Experimental unit (plot) — the smallest entity to which a treatment is applied (a plot, animal, person, machine).
Treatment — the procedure / level whose effect is to be measured (fertilizer brand, drug dose).
Replication — number of experimental units that receive the same treatment.
Yield (response) — the variable being measured (kg, score, mm).
Block — a group of homogeneous units (used in RBD/LSD).
Error — variation that cannot be explained by treatments or blocks.
2. Three Principles of Design of Experiments (Fisher)
PRINCIPLES
Replication — applying each treatment to several units to estimate experimental error and increase precision.
Randomization — assigning treatments at random to experimental units, ensuring unbiasedness and validity of statistical tests.
Local control — grouping homogeneous units into blocks to reduce experimental error (used in RBD, LSD; not in CRD).
3. Completely Randomised Design (CRD)
DEFINITION
The simplest design — each treatment is allotted to experimental units completely at random. Only two principles are used: replication and randomization. There is no local control / blocking.
When to use CRD?
Experimental units are homogeneous.
Experiment is conducted in a controlled lab/greenhouse.
Number of treatments is small.
Advantages
Most flexible — number of replications per treatment can vary.
Maximum number of degrees of freedom for error (highest precision when units are homogeneous).
Simple statistical analysis (one-way ANOVA).
Loss of one or more observations causes minimal complications.
Disadvantages
Inefficient when units are heterogeneous (variation gets pooled into error).
Not suitable for field experiments where soil fertility varies across plots.
Applications
Laboratory experiments where conditions are controllable.
Greenhouse pot experiments.
Animal feeding trials with homogeneous animals.
Industrial chemistry and pharmacy experiments.
4. Layout of CRD
Suppose 3 treatments T1, T2, T3 with 4 replicates each → 12 plots numbered 1–12. Assign treatments completely at random.
Fig 2.1 — A CRD layout: every plot's treatment is assigned completely at random
5. Statistical Analysis of CRD
The model and SS formulas are exactly those of the one-way ANOVA (Unit 1).
Mathematical Model
\[
y_{ij} = \mu + \alpha_i + \epsilon_{ij},
\]
where \(\alpha_i\) is the effect of the \(i\)-th treatment and \(\epsilon_{ij} \sim N(0, \sigma^2)\).
Hypotheses
\(H_0: \alpha_1 = \alpha_2 = \cdots = \alpha_k = 0\) vs \(H_1\): not all \(\alpha_i\) equal.
ANOVA Table
Source
df
SS
MS
F
Treatments
\(k-1\)
\(\text{SS}_{Tr}\)
\(\text{MS}_{Tr}\)
\(\text{MS}_{Tr}/\text{MS}_E\)
Error
\(N-k\)
\(\text{SS}_E\)
\(\text{MS}_E\)
—
Total
\(N-1\)
\(\text{SS}_T\)
Reject \(H_0\) if \(F > F_{\alpha,\, k-1,\, N-k}\).
Conclusion: F2 differs from both F1 and F3; F1 and F3 are statistically equal.
EXAMPLE 2 (CD for unequal n)
From Unit 1 Example 2 (unequal n): \(\text{MS}_E = 5,\; t_{0.025, 6} = 2.447\).
CD between V1 (n₁=3) and V2 (n₂=4): \(2.447 \sqrt{5(1/3 + 1/4)} = 2.447 \sqrt{2.917} = 4.18\). \(|22 - 31| = 9 > 4.18\) ⇒ differ.
7. Efficiency of CRD
Highest df for error → narrowest CI for treatment differences when units are homogeneous.
If units are heterogeneous, blocking (RBD or LSD) is more efficient — see Unit 5.
Key Take-aways
CRD uses two principles: replication & randomization; no blocking.
Layout: assign treatments completely at random to all units.
Analysis = one-way ANOVA. df: (k−1, N−k).
If F is significant, use CD to identify which treatment pairs differ.
Best when experimental units are homogeneous; max error df ⇒ high precision in that setting.
Extra Practical Problems
PRACTICE
Additional worked problems with step-by-step procedures to support self-study, matching this unit's topics.
STEP-BY-STEP PROCEDURE (CRD analysis)
CRD uses only replication and randomization (no local control); the experimental area must be homogeneous. With \(t\) treatments replicated \(r_1, r_2, \ldots, r_t\) times, total units \(N = \sum r_i\).
Grand total \(G\); correction factor \(C = G^2/N\).
\(\text{TSS} = \sum\sum x_{ij}^2 - C\).
\(\text{Treatment SS} = \sum \dfrac{T_i^2}{r_i} - C\) (divide each treatment total by its own replication count — this is the key step for unequal replication).
df: Treatment \(= t-1\), Error \(= N-t\), Total \(= N-1\). Build the ANOVA table and compute \(F = \text{MST}/\text{MSE}\).
Compare with \(F_\alpha(t-1, N-t)\). For comparing two treatment means,
\(\text{CD} = t_{\alpha,\text{error df}}\times\sqrt{\text{MSE}\left(\frac{1}{r_i}+\frac{1}{r_j}\right)}\).
Problem 1 — CRD with Unequal Replication
DATA
Mycelial growth (mm) of 5 R. solani isolates on PDA medium (unequal replications):
Isolate
Repl 1
Repl 2
Repl 3
Total \(T_i\)
Mean
RS 1
29.0
28.0
29.0
86.0
28.67
RS 2
33.5
31.5
29.0
94.0
31.33
RS 3
26.5
30.0
—
56.5
28.25
RS 4
48.5
46.5
49.0
144.0
48.00
RS 5
34.5
31.0
—
65.5
32.75
Grand total \(G = 446\), \(N = 13\). Correction factor \(C = 446^2/13 = 15301.23\).
Since \(F = 57.38 > 3.84\), the isolates differ significantly. For comparing RS 1 and RS 2:
SE \(=\sqrt{3.32\left(\frac13+\frac13\right)} = 1.49\); with \(t_{0.05,8} = 2.30\),
\(CD = 1.49\times 2.30 = 3.44\). The observed difference 2.66 < 3.44, so RS 1 and RS 2 do
not differ significantly, whereas RS 4 differs significantly from every other isolate.
KEY POINT
In CRD with unequal replication the treatment sum of squares uses
\(\sum \dfrac{T_i^2}{n_i}\) (each total divided by its own replication count), and the standard error
of a difference uses the two relevant \(n_i\): \(\text{SE} = \sqrt{\text{MSE}\left(\frac{1}{n_i}+\frac{1}{n_j}\right)}\).