When there are two factors to be considered, the material can be divided into homogeneous groups by the second factor so that each, experimental unit falls into one of the first factor groups and one of the second factor groups. Here we adopt the indigenous device of arranging the data known as ‘Latin Square Designs’. In this design the field is divided into homogeneous blocks in two ways.

The blocks in one direction are commonly known as rows and in other direction are known as column. In this design it is essential that each row and each column should contain the same application of treatments and hence, the variation between the row means and between the columns means can be assessed and eliminated, from the error increasing the precision of the estimates. This is also called Double Grouping.

**Main Features: **

In this design area is divided into plots such that the number of plots in each row is the same as the number of plots in each column. This number is being equal to the number of treatments. The plots are then assigned to different treatments such that every treatment occurs once and only once in each row and each column. The number of replications per treatment is also equal to the number of treatments.

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**Description: **

If the experimental material contains n = r^{2} experimental units arranged into homogeneous blocks of r units each in two ways commonly known as rows and columns and the r treatments are allocated to the experimental units in such a way that every treatment occurs just once in every row and in every column, the design so obtained is known as Latin Square Design.

**Applications: **

This design is particularly useful in agricultural and Industrial experimentation. In agricultural experiments this design is used, the fertility gradient is in two directions at right angles or when the fertility’ gradient is in one direction but not known.

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The latin square is applicable when there are variations of two factors to be considered (or controlled) and experimental material can be divided into homogeneous groups by one factor and into groups by the second factor so that each experimental unit can be one of the first factor groups and one of the second factor groups.

**Advantages: **

**The following are the advantages (merits) of LSD:**

(i) By means of grouping the experimental units in two-ways, the error variation is reduced by eliminating the two major sources of variation that are not relevant to the comparisons between the different treatments.

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(ii) Being an in complete 3-way design it is economic over the corresponding complete 3-way design.

**Disadvantages: **

**The following are the disadvantages (demerits) of LSD:**

(i) In this design the number of replications is necessarily equal to the number of treatments.

(ii) It necessitates approximately a square field (in agricultural experiment).

(iii) In case of 12 × 12 latin square, the square becomes too large that may not remain homogeneous.

(iv) The analysis of variance is difficult in comparison to other designs.

**Randomization of the Treatments: **

The treatments are to be randomized in such a way that every treatment occurs once and only once in each column and in each row.

**Suppose we are to test 5 varieties A, B, C, D, E of a particular crop as regards their yields, we may make the square as given here:**

**The other assumptions are:**

(i) x_{ijk} are independently normal with mean µ_{ijk} and variance O^{2} which implies that- (a) a random sample of size one is drawn from each of the m^{2} population, (b) all m^{2} populations are normal and (c) the variance of each of the m^{2} populations is the same.

Due to assumption of additively the interaction are zero.

**Least Squares Estimate:**

**The level squares estimates obtained by minimizing:**

**Null Hypotheses:**

**Porititioning of the Sum of Squares:**

**Squaring both sides and summing over all the observation we have:**

**These result are now represented in the analysis of variance table:**