Tests of Significance For Small Samples

So far we have discussed problems relating to large samples. When the size of sample is small the above tests are not applicable because the assumptions on which they are based generally do not hold good in case of small samples. In particular, it will no longer be possible for us to assume (a) that the random sampling distribution of a statistic is approximately normal, and (b) that values given by the sample data are sufficiently close to the population values and can be used in their place for the calculation of the standard error of the estimate.

The removal of these assumptions makes it necessary to use entirely now techniques to deal with the problems of small samples. The division between the theories of large and small samples is, therefore, a very real one though it is not always easy to draw a precise line of demarcation. It should be noted that as a rule, the methods and the theory of small samples are applicable to large samples, through the reverse is not true.


While dealing with small samples our main interest is not to estimate the population values as is true in large samples; rather our interest lies in testing a given hypothesis, i.e., in ascertaining whether observed values could have arisen by sampling fluctuations from some value given in advance.

It should be noted that the investigator who works with very small samples must know that his estimates will vary widely from sample to sample. Moreover, He must be satisfied with relatively wide confidence intervals. Precision of statement is less, of course, the wider the intervals employed. Each inference based on a much smaller sample.
   
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