Krishna
0

MODE:


Step 1: Recall the mode formula for the grouped data


                  Mode =   l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} * h


                        l - The lower boundary of the modal class,

                        h - The class size,

                       f_1 - The frequency of modal class,

                       f_0 - the frequency of the class preceding the modal class,

                       f_2 - the frequency of the class succeeding the modal class.


Step 2:  Identify the modal class and locate the values of the frequencies.

            NOTE: Locate a class with the maximum frequency, called the modal class.

                      (The mode is a value inside the modal class)


            Here the maximum class frequency is 40 , and the class corresponding

            to  this frequency is 1500 - 2000.

                      So, the modal class is 1500 - 2000.


                            lower boundary l = 1500

                    Class size h = 2000 - 1500 = 500

              The frequency of modal class f_1 = 40

              The frequency of the class preceding the modal class f_0 = 24

            The frequency of the class succeeding the modal class. f_2 = 33


Step 3: Substitute the values in the formula and calculate the mode.


              Mode = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} * h


                        = 1500+\frac{40-24}{2*40-24-33}*500


                      = 1500+\frac{16}{23}*500


                      = 1500 + 347.826


                      = 1847.826


                  Hence, mode = 1847.826

          Modal monthly expenditure = 1847.82


MEAN:

Step 4:  Calculate the mean of the given data

                

              

            Mean: The mean (or average) of observations is the sum of the values of

              all the observations divided by the total number of observations.


        Step deviation method

                   x_i is the class mark = Average of the boundaries(class intervals)

                     Assumed  value a = middle value of the observations = 2750

                            Deviation  d_i = x_i - a

                     Step deviation u_i = \frac{d_i}{h}

                  h = size of the class = 1500 - 1000= 500

                    

                    Sum of the values of all the observations  \Sigma f_iu_i=-35

                      Total number of observations \Sigma f_i=200


        Substitute all the values in the values in the step deviation formula

                        Formula: Mean =   a + \frac{\Sigma f_iu_i}{\Sigma f_i}*h


                                                  = 2750 + \frac{-35}{200}*500


                                                  = 2750 - 87.5


                                        Mean = 2662.50

                  Mean of the monthly expenditure = 2662.50