Question

The following data shows the number of family members living in different bungalows of a locality:
 

Number of Members	0−2	2−4	4−6	6−8	8−10	Total
Number of Bungalows	10	p	60	q	5	120

If the median number of members is found to be 5, find the values of p and q.

Hint:

Use the cumulative frequency method and median formula for grouped data to solve problems involving median.

Answer 1

Given Data:

Number of Members	0–2	2–4	4–6	6–8	8–10	Total
Number of Bungalows	10	p	60	q	5	120

Step 1: Use total number of bungalows
\[ 10 + p + 60 + q + 5 = 120 \] \[ p + q = 45 \]

Step 2: Find the median class
Median corresponds to \(\frac{N}{2} = \frac{120}{2} = 60\)
Cumulative frequency:
- Up to 0–2: 10
- Up to 2–4: \(10 + p = 10 + p\)
- Up to 4–6: \(10 + p + 60 = 70 + p\)
Since median is 5 (which lies in 4–6 class), cumulative frequency before median class \(F = 10 + p\),
Class width \(h = 2\),
Frequency of median class \(f = 60\),
Median formula:
\[ \text{Median} = l + \left(\frac{\frac{N}{2} - F}{f}\right) \times h \] Where \(l\) = lower limit of median class = 4.
Substitute values:
\[ 5 = 4 + \left(\frac{60 - (10 + p)}{60}\right) \times 2 \] \[ 5 - 4 = \frac{60 - 10 - p}{60} \times 2 \] \[ 1 = \frac{50 - p}{60} \times 2 \] \[ 1 = \frac{2(50 - p)}{60} = \frac{50 - p}{30} \] \[ 50 - p = 30 \] \[ p = 20 \]

Step 3: Find \(q\)
From Step 1, \(p + q = 45\)
\[ 20 + q = 45 \implies q = 25 \]

Final Answer:
\[ p = 20, \quad q = 25 \]