient and economical way.
The design of experiment indicates the size and number of the experimental units, the
procedure of assigning the treatments to units and also the appropriate grouping of the experimental
units.
It is the formulation of a set of rules and principles according to which experiment is to be
conducted to collect appropriate data whose analysis will lead to valid inferences for the problem
under investigation.
 PURPOSES OF EXPERIMENTAL DESIGN
1. To collect maximum amount of data at a minimum cost, time and resources.
2. To ensure the fulfillment of the requisite assumptions for analysis of the data.
3. To increase the accuracy of the results of an experiment.
 BASIC DESIGNS OF EXPERIMENT
1. Completely Randomized Design (CRD)
2. Randomized Complete Block Design (RCBD)
3. Latin Square Design (LSqD)
 STEPS OF EXPERIMENTAL DESIGN
 Choosing a set of treatments for comparisons.
 Selecting experimental units.
 Specify the number of experimental units.
 Selecting appropriate method of allocation of treatments in the units.
 Grouping of experimental units.
 Selecting parameters for data collection.
 Specify the appropriate method of data collection.
 STEPS IN EXPERIMENTATION
 Define and state the problem;
 Identify objectives and develop a hypothesis;
 Design and conduct experiments to test the hypothesis;
 Collect data
 Analyses the data
 Interpret the data
 Draw conclusions about the hypothesis.
 SOME TERMS
 Population: A population is a collection of items about which you want to make inferences.
Population is the entire pool from which a statistical sample is drawn for a study. Thus, any
selection of individuals grouped by a common feature can be said to be a population.
 Sample: A sample refers to a smaller, manageable version of a larger group. It is an analytic
subset containing the characteristics of a larger population. Samples are used in statistical
testing when population sizes are too large for the test to include all possible members or

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observations. A sample should represent the population as a whole and not reflect
any bias toward a specific attribute. Sample is a part of population.
 Data: Data are a set of numerical measurements or observations that are collected as a source
of information. There are a variety of different types of data, and different ways to represent
data.
 Variable: A variable is any characteristics, number, or quantity that can be measured or
counted. The characteristics which show variation from place to place, field to field, variety
to variety or from material to material are called variables. For example, plant height, number
of leaves per plant, mycelial growth of fungi, and number of fruits per plant etc.
 Parameter: A numerical characteristic of a population, as distinct from a statistic of a
sample. i.e. statistical parameter mean which gives the average measure of character. A
variable whose measure is indicative of a quantity or function that cannot itself be precisely
determined by direct methods.
[It refers to the characteristics that are used to define a given population. It is used to describe
a specific characteristic of the entire population. When making an inference about the
population, the parameter is unknown because it would be impossible to collect information
from every member of the population. Rather, we use a statistic of a sample picked from the
population to derive a conclusion about the parameter.]
 Test of significance: Once sample data has been gathered through an observational study or
experiment, statistical inference allows analysts to assess evidence in favor or some claim
about the population from which the sample has been drawn. The methods of inference used
to support or reject claims based on sample data are known as tests of significance.
A test of significance is a formal procedure for comparing observed data with a claim (also
called a hypothesis), the truth of which is being assessed.
Tests for statistical significance indicate whether observed differences between assessment
results occur because of sampling error or chance. Such "insignificant" results should be
ignored because they do not reflect real differences. ("Significance" here does not imply any
judgment about absolute magnitude or educational relevance. It refers only to the statistical
nature of the difference and indicates the difference is worth taking note of.)
Every test of significance begins with
 A null hypothesis, H0:
H0: there is no difference between the two fungicides in controlling late blight of potato.
 An alternative hypothesis, Ha:
Ha: there is difference between the two fungicides in controlling late blight of potato.

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 Degrees of freedom: The estimation of parameters can be based on the different
amounts of information. The number of independent information that goes into estimating a
parameter is called the degrees of freedom (df).
In general, the degrees of freedom of an estimate is equal to the number of
independent scores that go into the estimate minus the number of parameters estimated as
intermediate steps in the estimation of the parameter itself (which, in sample variance, is one,
since the sample mean is the only intermediate step).
Degrees of freedom refers to the maximum number of logically independent values,
which are values that have the freedom to vary, in the data sample. Once the degrees of
freedom quantity have been selected, specific data sample items must be chosen if there is
an outstanding requirement of the data sample.
 EXTERNAL FACTORS FOR VARIATION
1. Inherent variability in the experimental units.
2. Errors associated with the measurements made.
3. Lack of representative samples collected from the population.
4. Intellectual capacity differs from person to person.
5. Variation in measuring devices.
6. Day to day change in environment.
7. Error in measurement by the researcher.
COMPLETELY RANDOMIZED DESIGN (CRD):
A completely randomized design (CRD) is one where thetreatments are assigned completely
at random so that each experimental unit (such as plant, seed, animal, soil) has the same chance
of receiving any one treatment.
 Experimental error:
Any difference among the experimental units receiving the same treatments is
considered as experimental error.
 Conditions for CRD:
CRD is only appropriate for experiments with homogenous experimental units, such as
laboratory experiments, where environmental effects are relatively easy to control.
 Limitations:
For field experiments, where there is generally large variation among the experimental
plots, in suchenvironmental factors such as soil, the CRD is rarely used.
 Advantages of CRD:
1. Complete flexibility is allowed-any number of treatmentsand replicates may be used.
2. Relatively easy statistical analysis, even with variable replicates and variable experimental
errors for different treatments.
3. Analysis remains simple when data are missing.
4. Provides the maximum number of degrees of freedom for error for a given number of
experimental units and treatments.

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 Disadvantages of CRD:
1. Relatively low accuracy due to lack of restrictions which allows environmental variation to
enter experimental error.
2. Not suited for large numbers of treatments because a relatively large amount of experimental
material is needed which increases the variation.
 Examples of an experiment fit with CRD
In vitro activity of some plant extracts against late blight pathogen,Phytophthora infestans
Treatments:
T0= Control
T1= Neem Leaf Extract
T2= Garlic extract
T3= Allamonda Leaf extract
T4= Ridomil
Replications: 5
 Randomization and layout of a CRD
 Step 1: Determine the total number of experimental plots (n) as the product of the number of
treatments (t) and the number of replications (r); that is n = (r ) (t), For our example, n = (5) (5)
= 25
 Step 2: Assign a plot number to each experimental plot in any convenient manner; for
example, consecutively from 1 to n. For our example, the plot numbers 1------25 are
assigned to the 25 experimental plots.
 Step-3. Assign the treatments to the experimental plots by any of the following
randomized schemes:
A. By table of random numbers
 Step -1: Locate a starting point in table of random numbers by closing your eyes
and pointing a finger to any position in a page.
 Step 2: Using the starting point obtained in step-1, read downward vertically to
obtain n = 25 distinct three digit randomnumbers.
 Step-3: Rank the n random numbers obtained in step 2 in ascending or descending
order.
 Step 4: Divide the n ranks derived in step 3 into t groups, each consisting of r
numbers according to the sequence in whichthe random numbers appeared.
B. By drawing cards
 Step-1: From a deck of ordinary playing cards, draw n cards, one at a time, mixing the
remaining cards after every draw.
 Step-2: Rank the 25 cards drawn in Step 1 according to the suit rank and number of the card.
 Step-3: Assign the t treatments to the n plots by using the rank obtained in Step-2 as the plot
number.
C. By drawing lots
 Step-1: Prepare n identical pieces of paper and divide them into t groups, each groups with r
pieces of paper. Label each piece of paper of the same group with the same letter (or number)
corresponding to a treatment. Uniformly fold each of the n labeled pieces of paper, mix them
thoroughly, and place them in a container.
 Step-2: Draw one piece of paper at a time, without replacementand with constant shaking of

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the container after each draw to mix its content.
 Step-3: Assign the t treatments to the n plots based on the corresponding treatment label and
sequence, drawn in step 2.
 Analysis of variance of CRD:
Analysis of variance means partitioning of the total sum ofsquares into the components
associated with total variation. There are two sources of variation among n observations obtained
from a CRD trial.
 Treatment variation
 Experimental error
Treatment difference will be real if the treatment variation issufficiently larger than experimental
error.
 Steps involved in Analysis of variance
 Step-1: Group the data by treatments and calculate thetreatment totals (T) and grand total (G).
 Step-2: Construct an outline of the analyses of variance as follows:
 Step-3: Using t to represent the number of treatments and r, number
ofreplications, determine the degree of freedom (d. f.) for each source of
variation.
 Step-4: Using Xi to represent the measurement of the ith
plot, Ti as the total of
the ith
treatment, and n as the total number of experimental plots [i.e. N = (r) (t)],
calculate the correction factor and the various sums of squares (SS) as:
 Correction factor (C. F.) =
𝑮𝟐
𝑵
 Total SS = ∑Xi
2
- C.F.
 Treatment SS =
∑ 𝑻𝒊
𝟐
𝒓
 Error SS = Total SS - Treatment SS
 Step-5: Calculate the mean square (MS) for each source of variation by
dividing each SS by its corresponding d.f.:
Treatment MS =
𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝑆𝑆
𝑡−1
 Step-6: Calculate the F value for testing significance of thetreatment difference as:
F =
𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝑀𝑆
𝐸𝑟𝑟𝑜𝑟 𝑀𝑆
Source of
variation
Degrees of
Freedom
Sum of
Square, s
Mean
Square, e
Computed F Tabular F
5% 1%
Treatment t-1
Experimental
Error
t (r-1)
Total (r)(t)-1

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 Step-7: Obtain the tabular F value from the Appendix with f1= the treatment d. f.
= (t-1) and f2 = error d. f. = t (r-1).
 Step-8: Enter all the values computed in steps 3 to 7 in the outline of the
analysis of variance constructed in step 2.
 Step-9: Compare the computed F value of step 6 with the tabular F values of
step 7.
 Inferences on the significance difference among the treatments
 If the computed value of F is larger than the tabular value of F at 1% level of
significance, the treatment difference is said tobe highly significant which is indicated
by placing two asterisks (**) on the computed F value in the ANOVA.
 If the computed value of F is larger than the tabular value of F at 5% level of
significance but smaller or equal to the tabularF value at 1% level of significance, the
treatment difference is said to be significant which is indicated by placing one asterisk
(*) on the computed F value in the ANOVA.
 If the computed value of F is smaller or equal to the tabular Fvalue at 5% level of
significance, the treatment difference is said to be non-significant which is indicated
by placing NS orns on the computed F value in the ANOVA.
RANDOMIZED COMPLETE BLOCK DESIGN (RCBD):
Randomized complete block designs differ from the completely randomized design (CRD) in that the
experimental units are grouped into blocks according to known or suspected variation which is
isolated by the blocks.
RCBD design is characterized by blocks of equal sizes, each of which contains a complete set of all
the treatments.
 Conditions for RCBD:
RCBD design is commonly used for the field experiment where the experimental units are
heterogeneous and the number of treatments is not so large.
 Blocking technique:
 The primary purpose of blocking is to reduce the experimental error by eliminating the
contribution of known sources of variation among experimental units.
 This is done by grouping the experimental units into blocks such that variability within each
block is because only the variation within a block becomes part of the experimental error.
 Blocking is most effective when the experimental area has a predictable pattern of
variability.
 Important points:
 The selection of source of variability to be used as the basis for blocking. Examples are
soil heterogeneity, direction of insect migration, slope of the land
 The selection of the block shape and orientation.
 Unidirectional gradient:
 Use long and narrow blocks
 Orient the blocks length perpendicular to the directionof the gradient
 Two directional fertility gradient with one stronger than the other one: Ignore the

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 weaker one and follow the guideline for thecase of unidirectional gradient.
 Fertility gradient occurs in two directions with both gradientsequally
strong and perpendicular to each other:
 Use block that are square as possible
 Use long and narrow blocks with their length perpendicularto the direction
of one gradient
 Use Latin square design with two-way blockings, one for each gradient.
 When the pattern of variability is not predictable, blocks should be as square as possible.
 Advantages
 Complete flexibility. Can have any number of treatments and blocks.
 Provides more accurate results than the completely randomized design due to grouping.
 Relatively easy statistical analysis even with missing data.
 Allows calculation of unbiased error for specific treatments.
 Disadvantages
 Not suitable for large numbers of treatments because blocks become too large.
 Not suitable when complete block contains considerablevariability.
 Interactions between block and treatment effects increaseerror.
 If there are missing data, a RCBD experiment may be less efficient than a CRD
 Randomization and Layout
 Step-1: Divide the experimental area into r equal blocks, where r is the number of
replications, following technique as described above.
 Step-2: Subdivide the first block into t experimental plots,where t is the number of
treatments.
 Step-3: Assign the treatments to the plots by using the sequence number as the plot number
following lotterymethod.
 Step-4: Repeat step-3 completely for each of the remaining block.
Analysis of variance:
There are three sources of variation in RCBD design:
 Treatment
 Replication or block and
 Experimental error
 Step 1: Group the data by treatments and replications andcalculate the treatment totals,
replication totals and grandtotal.
 Step-2: Construct an outline of the analyses of variance as follows:
Source of
variation
Degrees of
Freedom
Sum of
Square, s
Mean
Square, e
Computed F Tabular F
5% 1%
Replication r-1
Treatment t-1
Experimental
Error
(t-1) (r-1)
Total (r)(t)-1

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 Step-3: Using r to represent the number of replications and t,the number of treatments,
determine the degree of freedom for each source of variation as:
Total d. f. = rt-1
Replication d. f. = r-1
Treatment d. f. = t-1
Error d.f. = (t-1) (r-1)
 Step-4: Compute the correction factor and the various sums of squares (SS) as follows:
 Correction factor (C. F.) =
𝑮𝟐
𝒓𝒕
 Error SS = Total SS - Replication SS - Treatment SS
 Step-5: Compute the mean of square for each source of variation by dividing each sum of
squares by its corresponding degree of freedom as:
 Replication MS=
𝑹𝒆𝒑𝒍𝒊𝒄𝒂𝒕𝒊𝒐𝒏 𝑺𝑺
𝒓−𝟏
 Treatment MS=
𝑻𝒓𝒆𝒂𝒕𝒎𝒆𝒏𝒕 𝑺𝑺
𝒕−𝟏
 Error MS=
𝑬𝒓𝒓𝒐𝒓 𝑺𝑺
(𝒓−𝟏)(𝒕−𝟏)
 Step-6: Compute the F value for testing the treatment difference
 F=
𝑻𝒓𝒆𝒂𝒕𝒎𝒆𝒏𝒕 𝑴𝑺
𝐄𝐫𝐫𝐨𝐫 𝐌𝐒
 Step-7: Compare the computed F value with the tabular F value.
 Block efficiency
 Blocking maximizes the difference among the blocks
 Minimize the difference as small as possible among the plots of the same block
To examine how these objectives has been achieved, the level of significance of the replication
variation need to be determined by computing the F value for replication as:
F (Replication) =
𝐑𝐞𝐩𝐥𝐢𝐜𝐚𝐭𝐢𝐨𝐧 𝐌𝐒
𝑬𝒓𝒓𝒐𝒓 𝑴𝑺
If the calculated replication F value is larger than the tabular F value at 5% level of significance, the
difference among the blocks is significant and if the calculated replication F value is larger than the
tabular F value at 1% level of significance, the difference among the blocks is highly significant.

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 Comparison between treatment means
Two types of pair comparison commonly used in agricultural research
 Planned pair comparison: Specific pair of treatments to be compared has been identified
before the start of the experiment. e. g. comparison of the control treatment with each of the
other treatments.
 Unplanned pair comparison: No specific comparison is chosen in advance. Instead, every
possible pair of treatment mean is compared to identify pairs of treatments that are
significantly different.
 Commonly used test procedures for pair comparison
 Least significant difference (LSD) test
 Duncan’s multiple range test (DMRT)
 Least significance difference (LSD)
The least significant difference (LSD) test is the simplest and the most commonly used procedure
for making pair comparisons of treatment means.
The procedure provides for a single LSD value, at a prescribed level of significance, which
serves as boundary between significant and non-significant differences between any pair of
treatments means.
 Conditions:
When the F test for treatment effect is significant and the number of treatments is not too large -
less than six.
 Computation of the LSD value at α level of significance
 STEP-1. Calculation of Standard error (SE)
Standard error (SE) of the mean difference for any pair of treatment means
=√
𝑬𝒓𝒓𝒐𝒓 𝑽𝒂𝒓𝒊𝒂𝒏𝒄𝒆 ×𝟐
𝑻𝒉𝒆 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏 𝒑𝒆𝒓 𝒎𝒆𝒂𝒏
 STEP-2. Calculation of LSD
LSD = SE between any two treatment means × value of t at .05 or .01 for the respective
degree of freedom (error d.f.)
LSD at 5% = 0.8 x 2.08 = 1.66
LSD at 1% = 0.8 x 2.26 = 2.26
 STEP-3. Arrange means of all the treatments in order of merit and use the LSD in the
following manner
20.75-2.26 = 18.49, it means that 20.75 to 18.49 is statistically similar. But the mean of B is
less than 18.49 and hence A is significantly superior.
Treatments Mean Yield
A 20.75
B 14.00
C 9.25
D 8.00
E 6.00
F 3.25
G 1.75

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14.00-2.26 = 11.74, it means that 11.74 to 14.00 is statistically similar. Hence B is
more efficient than C and others
 Thumb rule for mean comparisons
When the difference between the means of two treatments to be compared is greater than
20% of the mean of others treatments the difference will be significant.
For example:
The difference between the means of A and B is (20.75-14.00) = 6.75 which is 48%
(6.75x100/14.00) of the mean of B.
Therefore, A is significantly greater than B.
 Duncan’s Multiple Range Test (DMRT)
Experiments that require the comparisons of all possible pairs of treatment means,
these specific comparisons which is not possible by F-test and where LSD is not suitable for
treatments mean comparisons, for this in 1955, Duncan developed this new multiple range test.
For LSD only a single value is required for any pair of comparison at a prescribed
level of significance, the DMRT requires the computation of a series of values, each
corresponding to a specific set of pair comparisons.
 Computation of DMRT values for comparing all possible pair of means
 STEP-1. Compute the standard error of mean difference for any pair of treatment means
Standard error (SE) for the difference of two means of the treatments
=√
𝐸𝑟𝑟𝑜𝑟 𝑚𝑒𝑎𝑛 𝑠𝑞𝑢𝑎𝑟𝑒
𝑟
; Where r is the number of replication
SE =√
1.3
4
= 0.57
 STEP-2. Compute the values of the Least Significant Ranges using tabular value of Significant
Standardized ranges (SSR) for the 5% and 1% levels.
Each of these significant ranges is then multiplied by the value of SE to get the Least
Significant Ranges (LSR) which are then tabulated as follows:
LSR = SSR × SE
 STEP-3. Rank all the treatment means in
decreasing or increasing order
Value of P 2 3 4 5 6 7
SSR 2.94 3.09 3.18 3.25 3.30 3.33
LSR 1.68 1.76 1.81 1.85 1.88 1.90
Treatments Mean Yield
G 1.75
F 3.25
E 6.00
D 8.00
C 9.25
B 14.00
A 20.75

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 STEP-4. Test the differences in the following order
A. Largest minus smallest i. e. 20.75-1.75 = 19.00 > 1.90
Largest minus second smallest i. e. 20.75- 3.25 = 17.50 > 1.88
Largest minus third smallest i. e. 20.75-6.00 = 19.00 > 1.85
Largest minus fourth smallest i. e. 20.75-8.00 = 12.75 > 1.81
Largest minus fifth smallest i. e. 20.75-9.25 = 11.50 > 1.76
Largest minus second largest i. e. 20.75-14.00 = 6.75 > 1.68
B. Second Largest minus smallest i. e. 14.00-1.75 = 12.25 > 1.88
Second Largest minus second smallest i. e. 14.00- 3.25 =10.75 >1.85
Second Largest minus third smallest i. e. 14.00-6.00 = 8.00 > 1.81
Largest minus fourth smallest i. e. 14.00-8.00 = 6.00 > 1.76
Largest minus fifth smallest i. e. 14.00-9.25 = 4.75 > 1.68
C. Third Largest minus smallest i. e. 9.25-1.75 = 7.50 > 1.85
Third Largest minus second smallest i. e. 9.25- 3.25 = 6.00 >1.81
Third Largest minus third smallest i. e. 9.25-6.00 = 3.25 > 1.76
Third Largest minus fourth smallest i. e. 9.25-8.00 = 1.25 < 1.68
D. Fourth Largest minus smallest i. e. 8.00-1.75 = 6.25 > 1.81
Fourth Largest minus second smallest i. e. 8.00- 3.25 = 4.75 >1.76
Fourth Largest minus third smallest i. e. 8.00-6.00 = 2.00 > 1.68
E. Fifth Largest minus smallest i. e. 6.00-1.75 = 4.25 > 1.76
Fifth Largest minus second smallest i. e. 6.00- 3.25 = 2.75 >1.68
F. Sixth largest minus smallest i.e. 3.25-1.75 = 1.50 < 1.68
[Here ">" means that corresponding two treatment means are significantly different and "≤"
means that are similar]

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 LATIN SQUARE DESIGN (LSqD)
 Two known sources of variation among experimental units can be handled.
 Two independent blocking criteria, instead of only one as in the RCBD design are
considered.
 Blocking is done in two directions, known as Row and Column.
 Each row contains all the treatments and each column contain all the treatments.
Purposes: To estimate variation among row-blocks as wellas column-blocks and to remove them
from experimental error.
 Conditions for use of LSqD
This design is used when the experimental material is heterogeneous and where the variation
occurs in two directions or many directions that can't be minimized by RCBD. e. g. Sclerotial
population of Rhizoctonia solani and Sclerotium rolfsii, CFU of Ralstonia solanacearum, Population
of Fusarium spp. in the field may vary in two or many directions. The number of treatments should
be within 4 to 8 and the number of treatment is equal to the number of replications.
 Advantages of LSqD
1. Experimental error can be controlled in two directions.
2. Efficiency is high as compared to RCBD.
 Disadvantages of LSqD
1. The number of treatments must be equal to the number of replicates.
2. The experimental error is likely to increase with the size of the square.
3. Small squares have very few degrees of freedom for experimental error.
Example: An experiment was conducted with five fungicides to compare their efficacy against
cercospora leaf spot of banana. The fungicides were sprayed by five technicians with five sprayers.
Technicians were assigned in the rows and the sprayers were used in the columns.
 Randomization and Layout
A = Calyxin
B = Ridomil
C = Provax
D = Cupravit
E = Secure
Random number Sequence Rank
385 1 2
275 2 1
683 3 4
914 4 5
545 5 3
 Randomization for the rows
 Select five 3-digit random number from the random table.
 Give sequence number against each random number
 Rank the random numbers in order of merit
 Use the rank to represent the row number of the selected plan
(Figure 1)
 Use sequence to represent the row number of the new plan (Figure 2)

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Figure 1
Figure 2
 Randomization for the columns
 Select another five 3-digit random number from the random table.
 Give sequence number against each random number
 Rank the random numbers in order of merit
 Use sequence as the column number of Figure-2.
 Use the rank as the column number of Figure-3.
Figure-3
 Analysis of variance
There are four sources of variation in a LSqD:
 Row
 Column
 Treatment
 Experimental error
 STEP-1: Arrange the raw data according to their row and column designations, with
the corresponding treatment clearly specified for each observation as shown below:
 STEP-2: Compute row totals (R), column totals grand total as shown in the Table below
ROW 1 A B C D E
ROW 2 B C D E A
ROW 3 C D E A B
ROW 4 D E A B C
ROW 5 E A B C D
ROW 1 B C D E A
ROW 2 A B C D E
ROW 3 D E A B C
ROW 4 E A B C D
ROW 5 C D E A B
Random number Sequence Rank
870 1 4
965 2 5
838 3 2
853 4 3
450 5 1
COLUMN 1 COLUMN 2 COLUMN 3 COLUMN 4 COLUMN 5
ROW 1 E A C D B
ROW 2 D E B C A
ROW 3 B C E A D
ROW 4 C D A B E
ROW 5 A B D E C

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Plabon Saha
ID. No.: 22220506
Reg. No.: 54096
STEP-3: Outline the ANOVA
STEP-4: Using t to represent the number of treatments, determine the degree of freedom for
each source of variation as:
Total d. f. = t2
-1
Row d. f. = Column d. f. = Treatment d. f. = t-1
Error d. f. = (t-1) (t-2)
STEP-5: Compute the correction factor and the various sums of squares as:
Correction factor (C. F.) =
𝐺2
𝑡2
Total SS = ∑ 𝑋2
− C.F.
Row SS =
∑ 𝑅2
𝑡
; Column SS =
∑ 𝐶2
𝑡
; Treatment SS =
∑ 𝑇2
𝑡
Error SS = Total SS− Row SS− Column SS− Treatment SS
COLUMN 1 COLUMN 2 COLUMN 3 COLUMN 4 COLUMN 5 Row Total
ROW 1 E = 449 A = 444 C = 401 D = 229 B =292
ROW 2 D = 463 E = 375 B = 323 C = 264 A = 415
ROW 3 B = 393 C = 353 E = 278 A = 404 D = 425
ROW 4 C = 371 D = 241 A = 441 B = 410 E = 392
ROW 5 A = 258 B = 430 D = 450 E = 285 C = 347
Column Total Grand Total
Treatment Total Mean
A
B
C
D
E
Source of
variation
Degree of
Freedom
Sum of
Squares
Mean
Square
Computed
F
Tabular F
5% 1%
Row t-1
Column t-1
Treatment t-1
Error (t-1)(t-2)
Total t2
-1

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Plabon Saha
ID. No.: 22220506
Reg. No.: 54096
STEP-6: Compute the mean of square for each source of variation by diving the sum
of squares by its corresponding degree of freedom:
Row MS=
𝑅𝑜𝑤 𝑆𝑆
𝑡−1
; Column MS=
𝐶𝑜𝑙𝑢𝑚𝑛 𝑆𝑆
𝑡−1
; Treatment MS=
𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝑆𝑆
𝑡−1
Error MS=
𝐸𝑟𝑟𝑜𝑟 𝑆𝑆
(𝑡−1)(𝑡−2)
STEP-7: Compute the F value for testing the treatment effect
F =
𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝑀𝑆
𝐸𝑟𝑟𝑜𝑟 𝑀𝑆
SPLIT-PLOT DESIGN
Split-plot design is basically a factorial experiment in which one of the factors (known as main-plot
factor) whose effect is already known is arranged in the design in such a way so that some
information about its effect is sacrificed in order to obtain more information on other factors (Known
as subplot factors), including their interaction.
 Main plot factor
The factor which is sacrificed thus is called the main treatment and is allotted to main plots of the
experiments. The measurement of effects of the main-plot treatments (i. e. the levels of the main-
plot factors) is less precise than that obtainable with a randomized complete block design.
 Subplot factor
Each of the main plot is subdivided equally in which the second factor, called subplot factor, is
assigned. Thus, each main plot becomes a block for the subplot treatment (i.e. the levels of the
subplot factors). Since each main plot is splited up the design is called split-plot.
The measurement of the main effect of the subplot factor and its interactions with the main plot
factor is more precise than that obtainable with a RCBD.
 Appropriate use of split-plot designs
1. When the practical limit for plot size is much larger for one factor compared with the other,
e.g., in an experiment to compare irrigation treatments and population densities; irrigation
treatments require large plots and should, therefore, be assigned to the main plots while
population density should be assigned to the subplots.
2. When greater precision is desired in one factor relative to the other e.g., if several varieties
are being compared at different fertilizer levels and the factor of primary interest is the
varieties, then it should be assigned to the subplots and fertilizer levels assigned to the main
plots.

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Plabon Saha
ID. No.: 22220506
Reg. No.: 54096
 Advantages
1. Experimental units which are large by necessity or design may be utilized to compare
subsidiary treatments.
2. Increased precision over a randomized complete block design is attained on the subplot
treatments and the interaction between subplot and main plot treatments.
3. The overall precision of the split plot design relative to the randomized complete block
design may be increased by designing the main plot treatments in a Latin square design or in
an incomplete Latin square design.
 Disadvantages:
1. The main plot treatments are measured with less precision than they are in a randomized
complete block design.
2. When missing data occur, the analysis is more complex than for a randomized complete
block design with missing data.
3. Different treatment comparisons have different basic error variances which make the analysis
more complex than with the randomized complete block design, especially if some unusual
type of comparison is being made.
 Uses of Split Plot Design
1. Variable natural plot sizes for different factors in a factorial.
2. In situations where a second factor is brought into an experiment to increase its scope.
3. In situations where it is known that larger differences will occur for some factors than others.
4. Where greater precision is desired for some factors than others.
 Randomization and Layout
There are two separate randomization process ina split-plot design
1. One for main plot
2. Another for the sub plot
In each replication, main plot treatments are first randomly assigned to the main plots followed
by a random assignment of the subplot treatments within each main plot.
STEP-1: Divide the experimental area into r = 4 blocks, each of which is further divided
into a = 3 main plots

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Plabon Saha
ID. No.: 22220506
Reg. No.: 54096
STEP-2:
Following the RCBD randomization procedure with a = 3 treatments and r = 4 replications.
Randomly assign the three treatments (frequency) to the three main plots in each of four
blocks.
STEP-3:
Divide each of the (r)(a) = 12 main plots into b = 5 subplots and, following the RCBD
randomization procedure for b = 5 subplot treatments, (r)(a) = 12 replications. Randomly
assign the 5 subplot treatments (different levels of tilt concentration) in each of the 12 main
 Analysis of variance in Factorial Design
The sources of variation:
 Replication
 Treatment
 Factor A
 Factor B
 Interaction of two factors (A XB)
 Error
N1
N2
N3
N1
N2
N3 N1
N2
N3
N1
N2
N3

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Plabon Saha
ID. No.: 22220506
Reg. No.: 54096
STEP-1. Denote the number of replications by r, the level of factor A by ‘a’ and the
level of factor B by ‘b’. Construct the outline of the analysis of variance as:
STEP-2. Compute treatment totals (T), replication totals (R) and the grand total (G), and
compute the total SS, treatment SS and error SS:
Correction factor (C.F.) =
𝐺2
𝑟𝑎𝑏
Total SS = ∑ 𝑋2
− C.F.
Replication SS =
∑ 𝑅
2
𝑎𝑏
− C.F.
Treatment SS =
∑ 𝑇
2
𝑎𝑏
− C.F.
Error SS = Total SS − Replication SS − Treatment SS
Table 1. Preliminary analyses of variance
Source of
variation
Degree of
Freedom
Sum of
Squares
Mean
Square
Computed
F
Tabular F
5% 1%
Replication r-1
Treatment ab-1
Error (r-1) (ab-1)
Total rab-1
STEP-3. Construct the Factor A × Factor B two-way table as:
Incidence (%) N X K Factor B
(MP)
Total
N1 (60
kg/ha)
N2 (80
kg/ha)
K1 (40 kg/ha)
K2(70 kg/ha)
Factor A (Urea)
Total
Source of
variation
Degree of
Freedom
Sum of
Squares
Mean
Square
Computed
F
Tabular F
5% 1%
Replication r-1
Treatment
Factor A
Factor B
A × B
ab-1
a-1
b -1
(a-1) (b-1)
Error (r-1) (ab-1)
Total rab-1

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ID. No.: 22220506
Reg. No.: 54096
STEP-4. Compute the three factorial componentsof the treatment sum of squares as:
 Factor A SS =
∑ 𝐴
2
𝑟𝑏
− C.F
 Factor B SS =
∑ 𝐵
2
𝑟𝑎
− C.F
 Factor AB SS = Treatment SS − Factor A − SS Factor B SS
STEP-5. Compute the mean square (MS) for each source of variation by dividing the SS by its
d.f.:
 Factor A MS =
𝐹𝑎𝑐𝑡𝑜𝑟 𝐴 𝑆𝑆
𝑎−1
 Factor B MS =
𝐹𝑎𝑐𝑡𝑜𝑟 𝐵 𝑆𝑆
𝑏−1
 Factor A×B MS =
𝐹𝑎𝑐𝑡𝑜𝑟 𝐴×𝐵 𝑆𝑆
(𝑎−1)(𝑏−1)
 Error MS =
𝐸𝑟𝑟𝑜𝑟 𝑆𝑆
(𝑟−1)(𝑎𝑏−1)
STEP-6. Compute the F value for each of the three factorial components as:
 F (Factor A) =
𝐹𝑎𝑐𝑡𝑜𝑟 𝐴 𝑀𝑆
𝐸𝑟𝑟𝑜𝑟 𝑀𝑆
 F (Factor B) =
𝐹𝑎𝑐𝑡𝑜𝑟 𝐵 𝑀𝑆
𝐸𝑟𝑟𝑜𝑟 𝑀𝑆
 F (Factor A× B) =
𝐹𝑎𝑐𝑡𝑜𝑟 𝐴×𝐵 𝑀𝑆
𝐸𝑟𝑟𝑜𝑟 𝑀𝑆
STEP-7. Compare each of the computed F values with the tabular F value from the Appendix-
--- with
f1 = d. f. of the numerator MS and
f2 = d. f. of the denominator MS, at a prescribed level of significance.
STEP-8. Compute the coefficient of variation as:
 CV (%) =
√𝐸𝑟𝑟𝑜𝑟 𝑀𝑆 𝑜𝑟 𝐸𝑟𝑟𝑜𝑟 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒
𝐺𝑟𝑎𝑛𝑑 𝑀𝑒𝑎𝑛
× 100

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Plabon Saha
ID. No.: 22220506
Reg. No.: 54096
STEP-9. Enter all values obtained in steps 4-8 in the analysis of variance Table 2
Table 2: Analysis of variance
Source of
variation
Degree of
Freedom
Sum of
Squares
Mean
Square
Computed F Tabular F
5% 1%
Replication r-1
Treatment
Factor A
Factor B
A X B
ab-1
a-1
b -1
(a-1) (b-1)
Error (r-1) (ab-1)
Total rab-1

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Plabon Saha
ID. No.: 22220506
Reg. No.: 54096
Measures of dispersion
Various measures (such as arithmetic mean, median, mode, geometric and harmonic mean) of
location of a frequency distribution which shows us a typical value of the whole set around which
most of the values in the distribution tend to cluster.
The measure of dispersion i.e. variability among the values of a frequency distribution which tells us
how compactly the individual values are distributed around the average.
 Four important measures of dispersion
Range, Quartile deviation, mean deviation and Standard Deviation
 Standard deviation
The standard deviation (SD) is a measure that is used to quantify the amount of variation or
dispersion of a set of data values.
The standard deviation of a set of X1, X2, ----------Xn is given by
SD (x)= √
∑(𝑥𝑖−x̄)2
𝑛
SD (x)= √∑ 𝑥𝑖2−
(∑ 𝑥𝑖2)2
𝑛
𝑛
Calculation of arithmetic mean (AM)
Plant height, X Frequency fx
32.0 1 32.0
33.0 3 99.0
33.5 4 134.0
34.2 5 171.0
35.0 7 245.0
36.5 10 365.0
37.0 8 296.0
38.0 5 190.0
39.5 2 79.0
40.0 1 40.0
∑x = 358.7 ∑f = 46 ∑fx = 1651.0
Mean=
∑ 𝑓𝑥
∑ 𝑓
=
1651.0
46
= 35.89
𝑥𝑖2
= Raw sum of squares of the set
(∑ 𝑥𝑖2)2
𝑛
= Correction factor
∑(𝑥𝑖 − x̄)2
= Corrected sum of squares

22. 22 | P a g e
Plabon Saha
ID. No.: 22220506
Reg. No.: 54096
 Variance:
The variance of a set of data is defined as the square of the standard deviation.
i. e. V (x)= {Sd (x)}2
=
∑(𝑥𝑖−x̄)2
𝒏
 Coefficient of variation (cv):
The standard deviation can also be used to find out the another important measure of
variability, the coefficient of variation (cv) which is given by
cv (%) =
𝑆𝑑
𝑀𝑒𝑎𝑛
× 100
It expresses the experimental error as percentage of the mean. The higher cv value, the lower
is the reliability of the experiment.
CV indicates the degree of precision with which the treatments are compared and is a good
index of the reliability of the experiment.
Acceptable limit of the cv value
The range of cv values
 Variety trial: 6-8%
 Fertilizer trial: 10-12%
 Insecticide and Herbicide trials: 13-15%
 Standard Error (SE)
The standard deviation of a statistic is known as its standard error (SE). So the standard
deviation of x is known as its standard error and is denoted as SE (x).
SE (x)=
𝑆𝑑 (𝑥)
√𝑛
 Alternative calculation method:
The elaborate method of computing the
differences by Duncan’s method can be avoided
as follows:
Subtract the LSR below p = 7 from the
largest mean i.e. 20.75-1.90 = 18.85
Note: All means smaller than 18.85 shall be
significantly inferior to A.
Similarly, 14.00-1.88 = 12.12 and all means
smaller than this will be significantly inferior to
B.
 Factorial experiment
An experiment in which the treatments consist of all possible combinations of the selected
levels in two or more factors is referred to as a factorial experiment.
Thus, in factorial experiments the treatments are replaced by treatment combinations formed
by combining the different levels of the test factors.
Two factors are said to interact if the effect of one factor changes as the level of the other
factor changes. The interaction effect of two factors can be measured by factorial experiment.
Treatments Mean Yield Remarks
A 20.75 a
B 14.00 b
C 9.25 c
D 8.00 c
E 6.00 d
F 3.25 e
G 1.75 e
Treatments having common letters are statistically similar

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ID. No.: 22220506
Reg. No.: 54096
Based on the number of factor, experiments are said to be:
1) One factor experiment consists of one set of treatments
e.g. Comparative efficacy of five fungicides to control late blight of potato.
Basic designs will be: CRD, RCBD and LSqD.
Treatments: Fungicides
Number of treatments: 5
Factor: 1 (Fungicide)
2) Factorial experiment consists of two or more factors of treatments
e.g. Effect of different levels of nitrogen and potassium on the incidence of foot and root rot
of lentil.
Nitrogen: Factor A, Level: 2
N1 = 80 kg/ha
N2 = 120 kg/ha
Potassium: Factor B, Level: 2
K1 = 60 kg/ha
K2 = 90 kg/ha
Treatment combinations = levels of factor A × Levels of factor B, = 2 × 2 = 4
N1K1 = T1
N1K2 = T2
N2K1 = T3
N2K2 = T4
 Effect of treatments in a factorial experiment
The effect of treatments in a factorial experiment are broadly classified into three categories
as follows:
1. Main effect
2. Simple effect
3. Interaction effect
1. Main effect:
The difference between the levels of one factor within a single level of another factor is
called simple effect.
2. Simple effect:
The main effect of one factor is the average of the simple effects of the other factor.
Here, the main effect of Factor-A (Urea) is the average of the simple effects of the Factor-
B (Muriate of Potash), on the other hand, the main effect of Factor-B (MP) is the average
of the simple effects of the Factor-A (Urea).
Treatment Simple effect Simple effect Average (Main effect )
N 10 14 12
K 14 18 16
Factor B (MP) Factor A (Urea)
N1 (60 kg/ha) N2 (80 kg/ha) Simple effect
K1 (40 kg/ha) 30 40 10
K2 (70 kg/ha) 12 26 14
Simple effect 18 14

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ID. No.: 22220506
Reg. No.: 54096
3. Interaction effect:
Interaction effect is present due to relationship between the factors. The effect of one factor
depend on the other factor, the changes in the levels of one factor causes the change in the
effect of any level of the other factor, under this situation, interaction is present and the
factors are said to be interacted.
a) Interaction is present between factor-A and Factor-B when the simple effect of one factor
are not equal.
Factor B (MP) Factor A (Urea)
N1 (60 kg/ha) N2 (80 kg/ha) Simple effect
K (40 kg/ha) 30 40 10
K (70 kg/ha) 12 26 14
Simple effect 18 14
b) Interaction is absent between factor-A and Factor-B because the simple effect of one
factor is equal. Simple effect of N and K nutrient on disease incidence (%) showing no
interaction.
Factor B (MP) Factor A (Urea)
N1 N2 Simple effect
K1 30 44 14
K2 12 26 14
Simple effect 18 18
From the above Table,
Simple effect of MP at N1 is 18%
Simple effect of MP at N2 is 18%
Simple effect of Urea at K1 is 14%
Simple effect of Urea at K2 is 14%
Data presentation
Interaction effect of MP and N
Factor B (MP) Factor A (Urea)
N1 (60 kg/ha) N2 (80 kg/ha)
K1 (40 kg/ha) 30aA 40aB
K2(70 kg/ha) 12bA 26bB
Interaction effect of MP and N
Treatments Mean
K1N1 30b
K1N2 40a
K2N1 12d
K2N2 26c