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An incomplete distribution is given below:

If median value is $46$ and the total number of items is $230$ . What are the missing frequencies $f_{1}$ and $f_{2}$ ?

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40 46

34 46

37 50

51 46

answer is B.

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Detailed Solution

It is given the following frequency distribution table.
Question Image

We create a cumulative frequency table for the data as shown below:

Class Interval	Frequency(f)	Cumulative frequency
10 - 20	12	12
20 - 30	30	42
30 - 40	$f_{1}$	42+ $f_{1}$
40 - 50	65	107+ $f_{1}$
50 - 60	$f_{2}$	107+ $f_{1}$ + $f_{2}$
60 -70	25	132+ $f_{1}$ + $f_{2}$
70 - 80	18	150+ $f_{1}$ + $f_{2}$

We are given the median as 46.
We know that, median is given as:

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

From the data, we get that the median class is 40-50.
So, we get
l = 40

n = 230 ⇒ n 2 = 115

cf = 42 + f1
f = 65
h = 10
Putting these values in the formula, we get

46 = 40 + 115 − 42 + f 1 65 × 10 ⇒ 46 − 40 = 73 − f 1 65 × 10 ⇒ 6 × 65 10 = 73 − f 1 ⇒ f 1 = 73 − 39 ⇒ f 1 = 34

We are given that

n = 230 ⇒ 230 = 150 + f 1 + f 2 ⇒ 34 + f 2 = 230 − 150 ⇒ f 2 = 80 − 34 ⇒ f 2 = 46

Thus, the value of

f_{1}

is 34 and

f_{2}

is 46.
Hence, the correct option is 2.

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