Agricultural Statistics

CRD (Completely Randomised Design) — used when experimental units are homogeneous; treatments randomly assigned to all units; error df = N − t (total observations minus treatments); simplest design, highest df for error but no local control. RBD (Randomised Block Design) — units grouped into homogeneous blocks; one-way local control; error df = (t−1)(b−1) where t = treatments, b = blocks; most commonly used in field experiments. LSD (Latin Square Design) — two-way local control (rows and columns); requires equal number of treatments, rows, and columns (k×k square); error df = (k−1)(k−2); used when two sources of variation exist (e.g., row gradient + column gradient).

CV (Coefficient of Variation) = (Standard Deviation / Mean) × 100, expressed as a percentage. It measures the relative variability of an experiment — a dimensionless statistic used to compare precision across experiments with different means. In field experiments: CV <10% = highly precise; 10–20% = good; 20–30% = moderate; >30% = poor precision. ICAR JRF exams test CV calculation and interpretation. High CV indicates non-uniform experimental conditions — may require blocking in next experiment.

ANOVA (Analysis of Variance) partitions total variation in a dataset into components attributable to different sources. F-test (Fisher's test) = Mean Square (Treatment) / Mean Square (Error). If F-calculated > F-table value (at 5% or 1% significance), treatments differ significantly. In CRD: F = MS(Treatment)/MS(Error); df for treatment = t−1, error = N−t. In RBD: F = MS(Treatment)/MS(Error); df for treatment = t−1, block = b−1, error = (t−1)(b−1). F was developed by R.A. Fisher — also called Variance Ratio.

Chi-square (χ²) test is a non-parametric test used for categorical data. Two main uses: (1) Goodness of fit — tests whether observed frequencies match expected frequencies (e.g., Mendelian segregation ratios); (2) Test of independence — tests whether two categorical variables are independent (contingency table). Formula: χ² = Σ[(O − E)² / E] where O = observed, E = expected. df = (rows − 1)(columns − 1) for contingency table; df = k − 1 for goodness of fit (k = number of categories). Chi-square is always one-tailed and always positive.

Karl Pearson's correlation coefficient (r) measures the linear relationship between two continuous variables. Range: −1 to +1. r = +1 = perfect positive linear relationship; r = −1 = perfect negative; r = 0 = no linear correlation. Formula: r = Σ[(X − X̄)(Y − Ȳ)] / √[Σ(X − X̄)² × Σ(Y − Ȳ)²]. Interpretation: r² (coefficient of determination) = % variation in Y explained by X. Spearman's rank correlation (ρ) is the non-parametric alternative used when data are ordinal or not normally distributed.

Four main sampling methods: (1) Simple Random Sampling (SRS) — every unit has equal probability of selection; lottery method or random number tables; no prior information needed. (2) Stratified Random Sampling — population divided into homogeneous strata, SRS within each stratum; more precise than SRS when strata differ. (3) Systematic Sampling — every kth unit selected after a random start; k = N/n (sampling interval); efficient for crop cutting experiments. (4) Cluster Sampling — population divided into clusters (e.g., villages), clusters randomly selected, all units in chosen clusters included; used in large-scale surveys. Crop cutting experiments (CCE) for yield estimation use systematic or stratified random sampling.