5.   In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained
 varying number of mangoes.The following was the distribution of mangoes according to the number of boxes.

 

Number of mangoes
Number of boxes
50-52
15
53-55
110
56-58
135
59-61
115
62-64
25

 
Solution:


Therefore frequency table is given by:

 

Number of mangoes
Number of boxes ($f_i$)
Midpoint($x_i$)
$d_i=x_i-57$
$u_i= {x_i-a}/{h}$
$f_ix_i$
$f_id_i$
$f_iu_i$
50-52
15
51
-6
-3
765
-90
-45
53-55
110
54
-3
-1.5
5940
-330
-165
56-58
135
57
0
0
7695
0
0
59-61
115
60
3
1.5
6900
345
172.5
62-64
25
63
6
3
1575
150
75
Total
400
22875
75
37.5

Let assumed mean of $x_i$, $a$ = 57 and class size $h$ = 2

The Direct Method: $x̄$ = ${∑f_ix_i}/{∑f_i}$
                             = $22875/400$
							 =$57.1875$

The Assumed Mean Method: $x̄$ =   $a$ +${Σf_i d_i }/{Σf_i}$
                                   =  $57$ +${75 }/{400}$
                                   =  $57$ +$0.1875$
                                   =  $57.1875$


The Step Deviation Method:
 $x̄$  =$a$+$h({Σf_i u_i }/{Σf_i})$
 Let $u_i= {x_i-a}/{h}$

where $a$ is the assumed mean and $h$ is the class size. 

 $x̄$  =$a$+$h({Σf_i u_i }/{Σf_i})$
            =$57$+$2({37.5 }/{400})$
            =$57$+$0.1875$
            =$57.1875$