Module 4. Concepts of sampling methods

Lesson 14

SIMPLE RANDOM SAMPLING

14.1  Introduction

In the selection of a sample, it is desired to have the sample a true representative of the population. A large number of sampling schemes are available to achieve this objective. Various methods of sampling can be grouped under two broad heads; probability sampling (also known as random sampling) and non-probability (or non-random) sampling. Probability sampling methods are those in which every item in the universe has a known chance or probability of being chosen in the sample. This implies that the selection of sample item is independent of the person making the study-that is the sampling operation is controlled so objectively that the items will be chosen strictly at random.

14.2  Methods of Sampling

There are various methods of sampling that may be used singly or along with others. The choice of an appropriate sampling design is of paramount importance in the execution of a sample survey and is generally made keeping in view the objectives and scope of the enquiry and the nature of the population to be sampled. The sampling techniques may be broadly classified as follows.

(i)      Purposive or Subjective or Judgment Sampling

(ii)     Probability Sampling

(iii)  Mixed Sampling or Restricted Sampling

14.2.1  Purposive or subjective or judgment sampling

In this method of sampling the choice of sample items depends exclusively on the judgment of the investigator. In this method, a desired number of sample units are selected deliberately or purposively depending upon the object of the enquiry so that only the important items representing the true characteristics of the population are included in the sample. Purposive sampling is one in which the sample units are selected with definite purpose in view. This type of sampling suffers from the drawback of favouritism and nepotism depending upon the beliefs and prejudices of the investigator and thus does not give a true representation to the population.

14.2.2  Probability sampling

Probability sampling provides a scientific technique of drawing samples from the population according to some laws of chance in which each unit in the universe has some definite pre-assigned probability of being selected in the sample. The selection of the sample based on the theory of probability is also known as random selection and the probability sampling is also called Random Sampling. Different types of sampling are:

·         Each sample unit has an equal chance of being selected

·         Sampling units have varying probability of being selected

·         Probability of selection of a unit is proportional to the sample size.

14.2.3  Mixed sampling

Sampling design in which the sample units are selected partly according to some probability laws and partly according to a fixed sampling rule (no use of chance), is known as Mixed Sampling.

14.3  Simple Random Sampling

Simple random sampling (S.R.S.) is the technique in which sample is drawn in such a way that each and every unit in the population has an equal and independent chance of being included in the sample. Suppose we take a sample of size n from a finite population of size N. Then there are NCn possible samples. A S.R.S. is the technique of selecting the sample in which each of NCn samples has an equal chance or probability p = 1/ NCn of being selected.

14.3.1  Simple random sampling without replacement

If the unit selected or drawn one by one in such a way that a unit drawn in any draw is not replaced in the population before making the next draw, then it is known as simple random sampling without replacement (srswor). A very important and interesting feature of simple random sampling without replacement (srswor) is that, “the probability of selecting a specified unit of population at any given draw is equal to the probability of its being selected at the first draw.” This implies that in srswor from a population of size N, the probability that any sampling unit is included in the sample is 1/N and this probability remains constant throughout all the drawings. Mathematically, if Er is the event that any specified unit is selected at rth draw, then

               

where n is the sample size. In particular it implies

               

i.e. the chance of selection of any specified item is same at any subsequent draw as it was in the first draw, viz. 1/N.  Alternatively srswor can be defined as “If a sample of size n is drawn without replacement from a population of size N then there are NCn possible samples. Simple Random Sampling is the technique of selecting the sample so that each of these NCn samples has an equal chance or probability p = 1/ NCn of being selected in the sample.”

14.3.2  Simple random sampling with replacement

If the unit selected is drawn one by one in such a way that a unit drawn at a time is replaced in the population before the subsequent draw, then it is known as simple random sampling with replacement (srswr).In this type of sampling, a unit can be included more than once in a sample. Therefore, if the required sample size is n, the effective sample size is sometimes less than n due to inclusion of one or more units more than once. Thus, simple random sampling with replacement always amounts to sampling from an infinite population, even though the population is finite. If sampling is done with replacement, then there are Nn possible samples of size n. In this case, simple random sampling (srswr) gives equal chance P = 1/ Nn for each of the Nn samples to be selected.

Remark: With the idea that effective sample size should be adhered to, the simple random sampling without replacement is adopted.

14.4  Selection of Simple Random Sample

Generally, the method of selection should be independent of the properties of sampled population. Proper care should be taken to ensure that the selected sample is random. Human bias, which varies from individual to individual, is inherent in any sampling scheme administered by human beings. Random samples can be obtained by any of the following two methods:

(i)     Lottery method

(ii)   Use of tables of Random numbers  

14.4.1  Lottery method

The simplest method of drawing a random sample is the lottery system. Suppose we want to select ‘r’ individuals out of n. We assign the numbers (1 to n), one number to each individual and write these (numbers) 1 to n on n slips which are made as homogenous as possible in shape and size etc. These slips are then put in a bag and thoroughly shuffled and then ‘r’ slips are drawn one by one. The ‘r’ individuals corresponding to numbers on these slips drawn constitute the random sample. This method of selection is quite independent of the properties of population and is one of the most reliable methods of selecting a random number.

14.4.2  Use of tables of random numbers

The lottery method described above is quite time consuming and cumbersome to use if the population is sufficiently large. The most practical and inexpensive method of selecting a random sample consists in the use of random number tables. These tables have been constructed in such a way that each of the digits 0,1,2---, 9 appear with approximately same frequency and independent of each other. If we have to select a sample from a population of size N(≤ 99) then the numbers can be combined two by two to give pairs from 00 to 99.Simliarly if N(≤ 999) or N(≤ 9999) and so on, then combine the digits three by three (or four by four and so on ) .The method of drawing the random sample consists of the following steps :

 (i)      Identify the N units in the population with numbers from 1 to N.

(ii)      Select at random, any page of the random number tables and pick up the numbers in any row or column or diagonal at random.

(iii)       The population units corresponding to the numbers selected in step (ii) constitutes the random sample.  

The following random number tables are commonly used in practice:

1)   Tippet’s (1927) Random Numbers Tables: These tables consist of 10400 four-digited numbers giving in all 10400Χ4 i.e., 41600 digits taken from British Census reports.

2)     Fisher and Yates (1938) Tables: These tables comprise 15,000 digits arranged in twos.

3)     Kendall and Babington Smith’s (1939) Random Tables:  These tables consist of 1,00,000 digits grouped into 25, 000 sets of 4 digited random numbers.

4)     Rand Corporation (1955) random number tables consist of one million random digits consisting of 2,00,000 random numbers of 5 digits each.

14.5  Some Important Results on Simple Random Sampling

Let us consider a simple random sample (without replacement) of size n from a population of size N. Let the observations on the population units be denoted by Y1,Y2,Y3, … ,Yn and the observations on the sample units be denoted by X1,X2,X3, … ,Xn. Then, in the usual notation we have:

Population mean (μ) is given by:

               

Population variance (σ2) is given by:

                                                                                                                  ………(Eq. 14.1)           

Population mean squares (S2) is given by

                                                                                                                                                                                                ………(Eq. 14.2)

From equations 14.1 and 14.2 we get,

           

Sample mean is given by

                                                                                                                                                                            ………(Eq. 14.3) 

Sample variance is given by

                                                                                                                                                                                                                                             ………(Eq. 14.4) 

14.5.1 In simple random sampling without replacement (srswor), the sample mean is an unbiased estimate of the population mean

                                                                                                                                                                                                                          ………(Eq. 14.5) 

14.5.2 In srswor, the sample mean square is an unbiased estimate of the population mean square

                                                                                                                                                                                                 ………(Eq. 14.6) 

14.5.3  In srswor, the variance of the sample mean is given by

                                                                                                                                                         ………(Eq. 14.7)            

14.5.4  Comparison of  srswor with srswr

Simple random sampling with replacement (srswr) can be regarded as sampling from an infinite population and its variance is given by:

                                                                                                                                                                                          ………(Eq. 14.8)

Comparing it with equation 14.7 we get  i.e. the variance of the sample mean (as an estimate of ΅= ) is less in srswor as compared with its variance in the case of srswr. This implies that srswor provides a better (more efficient) estimator of the population mean ΅ relative to srswr.

14.6  Merits and Demerits of Simple Random Sampling

Merits

·            Since it is a probability sampling, it eliminates the bias due to personal judgment or discretion of the investigator. Accordingly, the sample selected is more representative of the population than in the case of judgment sampling.

·           Because of its random character, it is possible to ascertain the efficiency of the estimates by considering the standard errors of their sampling distributions.

·           The theory of random sampling is highly developed so that it enables us to obtain the most reliable and maximum information at the least cost, and results in savings in time, money and labour.

Demerits

·         Simple random sampling requires an up-to-date frame, i.e., a complete and up-to-date list of the population units to the sampled. In practice, since this is not readily available in many enquiries, it restricts the use of this sampling design.

·          In field surveys if the area of coverage is fairly large, then the units selected in the random sample are expected to be scattered widely geographically and thus it may be quite time consuming and costly to collect the requisite information or data.

·          If the sample is not sufficiently large, then it may not be representative of the population and thus may not reflect the true characteristic of the population.

·         The numbering of the population units and the preparation of the slips is quite time consuming and uneconomical particularly if the population is large therefore, this method cannot be used effectively to collect most of the data in social sciences.