Page 156 - DMGT404 RESEARCH_METHODOLOGY
P. 156

Research Methodology




                    Notes
                                            N   
                                               C  
                                   Q 1   L Q      4     h
                                              f Q 1
                                         1
                                   Here,  L  is lower limit of the first quartile class, h is its width,  f Q 1   is its frequency and C is
                                         Q
                                          1
                                   cumulative frequency of classes preceding the first quartile class.
                                   By definition, the second quartile is  median of the distribution. The third  quartile (Q ) of  a
                                                                                                          3
                                   distribution can also be defined in a similar manner.
                                   For a discrete distribution, Q  is that value of the variate such that at least 75% of the observations
                                                         3
                                   are less than or equal to it and at least 25% of the observations are greater than or equal to it.
                                   For a grouped frequency distribution, Q  is that value of the variate such that  area under the
                                                                   3
                                   histogram to the left of the ordinate at Q  is 75% and the area to its right is 25%. The formula for
                                                                   3
                                   computation of Q  can be written as
                                                 3
                                             3N  
                                                C
                                               
                                   Q   L      4      , h  where the symbols have their usual meaning.
                                    3   Q
                                         3    f
                                               Q
                                               3
                                   Deciles
                                   Deciles divide a distribution into 10 equal parts and there are, in all, 9 deciles denoted as  D , D ,
                                                                                                           1  2
                                   ...... D  respectively.
                                       9
                                   For a discrete distribution, the ith decile D  is that value of the variate such that at least (10i)% of
                                                                    i
                                   the observation are less than or equal to it and at least (100 – 10i)% of the observations are greater
                                   than or equal to it (i = 1, 2, ...... 9).
                                   For a continuous or grouped frequency distribution, D  is that value of the variate such that the
                                                                              i
                                   area under the histogram to the left of the ordinate at  D  is (10i)% and the area to its right is
                                                                                 i
                                   (100 - 10i)%. The formula for the ith decile can be written as
                                                      iN   
                                                          C
                                                         
                                               D   L      10     h (i = 1, 2, ...... 9)
                                               i   i D  f
                                                         i D
                                   Percentiles

                                   Percentiles divide a distribution into 100 equal parts and there are, in all, 99 percentiles denoted
                                   as P , P , ...... P , ...... P , ...... P , ...... P , respectively.
                                      1  2    25     40    60    99
                                   For a discrete distribution, the kth percentile P  is that value of the variate such that at least k%
                                                                        k
                                   of the observations are less than or equal to it and at least (100 –  k)% of the observations are
                                   greater than or equal to it.

                                   For a grouped frequency distribution, P  is that value of the variate such that the area under the
                                                                  k
                                   histogram to the left of the ordinate at P  is k% and the area to its right is (100 – k)%. The formula
                                                                  k
                                   for the kth percentile can be written as
                                                                         kN   
                                                                             C
                                                                            
                                                                P k   L  k P      100     , h  (k = 1, 2, ...... 99)
                                                                           f
                                                                            k P


          150                               LOVELY PROFESSIONAL UNIVERSITY
   151   152   153   154   155   156   157   158   159   160   161