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Sampling Theory

Dr. T. T. Kachwala

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Population Characteristics

There are two ways in which reliable data or information

for Population Characteristics may be obtained:

i. Complete Enumeration Survey (Census Method)

ii. Sampling Method

We use Complete Enumeration Survey to examine every

person or item in the population we wish to describe. This

method is accurate but costly & time consuming.

Sampling Method

We use sampling methods, when it is not possible to count or measure

every item in the population. Sampling is simply the process of

learning about the population on the basis of the sample drawn from it.

Thus, in the sampling technique instead of every unit of the

population, only a part of the population is studied and the conclusions

are drawn on that basis for the entire population.

A sample is not studied for its own sake. The basic objective of its

study is to draw inference about the population. In other words,

sampling is only a tool which helps to know the characteristics of the

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Characteristics of Population & Sample

Mathematically, we can describe the characteristics of samples and populations by using measures such as mean, standard deviation and proportion of success.

When these terms describe the characteristics of a sample, they are called statistics. When they describe the characteristics of a population, they are called parameters. i.e. A statistic is a characteristic of a sample & A parameter is a characteristic of a population. In both the cases the study objective is the same i.e. to predict the value of the population parameter ( or or p).

We use the following symbol to differentiate population characteristic and sample characteristic :

Parameters (Population) Statistics (Sample)

Population size = N Sample size = n

Population mean = Sample mean = X bar

Population S.D = Sample S.D = s

Population proportion of success = p Sample proportion of success = p bar

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Principles of Sampling

There are two important principles on which the theory of sampling is based:i. Principle of Statistical Regularityii. Principle of Inertia of Large Numbers

 Principle of Statistical Regularity: states that if the sample drawn is a random

sample, then the characteristics of the sample will be very close to the

characteristics of the population.

Random Sample is a sample such that each an every item of the population

has an equal and independent chance of being selected in the sample.

Principle of Inertia of Large Numbers: states that every thing else remaining the

same, larger the sample sizes, more accurate are the estimates of the Population

Parameters.

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Selection of Random Sample

To select a simple random sample one may use:Lottery methodRandom Number TablesExcel Functions

Apart from simple random sampling, there are other attempts of sampling to approximate simple random sampling. These are systematic sampling, stratified sampling and cluster sampling.

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Methods of Sampling

Systematic Sampling: In this method, elements are selected from the population at a uniform interval that is measured in time, order or space Example 1: We want to interview every tenth student in a class

of 100 students; we choose a random starting point in the first 10 names and then pick every 10th name thereafter; example - Roll Nos. 5, 15, 25, 35, 45 and up to 95}.

Example 2: In an audit in Bank for Cheque transactions (cheques are arranged in time chronology from 1st day to the last day), by selecting cheques at regular intervals we can ensure that cheques prepared at different times (days) are selected in our sample study

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Methods of Sampling

Stratified Sampling: We divide the population into relatively homogenous groups (strata). We select at random from each stratum a specified number of elements corresponding to the proportion of that stratum in the population as a whole. Example 1: Selection of students for group work based on academic

background Arts, Commerce, Science, Engineering, Other Disciplines. Example 2: Consumer Price Index is based on predefined groups

(strata) like food & beverages, light, housing, clothing etc. Example 3: Doctor wants to find out how many hours his patient

sleeps. He takes a random sample of four age groups: less than 19 years, 20-39, 40-59, 60 & above

Stratified Sampling is popular when the population is already divided in to groups of different sizes & we wish to acknowledge this fact

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Methods of Sampling

Cluster Sampling: We divide the population into clusters and then select a random sample of these clusters. After selecting a certain cluster, each element of this cluster would be examined.

Cluster Sampling is sometimes referred as Area sampling or Territory sampling. We divide the city map in to areas & then select a few areas randomly & then study each house in that area Example 1: Survey to find out the number of TV sets per house in

Mumbai city Example 2: Political polling for exit interviews in an election ward

Each of these methods have been developed for their precision, economy or physical ease. They are at best an approximation of simple random sampling. However simple random sampling procedure is assumed in all our discussions of statistical inferences.

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Thank you

Dr. T. T. Kachwala