Question

Find the missing frequencies p and q in the following frequency distribution, when sum of frequencies is 40 and mean is 19:

Class	0 – 5	5 – 10	10 – 15	15 – 20	20 – 25	25 – 30	30 – 35
Frequency	2	5	6	p	10	q	4

Answer 1

Step 1: Calculate mid-points (x_i) and f_i · x_i

The mid-point (x_i) of each class is the average of its upper and lower limits.

Class	Frequency (fi)	Mid-point (xi)	fi · xi
0 – 5	2	2.5	5
5 – 10	5	7.5	37.5
10 – 15	6	12.5	75
15 – 20	p	17.5	17.5p
20 – 25	10	22.5	225
25 – 30	q	27.5	27.5q
30 – 35	4	32.5	130
Total	Σfi = 27 + p + q		Σfixi = 472.5 + 17.5p + 27.5q

Step 2: Formulate the equations

We are given that the sum of frequencies is 40.

27 + p + q = 40

p + q = 13   ...(Equation 1)

Equation 2: Using the mean

The formula for the mean `(barx)` is `barx = (sumf_ix_i)/(sumf_i)`.

Given `barx = 19` and `sumf_i = 40`:

`19 = (472.5 + 17.5p + 27.5q)/40`

Multiply both sides by 40:

760 = 472.5 + 17.5p + 27.5q

17.5p + 27.5q = 287.5

To simplify, divide the entire equation by 2.5:

7p + 11q = 115   ......(Equation 2)

Step 3: Solve the System of Equations

From Equation 1, we can say p = 13 – q.

Substitute this into Equation 2:

7(13 – q) + 11q = 115

91 – 7q + 11q = 115

4q = 115 – 91

4q = 24

q = 6

Now, substitute q = 6 back into Equation 1:

p + 6 = 13

p = 7

The missing frequencies are p = 7 and q = 6.