Sampling Methods & Sampling Distribution | UGC NET Economics – Notes

Introduction

In statistical investigation, information about a group is required, and this group is called the population or universe. A population may be finite (having a countable number of units, such as the number of workers in a factory) or infinite (having an uncountably large number of units, such as the outcomes of tossing a coin repeatedly). When every single unit of the population is studied, the method is called census or complete enumeration. When only a part of the population, selected in a systematic and scientific manner, is studied and conclusions are drawn about the whole population, the method is called sampling. The selected part of the population is called a sample, and the number of units in the sample is called the sample size (denoted n), while the number of units in the population is the population size (denoted N).

Sampling is preferred over census for several practical and theoretical reasons. It results in reduced cost, since studying a fraction of the population requires fewer resources than studying the entire population. It leads to greater speed, as data collection, tabulation, and analysis take less time. It permits greater accuracy, because with a smaller number of units, more trained personnel, better supervision, and more sophisticated tools can be used, reducing non-sampling errors. Sampling is essential when the population is infinite or when the testing process is destructive in nature (for example, testing the lifespan of electric bulbs or crash-testing vehicles, where testing the entire population would destroy it).

The theoretical foundation of sampling rests on the Law of Large Numbers, which states that as the sample size increases, the sample statistic tends to approach the true population parameter, and on the Central Limit Theorem, which describes the behaviour of the sampling distribution of the mean.

Basic Terminology

A parameter is a numerical characteristic of the population (such as population mean μ, population variance σ², or population proportion P), and it is a fixed but usually unknown quantity. A statistic is a numerical characteristic computed from sample data (such as sample mean x̄, sample variance s², or sample proportion p), and unlike a parameter, a statistic is a random variable because its value changes from sample to sample.

The sampling frame is the actual list or record of all units of the population from which the sample is drawn. Sampling error arises because a sample, being only a subset, cannot perfectly represent the population, and this error diminishes as sample size increases and vanishes when the sample becomes the census. Non-sampling error, in contrast, arises from mistakes in data collection, recording, processing, or non-response, and these errors can occur even in a complete census and generally increase with sample size due to greater handling of data.

Types of Sampling

Sampling methods are broadly divided into two categories: probability sampling (also called random sampling) and non-probability sampling (also called non-random sampling).

Probability Sampling Methods

In probability sampling, every unit of the population has a known and non-zero probability of being selected in the sample. This is the scientifically preferred method because it allows the calculation of sampling error and permits statistical inference about the population.

(a) Simple Random Sampling (SRS):

Simple random sampling is a method in which every unit of the population has an equal chance of being selected, and every possible sample of a given size has an equal chance of being chosen. It can be conducted in two ways: sampling with replacement (SRSWR), where a selected unit is returned to the population before the next draw, so the same unit may be selected again, and sampling without replacement (SRSWOR), where a selected unit is not returned, so it cannot be chosen again.

For SRSWR, the total number of possible samples of size n from a population of size N is (N to the power n), while for SRSWOR, the number of possible samples is given by the combination formula:

$$
{}^{N}C_{n} = \frac{N!}{n!(N-n)!}
$$

Simple random sampling is typically carried out using a lottery method or a table of random numbers.

Example: If a college has 500 students and a researcher wants to select 50 students such that every student has an equal chance of selection, assigning each student a number from 1 to 500 and using a random number table to pick 50 numbers is an example of simple random sampling.