Question

Find the mean and the mode for the following data:

Class	Frequency
5 – 10	5
10 – 15	6
15 – 20	15
20 – 25	10
25 – 30	5
30 – 35	4
35 – 40	2
40 – 45	2

Answer 1

1. Calculation of mean

To find the mean `(barx)` for grouped data, we use the formula:

`barx = (sumf_ix_i)/(sumf_i)`

where f_i is the frequency and x_i is the class mark (mid-point) of each class.

Class	Frequency (fi)	Class Mark (xi)	fixi
5 – 10	5	7.5	37.5
10 – 15	6	12.5	75.0
15 – 20	15	17.5	262.5
20 – 25	10	22.5	225.0
25 – 30	5	27.5	137.5
30 – 35	4	32.5	130.0
35 – 40	2	37.5	75.0
40 – 45	2	42.5	85.0
Total	Σfi = 49		Σfixi = 1027.5

Mean = `1027.5/49`

= 20.969

= 20.97

2. Calculation of mode

The modal class is the interval with the highest frequency, which is 15 – 20 (frequency = 15).

The formula for the mode is:

Mode = `L + ((f_1 - f_0)/(2f_1 - f_0 - f_2))xx h`

L = 15   ...(Lower limit of modal class)

f₁ = 15   ...(Frequency of modal class)

f₀ = 6   ...(Frequency of preceding class)

f₂ = 10   ...(Frequency of succeeding class)

h = 5   ...(Class width)

Mode = `l + ((f_1 - f_0)/(2f_1 - f_0 - f_2)) xx h`

= `15 + ((15 - 6)/(2(15) - 6 - 10)) xx 5`

= `15 + (9/(30 - 16)) xx 5`

= `15 + (19/14) xx 5`

= `15 + 45/14`

= 15 + 3.214

= 18.21