Sampling Theory and Sampling Methods: Random, Stratified, Systematic and Cluster Sampling, Errors and Standard Error for SSC

Instead of studying a whole population, statisticians study a small part called a sample, and draw conclusions about the population. This guide on sampling theory and methods for SSC Statistics explains the key terms, the methods of probability and non-probability sampling, the errors in sampling and the standard error, with solved examples, practice questions and FAQs.

Basic Terms

Census versus Sample Survey

Probability (Random) Sampling Methods

Non-Probability Sampling Methods

Allocation in Stratified Sampling

In proportional allocation, the sample from stratum i is n_i = n × N_i / N, where n is the total sample size, N_i is the size of the stratum and N is the population size.

Errors in Surveys

Standard Error

• The standard error (SE) is the standard deviation of the sampling distribution of a statistic.
• SE of the sample mean = σ / √n.
• SE of the sample proportion = √(PQ / n), where Q = 1 − P.
• The standard error decreases as the sample size increases (it is inversely proportional to the square root of n).
• For a finite population, multiply by the finite population correction factor √((N − n)/(N − 1)).

Solved Examples

Example 1. A population has 1,000 units and a sample of 50 is required through systematic sampling. Find the sampling interval.
k = N/n = 1000/50 = 20.

Example 2. A company has 600 employees: 300 in A, 200 in B and 100 in C. A stratified sample of 60 is to be selected with proportional allocation. Find the sample from each stratum.
A: 60 × 300/600 = 30; B: 60 × 200/600 = 20; C: 60 × 100/600 = 10.

Example 3. The population standard deviation is 20 and the sample size is 100. Find the standard error of the sample mean.
20/√100 = 2.

Example 4. If the sample size is increased from 25 to 100, how does the standard error change?
SE ∝ 1/√n, so it becomes half (from σ/5 to σ/10).

Example 5. Which sampling method gives each unit an equal chance of selection?
Simple random sampling.

Quick Tips

• Learn the difference between parameter and statistic.
• Remember k = N/n for systematic sampling.
• Use stratified sampling when the population is heterogeneous.
• Remember that SE = σ/√n.

Quick Revision Points

• Parameter is for the population; statistic is for the sample.
• Systematic sampling: k = N/n.
• Stratified: random sampling from homogeneous strata.
• Standard error of the mean = σ/√n.
• Sampling error decreases as n increases.

Practice Questions With Answers

1. What is a numerical measure of the population called?
   Answer: Parameter
2. What is a numerical measure of a sample called?
   Answer: Statistic
3. In systematic sampling with N = 600 and n = 30, find k.
   Answer: 20
4. In stratified sampling, from the population of 800 (400, 300, 100), the sample is 80. How many from the second stratum?
   Answer: 30
5. The standard error of the mean when σ = 15 and n = 25 is?
   Answer: 3
6. Which sampling method divides the population into groups and surveys all units in selected groups?
   Answer: Cluster sampling
7. Which type of error decreases with a larger sample size?
   Answer: Sampling error
8. Is quota sampling a probability sampling method?
   Answer: No
9. What is the list of all units of the population called?
   Answer: Sampling frame
10. Which survey studies every unit of the population?
    Answer: Census

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Statistics notes page, read the previous topic Time Series Analysis and the next topic Hypothesis Testing and Estimation.

Frequently Asked Questions

What is a population?

The complete group of units about which information is wanted.

What is a sample?

A part of the population selected for study.

What is the difference between a parameter and a statistic?

A parameter describes the population and a statistic describes a sample.

What is simple random sampling?

A method in which every unit has an equal chance of being selected.

What is systematic sampling?

Selecting every kth unit of the list after a random start.

What is stratified sampling?

Dividing the population into homogeneous strata and drawing a random sample from each.

What is cluster sampling?

Selecting groups (clusters) at random and surveying all units in them.

What is a sampling error?

The difference between the sample result and the true population value because of studying only a sample.

What is the standard error?

The standard deviation of the sampling distribution of a statistic.

How does the sample size affect the standard error?

The standard error decreases as the sample size increases.

What is a census?

The study of every unit of the population.

Is sampling asked in SSC?

Yes. It is an important part of the SSC CGL Statistics syllabus.