Sampling Theory and Estimation Questions for SSC CGL

Basic Concepts: Population, Sample, Parameter, Statistic

A population is the complete group you want to study. A sample is a smaller subset drawn from it. A parameter is a numerical measure of the population (like population mean mu) while a statistic is a numerical measure of the sample (like sample mean x-bar). The goal of sampling is to estimate unknown population parameters using known sample statistics. Think of it like tasting one spoon of dal to judge the whole pot.

• Population parameter: fixed, usually unknown (mu, sigma, P)
• Sample statistic: calculated from data, used to estimate parameter (x-bar, s, p)
• Sampling frame: the list from which the sample is actually drawn
• Census vs Sample: Census covers all units; sampling is faster and cheaper
• Sampling error: difference between sample statistic and true parameter — always present in sampling

Types of Sampling Methods

Sampling methods are broadly divided into Probability Sampling (every unit has a known, non-zero chance of selection) and Non-Probability Sampling (selection is based on judgment or convenience, no fixed probability). SSC CGL tests your ability to identify which method is being described in a scenario.

• Simple Random Sampling (SRS): every unit has equal chance; done with or without replacement (SRSWR vs SRSWOR)
• Stratified Sampling: population divided into non-overlapping groups (strata) based on a relevant characteristic; sample drawn from each stratum — reduces variability
• Systematic Sampling: select every kth unit after a random start; k = N/n (sampling interval)
• Cluster Sampling: population divided into clusters (natural groups); entire clusters are randomly selected — cheap but higher sampling error
• Multistage Sampling: sampling done in stages (e.g., first select districts, then villages, then households)
• Quota Sampling and Purposive Sampling are Non-Probability methods — not used for formal inference

Standard Error (SE)

Standard Error measures how much a sample statistic varies from one sample to another. It is the standard deviation of the sampling distribution of a statistic. The smaller the SE, the more reliable the estimate. SE decreases as sample size increases — doubling sample size cuts SE by a factor of root 2, not 2. This is a very common exam trap.

• SE of sample mean (SRSWR): sigma divided by root n
• SE of sample mean (SRSWOR): (sigma / root n) multiplied by root of (1 - n/N) — the extra term is the Finite Population Correction (FPC)
• FPC = root of (N - n) / (N - 1) — reduces SE when sample is a large fraction of population
• SE decreases as n increases; increases as sigma increases
• SE is NOT the same as standard deviation of the data

Properties of a Good Estimator

An estimator is a formula or rule to estimate the population parameter from sample data. Not all estimators are equally good. SSC CGL Tier 2 tests these properties both theoretically (define them) and practically (which estimator is better). Remember the acronym CUBE: Consistency, Unbiasedness, Best (minimum variance), Efficiency.

• Unbiasedness: E(T) = theta — on average, the estimator equals the true parameter. x-bar is unbiased for mu
• Consistency: as n increases to N, the estimator converges to the true parameter value
• Efficiency: among all unbiased estimators, the one with smallest variance is most efficient — called UMVUE (Uniformly Minimum Variance Unbiased Estimator)
• Sufficiency: an estimator is sufficient if it uses ALL information in the sample about the parameter
• Bias = E(T) - theta; if Bias = 0, estimator is unbiased
• Sample variance s-squared = sum of (xi - x-bar)^2 / (n-1) is unbiased for sigma-squared; dividing by n gives biased estimate

Interval Estimation and Confidence Intervals

A point estimate gives a single value. An interval estimate gives a range within which the parameter likely falls. A 95 percent confidence interval means: if we took 100 samples and built 100 intervals, 95 of them would contain the true parameter. SSC CGL tests the formula and interpretation, not deep probability theory.

• 95% CI for mean: x-bar plus or minus 1.96 * (sigma / sqrt(n))
• 99% CI for mean: x-bar plus or minus 2.576 * (sigma / sqrt(n))
• Width of CI = 2 * Z * SE — wider CI means less precision but more confidence
• Confidence level and precision are inversely related: 99% CI is wider than 95% CI
• Key Z values to memorize: 90% = 1.645, 95% = 1.96, 99% = 2.576

• Confusing Standard Error with Standard Deviation: SE is the SD of the sampling distribution; it changes with sample size. SD of data does not change with sample size.
• Using n instead of n-1 when computing unbiased sample variance: dividing by n gives a biased (smaller) estimate of population variance — always use n-1 for unbiased result.
• Thinking that doubling n will halve the SE: SE is sigma / sqrt(n), so doubling n reduces SE by a factor of root 2 (about 1.41), not 2. To halve SE you need 4 times the sample size.
• Mixing up Cluster and Stratified Sampling: in stratified sampling you sample FROM each group; in cluster sampling you select entire groups. SSC loves to swap these in MCQ options.
• Interpreting confidence interval incorrectly: a 95% CI does NOT mean there is a 95% probability that the parameter is in THIS specific interval — the parameter is fixed; it is the interval that is random.

• CUBE for estimator properties: Consistency, Unbiasedness, Best (efficiency), Everything in the data (sufficiency)
• SSCC for sampling types: Simple random, Systematic, Cluster, Stratified — all are probability sampling. Add Quota and Purposive for non-probability.
• For CI Z-values use the phrase 9-1-6, 9-5-2, 9-9-3: 90% uses 1.645, 95% uses 1.96, 99% uses 2.576 — notice the last digit goes 6, 6, 6 roughly (mnemonically: three sixes, three confidence levels).
• Stratified = Stir the groups (mix within), Cluster = Collect whole buckets (take all of a group). Stratified reduces error; Cluster is economical but riskier.

• SSC CGL Tier 2 Stats paper typically has 4-6 questions on this chapter. Most are definitional or formula-based — direct questions like which estimator property means E(T) = theta, or what is the SE formula for SRSWOR.
• Numeric questions on SE and CI are very common. Keep the Z-values 1.96 and 2.576 memorized cold — you will need them in under 30 seconds.
• Scenario-based questions ask you to identify the sampling method: if a description says every 10th name is picked from a list, that is systematic sampling. Practice mapping descriptions to method names.
• Questions on bias frequently appear as: which of these is an unbiased estimator of sigma-squared — the answer is always the one with n-1 in the denominator.
• FPC (Finite Population Correction) questions test whether you know it applies only in SRSWOR and only when n/N is non-negligible (usually more than 5%). If n/N is small, FPC is approximately 1 and can be ignored.