Samples and Sampling

In Chapter 15 we dealt with the statistical properties of an arbi­trarily selected group of numbers, which can be considered a set. Every element of the set, in most cases, contributes toward deter­mining the required property. If the population of the set is too large to handle, or if economy in terms of time or money is cru­cial, it is normal practice in statistical studies to randomly select a smaller number of elements, forming a subset, known as a sample, and to predict or infer the statistical properties, such as average and deviation, of the whole population from those of the sample. The inferred properties thus obtained cannot be expected to give the corresponding property of the whole population with 100 percent accuracy for the simple reason that all the elements of the population are not allowed to participate in deriving such prop­erty. But such properties derived from the sample, known as sam­ple information, will bear probabilistic relations to the corresponding properties of the entire population. This explicit combination of probability and descriptive statistics is referred to as the Theory of Statistical Inference; some applications of this we will see in Chapter 19.

Because our context is experimental research, we may need to encounter a number of items within a class in various forms: experimental specimens, different parameters as causes in an experiment, the steps of variation of a given parameter, the com­bination of many parameters at many levels as causes, the read­ings of measuring instruments used to record the effects in an experiment, the enumeration of yes-or-no results as effects of an inquiry, and so on. In all such cases of laboratory experiments, the numbers involved are not necessarily large.

There are disciplines, however, in which the experiments are not confined to the laboratory, for example, psychology, educa­tion, and sports, in which surveys of various sizes are used, hence, in which large numbers of elements in a given class are fairly common. Sampling in such cases is a very important component of experimentation. Sampling has several contextual variations, for instance:

1. Collecting data, such as height from a small num­ber of male tenth graders to represent the data of all male tenth graders in the country
2. Selecting a small number of plants to represent all plants of that kind in a greenhouse, the pur­pose being to test the effect of a plant food (see Chapter 7)
3. Selecting a small number of manufactured items issued from an assembly line, the purpose being to test these items for performance as part of quality control

In the context of experimental research, sampling is often unavoidable, even in principle. For instance, the experiment on the benefit of a plant food discussed in Chapter 7 involved such questions as

1. Is the new plant food beneficial to the plants?
2. If yes, to what extent is it beneficial?
3. Are there any deleterious effect on the plants?

To answer such questions, it is reasonable that only a limited number of plants, a small percentage of the whole lot, be sub­jected to the test. When a new drug is to be tested on humans as a possible remedy for a specific disease, the situation is even more critical. Sampling in such circumstances is not only unavoidable, but it is even necessary because it embodies part of the hypothe­sis. The basis of all sampling is that out of an available set (also known as a lot, group, or population) of items, a smaller set needs to be selected. The selection should be done in such a way that every member of the original set has an equal chance of becoming a member of the smaller set. When this criterion is met, the proce­dure followed is known as random sampling. Thus, the idea or notion of a set plays an important role in sampling. The logical elaboration of this, expressed in mathematical terms, is known as set theory. A brief introduction to this follows.

Source: Srinagesh K (2005), The Principles of Experimental Research, Butterworth-Heinemann; 1st edition.