Students t Distribution
If we take a very large number of small samples from a population and calculate the mean for each sample and then plot the frequency distribution of these means the resulting sampling distribution would be the students t distribution.
The greatest contribution to the theory of small sample was made by Sir William Gossett and R. A Fisher published his discovery in 1905 under the pen name the students and it is popularly known as test or student distribution or student distribution.
When the sample size is 30 or less the population standard deviation is unknown we can use the t distribution.
The formula is t = ( X - µ) / s X √ n
Where s or sample standard deviation
if samples S. D is given without using n-1 denominator then
t= ( X- µ))/ S/ √ n-1
it should be noted that the only difference in the calculation of S in large and small samples is that whereas in the case of the former the sum of the squares of deviations of various items from the mean of ∑ ( X-X)2 is divided by n-1. ( the number of items ) in case of small samples it is divided by n-1 which are the degrees of freedom. The degree of freedom in such problems i s n-1 because one has the freedom to change only n-1 items as the last items has to be the difference between ∑X and the sum of n-1 items. Thus if there are 5 items in a sample with a total of 30. We can change only 4 items as we like. The last items will have to be 30 minus the sum of the remaining 4items. whose values have been changed.