Question

Find mean and mode of the following data:

Class	0 – 15	15 – 30	30 – 45	45 – 60	60 – 75	75 – 90	90 – 105
Frequency	4	6	8	10	12	7	3

Answer 1

1. Calculate the mid-points (x_i)

To find the mean, first determine the mid-point for each class interval using the formula `x_i = ("Lower Limit" + "Upper Limit")/2`

Class	Frequency (fi)	Mid-point (xi)	fi × xi
0 – 15	4	7.5	30.0
15 – 30	6	22.5	135.0
30 – 45	8	37.5	300.0
45 – 60	10	52.5	525.0
60 – 75	12	67.5	810.0
75 – 90	7	82.5	577.5
90 – 105	3	97.5	292.5
Total	Σfi = 50		Σfixi = 2670.0

2. Calculate the mean `(barx)`

The mean is calculated by dividing the sum of the products of frequencies and mid-points by the total frequency:

`barx = (sumf_ix_i)/(sumf_x)`

= `2670/50`

= 53.4

3. Identify the modal class

The modal class is the class interval with the highest frequency.

Maximum frequency = 12

Modal Class = 60 – 75

4. Calculate the mode

Use the formula for grouped data:

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

Where:

l = 60 (Lower limit of the modal class)

f₁ = 12 (Frequency of the modal class)

f₀ = 10 (Frequency of the preceding class)

f₂ = 7 (Frequency of the succeeding class)

h = 15 (Class width)

Substituting the values:

Mode = `60 + ((12 - 10)/(2(12) - 10 - 7)) xx 15`

= `60 + (2/(24 - 17)) xx 15`

= `60 + (2/7) xx 15`

= `60 + 30/7`

= 60 + 4.2857

= 64.2857

The mean of the data is 53.4 and the mode is approximately 64.29.