Sampling theory, Introduction and Reasons to Sample

Often we are interested in drawing some valid conclusions (inferences) about a large group of individuals or objects (called population in statistics). Instead of examining (studying) the entire group (population, which may be difficult or even impossible to examine), we may examine (study) only a small part (portion) of the population (entire group of objects or people). Our objective is to draw valid inferences about certain facts for the population from results found in the sample; a process known as statistical inferences. The process of obtaining samples is called sampling and theory concerning the sampling is called sampling theory.

Example: We may wish to draw conclusions about the percentage of defective bolts produced in a factory during a given 6-day week by examining 20 bolts each day produced at various times during the day. Note that all bolts produced in this case during the week comprise the population, while the 120 selected bolts during 6-days constitutes a sample.

In business, medical, social and psychological sciences etc., research, sampling theory is widely used for gathering information about a population. The sampling process comprises several stages:

• Defining the population of concern
• Specifying the sampling frame (set of items or events possible to measure)
• Specifying a sampling method for selecting the items or events from the sampling frame
• Determining the appropriate sample size
• Implementing the sampling plan
• Sampling and data collecting
• Data which can be selected

When studying the characteristics of a population, there many reasons to study a sample (drawn from population under study) instead of entire population such as:

1. Time: as it is difficult to contact each and every individual of the whole population
2. Cost: The cost or expenses of studying all the items (objects or individual) in a population may be prohibitive
3. Physically Impossible: Some population are infinite, so it will be physically impossible to check the all items in the population, such as populations of fish, birds, snakes, mosquitoes. Similarly it is difficult to study the populations that are constantly moving, being born, or dying.
4. Destructive Nature of items: Some items, objects etc are difficult to study as during testing (or checking) they destroyed, for example a steel wire is stretched until it breaks and breaking point is recorded to have a minimum tensile strength. Similarly different electric and electronic components are check and they are destroyed during testing, making impossible to study the entire population as time, cost and destructive nature of different items prohibits to study the entire population.
5. Qualified and expert staff: For enumeration purposes, highly qualified and expert staff is required which is some time impossible. National and International research organizations, agencies and staff is hired for enumeration purposive which is some time costly, need more time (as rehearsal of activity is required), and some time it is not easy to recruiter or hire a highly qualified staff.
6. Reliability: Using a scientific sampling technique the sampling error can be minimized and the non-sampling error committed in the case of sample survey is also minimum, because qualified investigators are included.

Every sampling system is used to obtain some estimates having certain properties of the population under study. The sampling system should be judged by how good the estimates obtained are. Individual estimates, by chance, may be very close or may differ greatly from the true value (population parameter) and may give a poor measure of the merits of the system.

A sampling system is better judged by frequency distribution of many estimates obtained by repeated sampling, giving a frequency distribution having small variance and mean estimate equal to the true value.

Sampling Unit

The population divided into a finite number of distinct and identifiable units is called sampling units. OR

The individuals whose characteristics are to be measured in the analysis are called elementary or sampling units. OR

Before selecting the sample, the population must be divided into parts called sampling units or simply sample units.

Sampling Frame

The list of all the sampling units with a proper identification (which represents the population to be covered is called sampling frame). The frame may consist of either a list of units or a map of area (in case sample of area is being taken), such that every element in the population belongs to one and only one unit.

The frame should be accurate, free from omission and duplication (overlapping), adequate, upto data and the units must cover the whole of the population and should be well identified.

In improving the sampling design, supplementary information for the field covered by the sampling frame may also be valuable.

Examples: Sampling Frame and Sampling Unit

1. List of household (and persons) enumerated in population census.
2. A map of areas of a country showing the boundaries of area units.
3. In sampling an agricultural crop, the unit might be a field, a farm or an area of land whose shape and dimensions are at out disposal.

An ideal sampling frame will have the following qualities/characteristics:

• all sampling units have a logical and have numerical identifier
• all sampling units can be found i.e. contact information, map location or other relevant information about sampling units is present
• the frame is organized in a logical and systematic manner
• the sampling frame has some additional information about the units that allow the use of more advanced sampling frames
• every element of the population of interest is present in the frame
• every element of the population is present only once in the frame
• no elements from outside the population of interest are present in the frame
• the data is up-to-date

A sampling frame can be classified subject to several types of defect as follows:

A frame may be inaccurate: where some of the sampling units of the population are listed inaccurately or some units which do not actually exist are included in the list.

A frame may be inadequate: when it does not include all classes of the population which are to be taken the survey.

A frame may be incomplete: when some of the sampling units of the population are either completely omitted or includes more than once.

A frame may be out of date: when it has not been updated according to the demand of the occasion, although it was accurate, complete and adequate at the time of construction.