BCS301 Module 5: Design of Experiments & ANOVA PDF

## Module-5: Design of Experiments & ANOVA **Principles of experimentation in design, Analysis of completely randomized design, randomized block design. The ANOVA Technique. Basic Principle of ANOVA, One-way ANOVA, Two-way ANOVA, Latin-square Design, and Analysis of Co-Variance. (12 Hours) (RBT Lev...

## Module-5: Design of Experiments & ANOVA **Principles of experimentation in design, Analysis of completely randomized design, randomized block design. The ANOVA Technique. Basic Principle of ANOVA, One-way ANOVA, Two-way ANOVA, Latin-square Design, and Analysis of Co-Variance. (12 Hours) (RBT Levels: L1, L2 and L3)** ### 5.1 The ANOVA Technique **Introduction:** The analysis of variance (ANOVA) is a statistical technique to test whether the means of three or more populations are equal or not. This technique was developed by R A Fisher. This technique is widely used in Professional Business and Physical Sciences. **In this technique, variance is splitted into two parts:** **(i)Variance between samples (Columns) (ii) Variance within samples (Rows)** A table showing the source of variation, the sum of squares, degrees of freedom, mean squares and the formula for the F ratio is called ANOVA table. - If the given data is classified according to one factor, the classification is called one way classification. Then ANOVA table for one-way classification is to be constructed. - If the given data is classified according to two factors, the classification is called two-way classification. Then ANOVA table for two-way classification is to be constructed. **Analysis of variance is based on the following assumptions:** - The samples are independently drawn the population. - Populations from which the sample are selected are normally distributed. - Each of the population have the same variance. ### ANOVA table for one-way classification: | Source of variation | Sum of squares | Degrees of freedom | Mean squares | F-Ratio | |---|---|---|---|---| | Between samples | SSC | c-1 | MSC=SSC/(c-1) | MSC/MSE | | Within samples | SSE | N-c | MSE=SSE/(N-c) | - | | Total | SST | N-1 | - | - | ### Expansion of abbreviations: * SSC - Sum of squares between samples (Columns) * SSE - Sum of squares within sample (Rows) * SST-Total sum of squares of variations * MSC - Mean squares of variations between samples (Columns) * MSE - Mean squares of variations within samples (Rows) ### Notations: * T-Total sum all the observations * N-Number of observations. * C-Number of columns. ### How to find SSC and SSE? $$ SSC = \frac{(\sum X_1)^2}{n_1} + \frac{(\sum X_2)^2}{n_2} + ... + \frac{(\sum X_c)^2}{n_c} - \frac{T^2}{N} $$ $$ SST = \sum X_1^2 + \sum X_2^2 + \sum X_3^2 + ⋯ + \sum X_N^2 - \frac{T^2}{N} $$ $$ SSE = SST - SSC $$ ### Working rule: 1. Assume Ho: $μ₁, μ₂, ..., μ_r$ all are equal. 2. Construct ANOVA tale for one-way classification. 3. Under Ho, $F = \frac{MSC}{MSE}$, if MSC > MSE, Reciprocate otherwise. 4. If calculated value < tabulated value, accept Ho. Reject otherwise.