Question

The median of the following frequency distribution of 165 observations is 38.2. Find the value of a and b.
Class
5-14
14-23
23-32
32-41
41-50
50-59
59-68

Frequency
5
11
a
53
b
16
10

Answer 1

Given: Median of the frequency distribution = 38.2

          Total Observation = 165

To Find: Value of 'a' and 'b'.

Step-by-step explanation:

Class                  Frequency                Cumulative Frequency

5-14                      5                                       5

14-23                    11                                       16

23-32                    a                                       16+a

32-41                    53                                      69+a

41-50                    b                                        69+a+b

50-59                   16                                       85+a+b

59-68                   10                                       95+a+b

                        N = 95 +a+b

Median = $l + (\frac{{\frac{N}{2}} - cf}{f})\times h$

where   l = Lower limit of the median class

            cf = cumulative frequency of the preceding median class

            f  = frequency of the median class

            h = class size

            $\frac{N}{2}$ = $\frac{165}{2} = 82.5$

Median Class = 32-41

l = 32

cf = 16+a

f = 53

h = 9

$38.2=32+(\frac{82.5-(16+a)}{53})\times 9\\38.2-32=(\frac{82.5-16-a}{53})\times 9\\6.2 = (\frac{66.5-a}{53})\times9\\6.2\times 53 = (66.5 -a )\times 9\\328.6= 598.5 - 9a\\9a = 598.5-328.6\\9a= 269.9\\a = \frac{269.9}{9}\\a= 29.98$

a = 30 (approx)

N = 165

95 + a + b = 165  (putting the value of a )

95+30+b= 165

125 + b = 165

b = 165- 125

b = 40