Sampling Theory for SSC CGL Tier 2 Paper 2

In SSC CGL Tier 2, SSC JSO, Sampling Theory is an important part of Statistics. Understanding sampling helps in making inferences about a population using a sample. This blog covers population, sample, sampling techniques, errors, and sampling distributions with easy explanations.

1. Population and Sample

In statistics, understanding the difference between a population and a sample is fundamental, as it forms the basis for collecting data and making inferences about a larger group.

Term	Definition	Example
Population	Complete set of items or individuals under study	All employees of a company
Sample	Subset of population selected for analysis	100 employees selected randomly

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2. Sampling Techniques

Sampling techniques are divided into probability and non-probability methods.

Type	Technique	Description	Use / Example
Probability	Simple Random Sampling	Every item has an equal chance	Drawing 50 names randomly
Probability	Systematic Sampling	Every kth item is selected	Every 10th student in a list
Probability	Stratified Sampling	Population divided into strata, samples from each	Sample of students by grade
Probability	Cluster Sampling	Population divided into clusters, select entire clusters	Randomly select 3 schools
Non-Probability	Convenience Sampling	Select easily available items	Survey friends in a class
Non-Probability	Judgmental / Purposive	Select based on researcher’s judgment	Expert opinion survey

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3. Sampling Errors

Sampling errors occur when a sample does not perfectly represent the population.

Error Type	Description	Example
Random Error	Due to chance variation in sample	Selecting 100 students randomly may slightly differ from population mean
Systematic Error / Bias	Consistent error in one direction	Survey conducted only in morning → misses absent students
Non-Sampling Error	Errors not related to sampling	Data entry mistakes, misreporting

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4. Sampling Distribution

Definition: Probability distribution of a statistic (like mean, proportion) calculated from a sample.

• Central Limit Theorem (CLT):
  If sample size nnn is large, the sampling distribution of the sample mean is approximately normal regardless of population distribution.

Xˉ∼N(μ,σ2n)\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)Xˉ∼N(μ,nσ2​)

• Standard Error (SE): Standard deviation of sampling distribution

SE=σnSE = \frac{\sigma}{\sqrt{n}}SE=n​σ​

Example:
Population mean μ=50\mu = 50μ=50, population SD σ=10\sigma = 10σ=10, sample size n=25n = 25n=25 SE=10/25=2SE = 10 / \sqrt{25} = 2SE=10/25​=2

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5. Key Formulas for SSC CGL

In SSC CGL Tier 2, memorizing key formulas for sampling theory helps quickly calculate sample statistics, standard errors, and make accurate inferences about the population.

Concept	Formula	Use
Sample Mean	Xˉ=∑X/n\bar{X} = \sum X / nXˉ=∑X/n	Estimate population mean
Sample Variance	S2=∑(X−Xˉ)2/(n−1)S^2 = \sum (X – \bar{X})^2 / (n-1)S2=∑(X−Xˉ)2/(n−1)	Estimate population variance
Standard Error	SE=σ/nSE = \sigma / \sqrt{n}SE=σ/n​	Measure of sampling variability
Proportion SE	SEp=p(1−p)/nSE_p = \sqrt{p(1-p)/n}SEp​=p(1−p)/n​	Estimate proportion from sample

Key Takeaways

• Population is the whole group, sample is a subset.
• Probability sampling → each element has known chance; Non-probability → chance unknown.
• Sampling errors can be random or systematic.
• Sampling distribution → distribution of sample statistics; CLT allows normal approximation for large samples.
• Standard error quantifies sample variability.

FAQs on Sampling Theory

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