Question

The median of the following data is 50. Find the values of p and q, if the sum of all the frequencies is 90.

Marks:	20 -30	30-40	40-50	50-60	60-70	70-80	80-90
Frequency:	P	15	25	20	q	8	10

The median of the following data is 50. Find the values of ‘p’ and ‘q’, if the sum of all frequencies is 90. Also find the mode of the data.

Marks obtained	Number of students
20 – 30	p
30 – 40	15
40 – 50	25
50 – 60	20
60 – 70	q
70 – 80	8
80 – 90	10

Answer 1

The given series is in inclusive form. Converting it to exclusive form and preparing the cumulative frequency table, we have

Class interval	Frequency (fi )	Cumulative Frequency (c.f.)
20 – 30	p	p
30 – 40	15	p + 15
40 – 50	25	p + 40
50 – 60	20	p + 60
60 – 70	q	p + q + 60
70 – 80	8	p + q + 68
80 – 90	10	p + q + 78
	78 + p + q = 90

Median = 50 It lies in the interval 50 – 60, so the median class is 50 – 60.

Now, we have

l = 50, h = 10, f = 20, F = p + 40, N = 90

We know that

Median = `"l" + {("N"/2 - "f")/"f"} xx "h"`

`50 = 50 + (45 - ("p" + 40))/20xx10`

⇒ 0 = `(5 - "p")/2`

⇒ p = 5

And,

p + q + 78 = 90

⇒ p + q = 12

⇒ q = 12 − 5 = 7

Mode = `"l" + ("f"_1 - "f"_0)/(2"f"_1 - "f"_0 - "f"_2)."h"`

= `40 + (25 - 15)/(2(25) - 15 - 20)xx10`

= `40 + 100/15`

= 40 + 6.67

= 46.67