8.17 Test of significance for small samples
So far we have discussed problems belonging to large samples. When a small sample (size < 30) is considered, the above tests are inapplicable because the assumptions we made for large sample tests, do not hold good for small samples. In case of small samples it is not possible to assume (i) that the random sampling distribution of a statistics normal and (ii) the sample values are sufficiently close to population values to calculate the S.E. of estimate.
Thus an entirely new approach is required to deal with problems of small samples. But one should note that the methods and theory of small samples are applicable to large samples but its converse is not true.
Degree of freedom ( df ): By degree of freedom
we mean the number of classes to which the value can be assigned
arbitrarily or at will without voicing the restrictions or limitations
For example, we are asked to choose any 4 numbers whose total
is 50. Clearly we are at freedom to choose any 3 numbers say 10,
23, 7 but the fourth number, 10 is fixed since the total is 50 [50
- (10 + 23 + 7) = 10]. Thus we are given a restriction, hence the
freedom of selection of number is 4 - 1 = 3.
The degree of freedom ( df ) is denoted by n (nu) or df and it is given by n = n - k, where n = number of classes and k = number of independent constrains (or restrictions).
In general for a Binomial distribution, n = n - 1
For Poisson distribution, n = n - 2 (since we use total frequency and arithmetic mean).
For normal distribution, n = n - 3 (since we use total frequency, mean and standard deviation) etc.