Non-Parametric Statistics

1. Meaning and Definition

• Non-parametric statistics are statistical methods that do not assume any specific population distribution (e.g., normal distribution).
• They are also called distribution-free tests because they work even when population parameters (mean, variance, etc.) are unknown.
• Useful for ordinal, nominal, or ranked data.

Definition (Siegel & Castellan):
“Non-parametric tests are statistical procedures that do not make strong assumptions about the form of the population distribution.”

2. When to Use Non-Parametric Tests

• Data is ordinal/nominal (ranks, categories, preferences).
• Sample size is small.
• Assumptions of normality and homogeneity of variance are violated.
• Outliers or skewed data are present.
• When dealing with attitudes, preferences, perceptions, opinions in social research.

3. Types of Non-Parametric Tests

(A) Tests for One Sample

• Chi-Square (χ²) Goodness-of-Fit Test; Tests whether sample distribution fits a theoretical distribution. Example: Do farmers’ preferences for crops (rice, wheat, maize) follow an expected proportion?
• Sign Test; Compares median of a sample to a hypothesized value. Example: Checking if the median score of knowledge is higher after training.
• Run Test; Tests randomness of data sequence.

(B) Tests for Two Independent Samples

• Mann-Whitney U Test; (equivalent of independent t-test) Compares whether 2 groups differ in their ranks. Example: Compare adoption levels of male vs. female farmers.
• Chi-Square Test of Independence; Tests association between two categorical variables. Example: Is there an association between education level and technology adoption?
• Kolmogorov-Smirnov Two-Sample Test; Compares the distribution of two samples.

(C) Tests for Two Related Samples

• Wilcoxon Signed-Rank Test (equivalent of paired t-test), Compares before-after scores for same group. Example: Farmers’ knowledge score before vs. after training.
• McNemar Test; For paired nominal data (yes/no type). Example: Did training change the number of “yes” adopters?

(D) Tests for More than Two Independent Samples

• Kruskal-Wallis H Test (equivalent of one-way ANOVA); Compares 3 or more groups based on ranks. Example: Compare adoption levels across three villages.
• Median Test; Tests if several groups have the same median.

(E) Tests for More than Two Related Samples

• Friedman Two-Way ANOVA by Ranks (equivalent of repeated measures ANOVA). Compares 3+ related groups. Example: Comparing farmers’ preference for three extension teaching methods (lecture, demonstration, film).

(F) Measures of Association (Ordinal/Rank Data)

• Spearman’s Rank Correlation (ρ); Measures correlation between two ranked variables. Example: Correlation between education rank and adoption rank.
• Kendall’s Tau (τ); Similar to Spearman’s, but more accurate with small samples.

4. Advantages of Non-Parametric Tests

• No strict assumptions about normality or variance.
• Suitable for small samples
• Can be used with ordinal and nominal data.
• Robust against outliers and skewed data.
• Simple to understand and apply.

5. Limitations

• Less powerful than parametric tests (require larger samples to detect effect).
• Provide less detailed information (usually medians/ranks, not means/variances).
• May not generalize as strongly as parametric tests.

6. Applications in Social Research

• Farmer’s preference ranking for crop varieties (Kendall’s Tau, Friedman test).
• Measuring association between education and adoption (Chi-square test).
• Studying before-after training impact on farmers’ knowledge (Wilcoxon Signed-Rank).
• Ranking effectiveness of different communication channels (Friedman test).
• Correlation between income level and extension contact frequency (Spearman’s rho).

7. Common Non-Parametric Tests (Exam Quick Table)