State statistical services: Statistical Officer / Assistant Statistical Officer (State PSCs and selection boards) · India · State Public Service Commissions and Subordinate Service Selection Boards
Original practice questions written to the published syllabus · not past papersMethods that hold without a normal model: the empirical distribution, goodness-of-fit, sign and rank tests for one, two and many samples, rank correlation, and tests used for scored or ranked agricultural data such as disease ratings and sensory panels.
In the exams: AICE-JRF Ag Stat Unit 2 (heavy: run, sign, rank, Wilcoxon, Kruskal-Wallis, Mann-Whitney, Cochran, Friedman named); GATE ST 'Non-parametric Statistics' (heavy); UPSC ISS Statistics II (present)
State statistical services and SSC JSO
Sign test, run test, rank correlation and chi-square goodness of fit computations.
How our questions for State statistical services spread across the units of Nonparametric & Rank-Based Methods. Difficulty: 18 easy, 12 medium, 8 hard.
| Kruskal-Wallis test | 5 | |
| Empirical distribution function and its properties | 4 | |
| Goodness of fit: chi-square and Kolmogorov-Smirnov tests | 4 | |
| Spearman's and Kendall's rank correlations | 4 | |
| Asymptotic relative efficiency | 4 | |
| Analysing ordinal scores: disease ratings and sensory panels | 4 | |
| Run test for randomness | 3 | |
| Sign test and Wilcoxon signed-rank test | 3 | |
| Cochran's Q test | 3 | |
| Mann-Whitney U test and two-sample Kolmogorov-Smirnov test | 2 | |
| Friedman's test for randomised blocks | 2 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
In a chi-square goodness-of-fit test of insect counts, several classes have expected frequencies below 5. What is the usual remedy?
B. Pool adjacent classes, then count df after pooling
Small expected counts make the χ² approximation unreliable; adjacent classes are pooled and the df counted on the pooled classes. Dropping classes discards data and biases the test.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
In a runs test for randomness, what do far too many runs suggest?
D. Alternation or negative dependence
Many short runs mean the two symbols alternate more than chance allows, as with regular spacing or negative serial correlation. Too few runs indicate clustering or trend.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Under randomness, how does the number of runs R behave for large n₁ and n₂?
C. Approximately normal with mean 2n₁n₂/N + 1
R is asymptotically normal (Wald and Wolfowitz), with the stated mean and variance 2n₁n₂(2n₁n₂ − N)/[N²(N − 1)], which justifies the Z approximation.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Two independent samples have sizes 6 and 6, and sample 1's rank sum is W₁ = 25. What is the Mann-Whitney statistic U₁?
A. 4
U₁ = W₁ − n₁(n₁ + 1)/2 = 25 − 21 = 4. U₂ = n₁n₂ − U₁ = 32. 14 = 25 − 11 omits the correct offset.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Which design is Friedman's test intended for?
C. An RCBD with one observation per treatment per block
Friedman ranks treatments within blocks, the rank analogue of the RCBD ANOVA. Binary blocked data use Cochran's Q.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
What is the null hypothesis of Cochran's Q test?
D. The success probability is the same for all k treatments
Q tests equality of the treatment marginal success proportions, allowing blocks to differ. It does not assume p = 0.5 or test blocks.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Hodges and Lehmann proved a lower bound for the ARE of the Wilcoxon (or Mann-Whitney) test relative to the t-test, over all continuous symmetric distributions. What is it?
A. 0.864
108/125 = 0.864; it can never be lower, while it can be arbitrarily large for heavy-tailed distributions. 0.955 is the value at the normal, not the minimum.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Ten panellists each rank four mango samples for sweetness. Which statistic summarises their agreement?
D. Kendall's coefficient of concordance W
W = χ²_F/[b(k − 1)] measures concordance among b judges ranking k items. Spearman's ρ compares two rankings only.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Farmers in a participatory trial each rank the same five varieties. Which test compares the varieties?
D. Friedman's test, with farmers as blocks
Each farmer ranks all varieties, so farmers are blocks and Friedman's test applies; Kendall's W measures agreement among farmers. Kruskal-Wallis assumes independent groups.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
In the Wilcoxon signed-rank test, how is the null variance of T⁺ adjusted when some |dᵢ| are tied and given mid-ranks?
A. It is reduced by Σ(t³ − t)/48 over tie groups of size t
Mid-ranks keep E(T⁺) but lower the variance of the ranks, so Var(T⁺) = n(n + 1)(2n + 1)/24 − Σ(t³ − t)/48.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
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