Question

Find the missing frequency in foll0wing distribution if N is60 and median is 40

Answer 1

The missing frequencies f₁ and f₂ are 15 and 30 respectively.

Given:

N = 60

Median = 40

Marks          0-10   10-30   30-60    60-80     80-90

Frequency    5          f₁           f₂            8             2

To Find:

The values of  f₁ and f₂

Solution:

Cumulative frequency

5

5 + f₁

5 + f₁ + f₂

13 + f₁ + f₂

15 + f₁ + f₂

Since, N = 60

5 + f₁ + f₂ + 8 + 2 = 60

f₁ + f₂ = 60 - 15

f₁ + f₂ = 45 ------------(1)

Median size = N/2 = 60/2 = 30th term

Since median = 40

The median class will be 30-60

The formula to find the median of grouped data is given by

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

where, l = lower limit of the middle class

N = Total number of observations

f = frequency of each class

h = class size

c = cumulative frequency of preceding class

h = upper limit - lower limit

Median = $30 + ( \frac {\frac{60}{2}- (5+f_{1} ) }{f_{2} })30$

40 = $30 +( \frac{30 - 5 - f_{1} }{f_2} )30$

40 = $\frac{30f_2 + 750 -30f_1}{f_2}$

40f₂ = 30f₂ + 750 - 30f₁

30f₁ + 10f₂ = 750

3f₁ + f₂ = 75---------- (2)

Equation (2) - Equation (1)

3f₁ + f₂ - (f₁ + f₂) = 75 - 45

3f₁ + f₂ - f₁ - f₂ = 30

2f₁ = 30

f₁ = 15

Putting the value of f₁ = 15 in equation (1), we get

15 +  f₂ = 45

f₂ = 30

Hence, the missing frequencies f₁ and f₂ are 15 and 30 respectively.

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