6 Completely Randomized Design (CRD)

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Design and Analysis of
Single Factor
Experiments

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Design and Analysis of Single Factor
Experiments:
A single factor experiment is a simple
experiment that consists of limited number of
treatments belonging to a certain factor.

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Examples of single factors:
Different varieties of a crop:
Different levels of a pesticide.
Different genotypes of animals.
Different strains of bacteria or fungi.
Different types of harvesters or tractors.

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Completely
Randomized Design
(CRD)

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Completely randomized design is the simplest,
most easily understood, and most easily analyzed
designs. In CRD, treatments are assigned
randomly to homogenous experimental units
without any condition.

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Total variation
Treatment
variation
Experimental
error variation

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Advantages of CRD:
Simple to use.
Flexible.
Easy statistical analysis.
Lost experimental unit don’t complicate
statistical analysis.
Degree of freedom of experimental error is
higher compared to other designs.

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Disadvantages of CRD:
Needs homogenous experimental units.
The size of experimental error is relatively
large to other designs.

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Uses of CRD:
Laboratory experiments.
Greenhouse experiment.
Animal experiment.
It is very rarely used in field experiments.

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Mathematical model of CRD:
Where:
Character measured
Grand mean
Treatment effect
Experimental error effect

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An example:
Research problem:
Papaya (Carica
papaya
) is generally propagated
from seeds but seedlings are variable in both
plant and fruit characteristic. Vegetative
propagation of papaya is preferred sine it
produces true-to type plant.
So, a researcher in the field of tissue culture
wanted to study the effect the growth regulator,
BAP, on micro-propagation of papaya in the
tissue culture laboratory.

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Shoot tip explants of about one cm long were
cultured on a medium supplement with different
concentrations of BAP (0, 0.5, 1.0, 1.5 and 2.0
mg/l) in test tubes.
Each concentration repeated 4 times.
All cultures were maintained at temperature of
25-27
0
C and 16 hour photoperiod. Length of
roots was measured 4 weeks later.

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Objective of the experiment:
Effect BAP on Micro-propagation of Papaya
(Carica
papaya
L.)

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Hypothesis:
Null hypothesis: There are no significant differences in
root length of papaya in response to different
concentrations of BAP.
Alternative hypothesis: There are significant differences
in root length of papaya in response to different
concentrations of BAP.

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Randomization and layout of experiment:
•Number of treatments (T): 5 concentrations of
BAP
•Number of replicates (R): 4 times
•Total number of experimental units (N):
•N = T* R = 5*4 = 20

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20 test tubes

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Treatments were randomly distributed to
experimental units

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Data collection and organization
Concentrations
of BAP (mg/l)
Replication
(root length in cm)
Treatment
total
Treatment
mean
I II III IV
T1: 0.0 mg/l 12 13 12 14 51 12.75
T2: 0.5 mg/l 13 13 12 15 53 13.25
T3: 1.0 mg/l 14 15 14 13 56 14.00
T4: 1.5 mg/l 16 17 14 16 63 15.75
T5: 2.0 mg/l 17 18 16 19 70 17.50
Grand total 293
Grand mean 14.65

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Testing hypothesis: Steps for analysis of
variance:
1)Sum of squares
2)Degree of freedom
3)Mean squares
4)F values
5)Analysis of variance Table (ANOVA table)
6)Testing significance
7)General conclusion
8)Summary of results

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1) Sum squares:
Before calculation of sum of squares, the
correction factor is, firstly, calculated as follow:

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Sum of squares for total:

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Sum of squares for treatments:

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Sum of squares for error:

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2) Degree of freedom:
Degree of freedom for total:

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Degree of freedom for treatments:
Degree of freedom for error:

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3) Mean squares:
Mean squares for treatments:
Mean squares for error:

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4) F values
Calculated F value:

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Tabulated F value
Level of significance or probability: 1% or 5%
Degree of freedom for treatments: 4
Degree of freedom for error: 15

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Table: distribution of F: alpha = 0.05
DFE
Number of Means
1 2 3 4 5
1 161.469 199.493 215.737 224.500 230.066
2 18.5128 18.9995 19.1642 19.2467 19.2969
3 10.1278 9.5522 9.2767 9.1173 9.0133
4 7.7087 6.9444 6.5915 6.3882 6.2561
5 6.6080 5.7861 5.4095 5.1922 5.0503
6 5.9874 5.1433 4.7570 4.5337 4.3874
7 5.5914 4.7374 4.3469 4.1204 3.9715
8 5.3177 4.4590 4.0662 3.8378 3.6875
9 5.1174 4.2565 3.8626 3.6331 3.4817
10 4.9647 4.1028 3.7083 3.4781 3.3258
11 4.8443 3.9823 3.5875 3.3567 3.2039
12 4.7472 3.8853 3.4903 3.2592 3.1059
13 4.6672 3.8055 3.4106 3.1791 3.0255
14 4.6001 3.7389 3.3439 3.1122 2.9582
15 4.5430 3.6824 3.2874 3.0555 2.9013

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Table: distribution of F: alpha = 0.01
DFE
Number of Means
1 2 3 4 5
1 4063.25 4992.22 5404.03 5636.51 5760.41
2 98.5043 98.9977 99.1505 99.2654 99.3038
3 34.1141 30.8155 29.4553 28.7098 28.2359
4 21.1977 17.9994 16.6939 15.9766 15.5219
5 16.2582 13.2737 12.0599 11.3919 10.9670
6 13.7450 10.9246 9.7796 9.1482 8.7460
7 12.2466 9.5467 8.4512 7.8465 7.4605
8 11.2586 8.6492 7.5911 7.0060 6.6319
9 10.5612 8.0216 6.9918 6.4221 6.0570
10 10.0444 7.5594 6.5523 5.9943 5.6363
11 9.6461 7.2057 6.2167 5.6684 5.3160
12 9.3302 6.9266 5.9526 5.4120 5.0643
13 9.0737 6.7009 5.7394 5.2054 4.8616
14 8.8615 6.5148 5.5639 5.0354 4.6950
15 8.6830 6.3589 5.4170 4.8932 4.5556

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5) ANOVA table:
SOV SS DF MS F-cal
F-tab
0.05 0.01
Treatment 61.33 4 15.325 11.942

3.08 4.89
Experimental
error
19.25 15 1.283
Total 80.55 19

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6) Testing significance
Since the calculated f value (11.942) is greater
than tabulated f value at 0.05 (3.08) and
0.01(4.89) levels of significance, null
hypothesis is rejected.

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7) General conclusion:
There are significant differences in root length
of papaya in response to different concentrations
of BAP.

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8) Summary of results:
Standard error (SE±) =
Coefficient of variation (CV%) =

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CV % is used to measure accuracy in which
experiment is carried out
CV % in laboratory experiment should be less
than 20%.
CV % in Field experiment should be less than
40 %.
CV % in surveys experiment should be less
than 60 %.

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Table. 1 Effect BAP on Micro-propagation of
Papaya (Carica
papaya

L.)
Concentrations
of BAP (mg/l)
0.0 mg/l 0.05 mg/l 1.0 mg/l 1.5 mg/l 2.0 mg/l
Mean of root
length (cm)
12.75 13.25 14.00 15.75 17.50
SE± 0.566
CV % 7.73

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12.75
13.25
14
15.75
17.5
0
5
10
15
20
0.0 mg/l 0.05 mg/l 1.0 mg/l 1.5 mg/l 2.0 mg/l
Mean of root length (cm)
Concentrations of BAP )mg/l)
Fig. 1. Effect BAP on Micro-propagation of Papaya
(Carica
papaya

L.)

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Reference:
Dafaallah, A. B. (2017).Fundamentals of Design and
Analysis of Agricultural. Experiments
(Observation – Experimentation –Discussion), Part
One. First Edition. University of Gezira House for
Printing and Publishing, Wad Medani, Sudan. Pp
246.
Dafaallah, A. B. (2017). Fundamentals of Design and
Analysis of Agricultural Experiments (Observation
– Experimentation –Discussion), Part Two. First
Edition. University of Gezira House for Printing
and Publishing , Wad Medani, Sudan. Pp 204.

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Thanks
Dr. Awadallah Belal Dafaallah
E-mail: [email protected];
[email protected]
Tel: [phone]