# Random Sample From an Infinite Universe

It is relatively difficult to explain the concept of random sample from an infinite population. However, a few examples will show the basic characteristics of such a sample. Suppose we consider the 10 throws of a fair dice as a sample from the hypothetically infinite population that consists of the results of all possible throws of the dice. If the probability of getting a particular number, say 7, is the same for each throw and the 10 throws are all independent, then we say that the sample is random. Similarly, it would be said to be sampling from an infinite population if we sample with replacement from an infinite population and our sample would be considered as a random sample if in each draw all elements of the population have the same probability of being selected and successive draws happen to be independent. In brief, one can say that the selection of each item in a random sample from an infinite population is controlled by the same probabilities and that successive selections are independent of one another.

In other words, if we have to take a sample of grain from a bag, it is not possible to assign a number to each grain or particle constituting the universe and as such the methods of constructing card population or of random sampling numbers cannot be used. In such cases a thorough mixing of the grain may be done and by dividing and sub-dividing the lot in parts, a sample of an adequate size can be obtained. The contents of the bag after thorough mixing may be divided in two equal parts of which one may be selected and this may further be divided in two parts after mixing. In this way the process can be continued till one of the sub-divisions is equal to the size of the desired sample.

# Complex Random Sampling Designs

Complex random sampling designs are probability sampling done with restricted sampling techniques. They are also called mixed sampling designs as they tend to combine probability and non-probability sampling procedures during sample selection.

### Some of the popular complex random sampling designs are as follows:

(i) Systematic sampling: The researchers sometimes select every ith item from a list, this is known as systematic sampling. The first unit is a random number and the next unit onwards they are selected at the same fixed intervals.

(ii) Stratified sampling: In a very diverse universe stratified sampling is used were the population is divided into several groups that are more similar and then items are selected from each strata as a sample. The strata is a subjective choice of the researcher based on his experience and judgment by using simple random sampling.

(iii) Cluster sampling: In cluster sampling within the population there might be similar groups these are divided into a number of small homogeneous subdivisions then some of these clusters are randomly selected as sample. Cluster sampling is highly economic. The difference between stratified sampling and cluster sampling is that in stratified sampling a random sample is drawn from each of the strata, whereas in cluster sampling only the selected clusters are studied.

(iv) Area sampling: In area sampling a large area is divided into smaller parts and then samples are selected randomly.  This is a type of cluster sampling were the cluster of units is based on geographic area.

(v) Multi-stage sampling: Multi-stage sampling is a complex type of cluster sampling. Multi-stage sampling is used in researches where the entire universe is very large, for example the entire country; the researcher selects samples in various levels. The researcher after selecting clusters from all universe than randomly selects elements from each cluster. This type of sampling is cost effective and easy to administer.

(vi) Probability proportional to size (PPS) sampling: Probability proportional to size (PPS) sampling: Sometimes cluster sampling units lack equal number of elements; in such cases the researcher uses a random selection process where the probability of selection of each sub group is proportional to the size of the cluster. The actual numbers selected are indicative of the clusters chosen and selected. PPS avoids under representation of any one group.

(vii) Sequential sampling: This is a complex sampling design was the size of the sample is not fixed earlier but is determined according the need of the researcher. In this type of sampling method, the researcher does his research on a particular sample if not satisfied takes another sample unit and so on. The researchers keeps fine tuning the experiment and decides only after doing the experiment whether more samples are needed or not.

# Types of Sampling

Sampling can be basically categorized into probability and non-probability sampling. In probability sampling, each and every element of the population has a probability of being selected in the sample, i.e., the probability can be accurately measured. Whereas, in non-probability sampling, not all elements have a chance of being selected in the sample, i.e., their probability cannot be accurately measured.

### The commonly used sampling methods are given below:

»  Deliberate sampling: It is a non-probability sample design in which the researcher purposively or deliberately selects certain units of the universe to form a sample that would represent the universe. In other words, it is a sampling with a purpose. It is also known as purposive sampling.

»  Simple random sampling: It is a probability sample design where each and every element has an equal probability of being selected in the sample. It is also known as chance sampling.

»  Systematic sampling: In this method, elements from a large population are selected at periodic intervals according to a random starting point, i.e., every nth element is selected for the sample, where n can be any random position of an element.

»  Stratified sampling: In this method, the researcher divides the entire population into different subgroups or strata, and then randomly selects elements proportionally from the strata to include in the sample.

»  Quota sampling: It is a non-probability sample in which the researcher selects random units for a sample according to certain given criteria or quota. In other words, elements are selected according to pre-specified criteria in such a way that the sample represents the same characteristics of the population under study.