**Subject :**Math -3

**Unit :**Statistical Techniques - II

## Sampling Theory

**Universe of Population:**

In any statistical investigation often called statistical survey, observations are made on a group of objects or individuals called elementary units as they are without any interference. The aggregate of individuals under study in any statistical survey is called a population.** According to A.C. Rosander: **A Population is the totality of objects under consideration.

**Types of Population:**

There are two bases to classify populations:

**1**. Population Based on number of objects

- Finite
- Infinite

**2**. Population Based on existence of objects

- Real
- Hypothetical

**Finite or Infinite Population:**

A population is said to be finite if the number of individuals is fixed, i.e., finite. A population is said to be infinite if it is composed of infinitely large number of individuals. For example: Students of your university constitute a finite population, leaves of a tree constitute an infinite population (here infinite means very large).

**Real or Hypothetical Population:**

A population is said to be real, true or existent if it contains concrete and existing objects. For example, the workers in a factory, etc. A population is said to be hypothetical or artificial if it is imaginary or it is constructed hypothetically on a paper by the statistician in his laboratory. This is done to illustrate certain statistical principles. For example: The possible outcomes of heads and tails in tossing a coin.

**Sample:**

A part of the population selected to know some thing about the population is called a Sample. The number of individuals selected in a sample is called its size.** According to G. W. Snedecore and W.G. Cochran: **A sample consists of a small collection from a larger aggregates about which we seek information. The statistical procedure of drawing a sample from the population is called sampling. the statistical procedure which are used for drawing inferences or conclusions about the population from the sample data are covered under inferential statistics or statistical inference. Thus sampling theory is the basis of statistical inference in which we wish to obtain maximum information about the population with minimum effort and maximum precession. The main object of the study of sample or sampling is to get maximum information about the population under consideration at reduced cost, time and energy. According to weather burn, The theory of sampling is concerned first, with estimating the parameters of the population from those of the sample and secondly with gauzing the precision of the estimates.

**Principles of Sampling:**

The following are two important principles which determine the possibility of arriving at a valid statistical inference about the features of a population or process:

- Principle of statistical regularity
- Principle of inertia of large numbers

**Principle of statistical Regularity:
**

This principle is based on the mathematical theory of probability. According to Professor King The low of statistical regularity lays down that a moderately large number of items chosen at random from a large group are almost sure on the average to process the characteristic of the large group. This Principle on emphasizes on two factors:

- Sample size should be Large
- Samples must be Drawn randomly

**Principle of Inertia of Large Numbers:
**

This principle is a corollary of the principle of statistical regularity and plays a significant role in the sampling theory. This principle states that under similar conditions as the sample size get large enough the statistical inference is likely to be more accurate and stable. For example, if a coin is tossed a large number of times then relative frequency of occurrence of head and tail is expected to be equal.

**Large Sample and Small Sample:**

On the study of statistical research methods it is clear the sample size affects the reliability of sample information. The general assumption is that large sample is more reliable. So far test of significance purpose, we divide sample into two groups.

**1**. Large Samples

**2**. Small Samples

The basic assumptions in large sample and small sample are different that is why we use different methods of tests of significance. There is no definite line between large sample and small sample, but when sample size n is 30 or more (n ≤ 30), it is a large sample and when n < 30 it is a small sample.