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UPSC Statistics — Previous Year Questions

Every question from Statistics papers in UPSC Civil Services Mains (2021–2026), organised by year. 96 questions in total, each with directive-word analysis, model answer points, and a full evaluation rubric.

96Questions
6Years covered

2026 Mains

16 questions · 800 marks

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Paper I 8 Qs · 400 marksPaper II 8 Qs · 400 marks
All 16 questions from 2026

Paper I

  1. Q1 (a) (i) Show that for n events A₁, A₂, ⋯, Aₙ P( ⋂ᵢ₌₁ⁿ Aᵢ ) ≥ Σᵢ₌₁ⁿ P(Aᵢ) − (n − 1) (7 marks) (ii) For two mutually… 50 marks
  2. Q2 (a) (i) Find the moment generating function (m.g.f.) of a random variable X with probability density function (p.d.f.)… 50 marks
  3. Q3 (a) Let the joint probability density function of X and Y be f(x, y) = 3/4 ( xy + x²/2 ) ; 0 < x < 1, 0 < y < 2 Find—… 50 marks
  4. Q4 (a) Let X₁, X₂, ⋯, Xₙ be i.i.d. random variables following N(μ, σ²). Find the maximum likelihood estimator for σ² and… 50 marks
  5. Q5 (a) Let Y₁, Y₂, ⋯, Yₙ be n independent observations from a population with mean μ and variance σ². Obtain the best… 50 marks
  6. Q6 (a) Let l′β be any estimable linear function of parameters β₁, β₂, ⋯, β_p. Then prove that— (i) there exists a unique… 50 marks
  7. Q7 The following data are from a two-factor random effects model : | Factor B Factor A | B₁ | B₂ | B₃ | | A₁ | 6, 8 | 10… 50 marks
  8. Q8 (a) Define multivariate normal distribution and write down its important properties. Let X₁, X₂, X₃ and X₄ are mutually… 50 marks

Paper II

  1. Q1 (a) Suppose that a given individual in a population has a survival time which is exponential with a hazard rate θ. Each… 50 marks
  2. Q2 (a) (i) Describe the construction of a CUSUM chart with Decision interval and Reference value. (10 marks) (ii) One… 50 marks
  3. Q3 (a) How will you draw the OC curve for a single sampling plan with sample size 5, if the acceptance number is c = 2… 50 marks
  4. Q4 (a) (i) Discuss the theoretical basis of the sequential probability ratio test (SPRT) in acceptance sampling. Explain… 50 marks
  5. Q5 (a) Explain the 2-stage least squares (2SLS) method of estimation for the following simultaneous equation systems : C_t… 50 marks
  6. Q6 (a) Define autocorrelation of order k and correlogram. For an infinite series generated by the moving average of random… 50 marks
  7. Q7 (a) (i) Define a life table and mention its uses. (5 marks) (ii) The number of persons dying at age 75 is 476, and the… 50 marks
  8. Q8 (a) Consider the following structural equation model assuming Y's as endogenous and X's as pre-determined variables … 50 marks

2025 Mains

16 questions · 800 marks

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Paper I 8 Qs · 400 marksPaper II 8 Qs · 400 marks
All 16 questions from 2025

Paper I

  1. Q1 (a) Let E, F and G be three pairwise independent events such that P(E∩F) = 0·1 and P(F∩G) = 0·3. Prove that P(Eᶜ∪G) ≥… 50 marks
  2. Q2 (a) Let X be a continuous random variable having probability density function f(x) = (2/25)(x+2), -2 ≤ x ≤ 3; 0… 50 marks
  3. Q3 (a) Let probability of obtaining Head on a biased coin be 4/5 and X be the number of heads obtained in a sequence of 25… 50 marks
  4. Q4 (a) Let X₁, X₂, ... be a sequence of random variables from Bernoulli distribution with mean θ, 0<θ<1. Derive SPRT for… 50 marks
  5. Q5 (a) For a two variable linear regression model Yᵢ = a + bXᵢ + eᵢ, where E(eᵢ) = 0, Var(eᵢ) = σ²ₑ, Cov(eᵢ, eⱼ) = 0 for i… 50 marks
  6. Q6 (a)(i) If (X, Y) follows bivariate normal BN(μ₁, μ₂, σ₁², σ₂², ρ), then obtain (A) E(e^X) (B) E(e^(X+Y)) (C) Var(e^X)… 50 marks
  7. Q7 (a) Analyse and interpret the following data concerning output of wheat per field obtained as a result of experiment… 50 marks
  8. Q8 (a)(i) What are principal components ? Show that the principal components are uncorrelated. (10 marks) (a)(ii) Obtain… 50 marks

Paper II

  1. Q1 (a) State the significance of operating characteristic (OC) curves in control chart analysis. Obtain the general… 50 marks
  2. Q2 (a) (i) What are control charts by variables and control charts by attributes? 5 marks (ii) Derive the control limits… 50 marks
  3. Q3 (a) A company manufactures 30 items per day. The sale of those items depends upon demand which has the following… 50 marks
  4. Q4 (a) Solve the game whose payoff matrix is -1 & -2 & 8 7 & 5 & -1 6 & 0 & 12 (15 marks) (b) Use the penalty (Big M)… 50 marks
  5. Q5 (a) Explain the multicollinearity problem in a regression model. What are its consequences? State the different… 50 marks
  6. Q6 (a) On the basis of the figures given below, calculate the age-specific death rates (ASDRs) for all the age groups… 50 marks
  7. Q7 (a) Explain the concept of index number. Calculate the Fisher's ideal index number from the following data and verify… 50 marks
  8. Q8 (a) Define time series. For a moving-average process with weights {a₁, a₂, ..., aₘ} of random components {eᵢ, i = 1, 2… 50 marks

2024 Mains

16 questions · 800 marks

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Paper I 8 Qs · 400 marksPaper II 8 Qs · 400 marks
All 16 questions from 2024

Paper I

  1. Q1 (a) Two events A and B are such that P(A) = 1/3, P(B) = 1/4 and P(A|B) + P(B|A) = 2/3. Evaluate the following: (i)… 50 marks
  2. Q2 (a) Let the joint probability density function of two random variables X and Y be f(x, y) = x/3, for 0 < 2x < 3y < 6… 50 marks
  3. Q3 (a) Let moment generating function of random variable X exist in the neighbourhood of zero and if E(Xⁿ) = 1/5 + (-1)ⁿ… 50 marks
  4. Q4 (a) Find the most powerful test of size α(= 0·05) for testing H₀: μ = 0 vs. H₁: μ = 1, given a random sample of size 25… 50 marks
  5. Q5 (a) How will you justify the usage of the principle of least squares in estimating the parameters of a linear… 50 marks
  6. Q6 (a) (X, Y) has bivariate normal distribution BN(μ₁, μ₂, σ₁², σ₂², ρ). (i) Show that X and Y are independent if and only… 50 marks
  7. Q7 (a) A very big population is divided into two strata. The allocation of units of stratified random sample of size n for… 50 marks
  8. Q8 (a) A 2²-factorial design was used to develop the yield of a crop. Two factors A and B were used at two levels: low… 50 marks

Paper II

  1. Q1 (a) Consider a system consisting of three identical units connected in parallel. The unit reliability factor is 0·90… 50 marks
  2. Q2 (a) Obtain the control limits for X̄-chart and R-chart and describe the significance of joint study of these charts. 20… 50 marks
  3. Q3 (a) With respect to a given Linear Programming Problem (LPP), explain the following concepts : 15 marks (i) Extreme… 50 marks
  4. Q4 (a) A manually handled toll-booth has two tellers, who are each capable of handling an average of 60 vehicles per hour… 50 marks
  5. Q5 (a) Discuss the Indian Statistical System. State some important organisations and explain the main working of the… 50 marks
  6. Q6 (a) For the linear model Y = Xβ + u, obtain the expressions for Ordinary Least Squares (OLS) and Generalised Least… 50 marks
  7. Q7 (a) Explain the problem of identification with a suitable example. Also discuss the conditions of identification. Check… 50 marks
  8. Q8 (a) Four items are to be constructed so that they are equispaced on the difficulty scale. If the easiest item is passed… 50 marks

2023 Mains

16 questions · 800 marks

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Paper I 8 Qs · 400 marksPaper II 8 Qs · 400 marks
All 16 questions from 2023

Paper I

  1. Q1 (a) Out of 1000 persons born, only 900 reach the age of 15 years, and out of every 1000 who reach the age of 15 years… 50 marks
  2. Q2 (a) Let X, Y, Z be three mutually independent standard exponential variates and W₁ = X + Y + Z, W₂ = (X + Y)/(X + Y +… 50 marks
  3. Q3 (a) (i) If X is a random variable with finite variance, show that lim (n→∞) n² P{|X| > n} = 0. (ii) In a certain… 50 marks
  4. Q4 (a) What is the role of properties of completeness and sufficiency in Statistical Inference ? Explain. In U (0, θ)… 50 marks
  5. Q5 (a) (i) If **X** = (X₁ X₂ X₃)' is distributed as N₃ (μ, Σ), find the distribution of [(X₁ – X₂) (X₂ – X₃)]'. (5 marks)… 50 marks
  6. Q6 (a) Let **X** = (X₁ X₂ X₃)' be distributed as N₃ (μ, Σ), where μ = (2 −1 3)' and Σ = [[4, 1, 0], [1, 2, 1], [0, 1, 3]]… 50 marks
  7. Q7 (a) Discuss the difference between sampling for variables and sampling for attributes with examples. For a qualitative… 50 marks
  8. Q8 (a) Differentiate between randomised block design and balanced incomplete block design. In usual notations, for a… 50 marks

Paper II

  1. Q1 (a) What do you understand by Statistical Quality Control (SQC)? Discuss briefly its need and utility in Industry… 50 marks
  2. Q2 (a) What do you understand by control chart for fraction defective? Explain its construction. Give the theoretical… 50 marks
  3. Q3 (a) Solve the following Linear Programming problem using Two Phase method : Maximize Z = 3x₁ - x₂ Subject to 2x₁ + x₂ ≥… 50 marks
  4. Q4 (a) A Company ships truckloads of grain from three silos to four mills. The supply (in truckloads) and the demand (also… 50 marks
  5. Q5 (a) State and discuss the important points which have to be considered in the construction of index numbers. (10 marks)… 50 marks
  6. Q6 (a) Explain the principle of least squares. How it is used in fitting trend in time series analysis ? Explain the… 50 marks
  7. Q7 7.(a) Derive, by starting from a suitable functional form for lₓ, the formula (i) Lₓ = (lₓ + lₓ₊₁)/2 and (ii) Lₓ = (lₓ… 50 marks
  8. Q8 8.(a) Explain the method of collection of agriculture data. Describe the (i) official publications for data collection… 50 marks

2022 Mains

16 questions · 800 marks

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Paper I 8 Qs · 400 marksPaper II 8 Qs · 400 marks
All 16 questions from 2022

Paper I

  1. Q1 (a) Let X and Y be independent random variables with exponential distribution having respective means 1/(λ₁) and… 50 marks
  2. Q2 (a) Let a random variable X have exponential distribution with mean 1/θ, θ > 0. To test H₀ : θ = 3 against H₁ : θ = 2… 50 marks
  3. Q3 (a) (i) How large a sample must be taken in order that the probability will be at least 0.90 that the sample mean will… 50 marks
  4. Q4 (a) Consider Poisson distribution P_θ(X = j) = e^(−θ) θ^j / j! = pⱼ, j = 0, 1, 2, .... Let fⱼ be the frequency for X =… 50 marks
  5. Q5 (a) Define general linear model with usual assumptions. If y₁ = β₁ + u₁, y₂ = –β₁ + β₂ + u₂, y₃ = –β₂ + u₃, where u₁… 50 marks
  6. Q6 (a) In a set of two-way classified data according to k levels of factor A and r levels of factor B, there is one… 50 marks
  7. Q7 (a) Consider the following data given for a BIBD with v = b = 4, r = k = 3, λ = 2 and N = 12 : Block : 1 2 3 4… 50 marks
  8. Q8 (a) (i) What are orthogonal polynomials ? How do you fit an orthogonal polynomial of degree 'p' ? (ii) For the model… 50 marks

Paper II

  1. Q1 (a) Distinguish between process control and product control. Explain the various sources of variation encountered in a… 50 marks
  2. Q2 (a) Samples of size n = 5 units are taken from a process every hour. The x̄ and R̄ values for a particular quality… 50 marks
  3. Q3 (a) It is planned to raise a research team to a strength of 50 chemists, which is to be maintained. The wastage of… 50 marks
  4. Q4 (a) Use penalty method to solve the following linear programming problem : Maximize Z = x₁ + 2x₂ + 3x₃ - x₄ subject to… 50 marks
  5. Q5 (a) Apply the method of link relatives to the following data and calculate the seasonal indices : Price of Rice (in ₹… 50 marks
  6. Q6 (a) Explain Akaike information criterion for order selection in an ARMA (p, q) process. 15 marks (b) Define… 50 marks
  7. Q7 (a) What are the various indices of mortality measure? Explain the purpose and procedure for standardizing them. (20… 50 marks
  8. Q8 (a) How does the concept of wholesale price index work? Describe the major components of wholesale price index. Explain… 50 marks

2021 Mains

16 questions · 800 marks

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Paper I 8 Qs · 400 marksPaper II 8 Qs · 400 marks
All 16 questions from 2021

Paper I

  1. Q1 (a) A production unit manufacturing surgical masks is concerned about the quality of their masks. A random sample of n… 50 marks
  2. Q2 (a) Let Y₁, Y₂, Y₃, ... be independent and identical Poisson random variables with parameter 1. Use central limit… 50 marks
  3. Q3 (a) Let X and Y be two independent random variables following exponential distribution with mean 1/(λ) and 1/(μ)… 50 marks
  4. Q4 (a) Let X₁, X₂, ..., Xₙ be a random sample from Poisson distribution with mean λ > 0. Define a statistic W = (1 −… 50 marks
  5. Q5 (a) For a simple linear regression model Y = β₀ + β₁Xᵢ + εᵢ, i = 1, ..., n (i) Derive the least square estimators of β₀… 50 marks
  6. Q6 (a) For a multiple linear regression model with three covariates X₁, X₂ and X₃, let rᵢⱼ denote the correlation… 50 marks
  7. Q7 (a) (i) What is confounding in factorial experiments ? (ii) A 2^6factorial experiment is conducted in blocks of size2³… 50 marks
  8. Q8 (a) (i) In stratified sampling under optimum allocation, how will you proceed to select units from different strata, if… 50 marks

Paper II

  1. Q1 Explain Single Sampling Plan with the help of an example. Also, write the importance of an Operating Characteristics… 50 marks
  2. Q2 A manufacturer finds that on the average, a television set is used 1.8 hours per day. A one year warranty is offered on… 50 marks
  3. Q3 (a) Explain M|G|1 queuing system. Obtain Pollaczek-kinchine formula. (15 marks) (b) Use MODI method to solve the above… 50 marks
  4. Q4 (a) Solve the following linear programming problem: Maximize z = 3x₁ + 5x₂ subject to the constraints 3x₁ + 2x₂ ≤ 18 x₁… 50 marks
  5. Q5 (a) Explain Zellner's seemingly unrelated regression model and the feasible generalized least squares method of… 50 marks
  6. Q6 (a) Explain Box-Jenkins methodology to build ARIMA models. (15 marks) (b) Prepare the cost of living index for 2006 on… 50 marks
  7. Q7 (a) If c(x, t) denote observed proportion of females in the age group (x, x+t) and f(x, t) is the observed proportion… 50 marks
  8. Q8 (a) What is autocorrelation ? What are its consequences ? Explain the Goldfeld-Quandt test and Glesjer test for… 50 marks

How Statistics answers are marked

The shape the examiner expects, what earns the marks and the traps that lose them: the same card our rubrics for this subject are written from.

Answer skeleton

  1. Setup
  2. Assumptions
  3. Derivation or test
  4. Computation and interpretation

What earns the marks

  • State distributional assumptions explicitly
  • Name the test and its conditions
  • Write the formula before substituting
  • Interpret the number in context, not just report

What earns the extra mark

  • Clean table
  • Correct notation for estimator versus parameter

Traps that cost marks

  • Computation without interpretation
  • Unstated assumptions
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