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.