Question

The median value for the following frequency distribution is 35 and the sum of all the frequencies is 170. Using the formula for median, find the missing frequencies.

Class	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60	60 – 70
Frequency	10	20	?	40	?	25	15

Answer 1

1. Formulate the frequency equation

Let the missing frequencies for the classes 20 – 30 and 40 – 50 be f₁ and f₂ respectively.

We are given that the sum of all frequencies is 170.

10 + 20 + f₁ + 40 + f₂ + 25 + 15 = 170

110 + f₁ + f₂ = 170

f₁ + f₂ = 60   ...(Equation 1)

2. Set up the cumulative frequency table

To use the median formula, we calculate the cumulative frequency (cf) for each class:

Class Interval	Frequency (f)	Cumulative Frequency (cf)
0 – 10	10	10
10 – 20	20	30
20 – 30	f1	30 + f1
30 – 40	40	70 + f1
40 – 50	f2	70 + f1 + f2
50 – 60	25	95 + f1 + f2
60 – 70	15	110 + f1 + f2

3. Identify the median class parameters

Since the given Median is 35, it lies within the class interval 30 – 40. Thus, 30 – 40 is our median class.

Lower limit of the median class (L) = 30

Frequency of the median class (f) = 40

Cumulative frequency of the preceding class (cf) = 30 + f₁

Class width (h) = 10

Total frequency (N) = 70 ⇒ `N/2 = 85`

4. Solve for the first missing frequency (f₁)

Using the continuous grouping median formula:

Median = `L + ((N/2 - cf)/f) xx h`

Substitute the known parameters:

`35 = 30 + ((85 - (30 + f_1))/40) xx 10`

`35 - 30 = (85 - 30 - f_1)/4`

`5 = (55 - f_1)/4`

20 = 55 – f₁

f₁ = 35

5. Solve for the second missing frequency (f₂)

Substitute the value of f₁ = 35 into Equation 1:

35 + f₂ = 60

f₂ = 60 – 35

f₂ = 25

The values of the missing frequencies are f₁ = 35 and f₂ = 25.