Question

The median of the following data is 525. If the sum of all the frequencies is 100, find the values of p and q.

Class	Frequency
0 – 100	2
100 – 200	p
200 – 300	9
300 – 400	12
400 – 500	17
500 – 600	20
600 – 700	15
700 – 800	9
800 – 900	q
900 – 1000	4

Answer 1

1. Cumulative frequency table

First, let’s calculate the cumulative frequency (cf) for each class:

Class	Frequency (f)	Cumulative Frequency (cf)
0 – 100	2	2
100 – 200	p	2 + p
200 – 300	9	11 + p
300 – 400	12	23 + p
400 – 500	17	40 + p
500 – 600	20	60 + p
600 – 700	15	75 + p
700 – 800	9	84 + p
800 – 900	q	84 + p + q
900 – 1000	4	88 + p + q

2. Using total frequency

We are given that the sum of all frequencies (N) is 100.

88 + p + q = 100

p + q = 100 – 88

p + q = 12   ...(Equation 1)

3. Finding p using the median formula

The median is given as 525, which falls in the class interval 500 – 600.

Therefore, the median class is 500 – 600.

Median Formula:

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

Where:

l (lower limit of median class) = 500

N (total frequency) = 100

cf (cumulative frequency of the preceding class) = 40 + p

f (frequency of the median class) = 20

h (class size) = 100

Substitute the values:

`525 = 500 + ((100/2 - (40 + p))/20) xx 100`

`525 - 500 = ((50 - 40 - p)/20) xx 100`

25 = (10 – p) × 5

Divide both sides by 5:

5 = 10 – p

p = 10 – 5

p = 5

4. Finding q

Now, substitute the value of p into equation 1:

5 + q = 12

q = 12 – 5

q = 7

The value of p is 5 and the value of q is 7.