Question

Find the mean, median and mode of the following data.

Class	0 – 20	20 – 40	40 – 60	60 – 80	80 – 100	100 – 120	120 – 140
Frequency	6	8	10	12	6	5	3

Answer 1

To find the mean, let us put the data in the table given below:

Class	Frequency `(f_i)`	Class mark `(x_i)`	`f_i x_i`
0 – 20	6	10	60
20 – 40	8	30	240
40 – 60	10	50	500
60 – 80	12	70	840
80 – 100	6	90	540
100 – 120	5	110	550
120 – 140	3	130	390
Total	`Ʃ f_i` = 50		`Ʃ f_i x_i` = 3120

Mean =`(Ʃ _i  f_i x_i)/(Ʃ_i  f_i)`

=`3120/50`

= 62.4

Thus, the mean of the given data is 62.4.

Now, to find the median, let us put the data in the table given below:

Class	Frequency` (f_i)`	Cumulative Frequency (cf)
0 – 20	6	6
20 – 40	8	14
40 – 60	10	24
60 – 80	12	36
80 – 100	6	42
100 – 120	5	47
120 – 140	3	50
Total	`N = Σ f_i` = 50

Now, N = 50 ⇒`N/2 =25`

The cumulative frequency just greater than 25 is 36, and the corresponding class is 60 – 80.

Thus, the median class is 60 – 80.

∴ l = 60, h = 20, N = 50, f = 12 and cf = 24.

Now,

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

=`60+ ((25-24)/12) xx 20`

= 60 + 1.67

= 61.67

Thus, the median is 61.67.

We know that,

The modal class is 60–80 (highest frequency = 12).

Using the formula,

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

where: l = 60, f₁ = 12, f₀ = 10, f₂ = 6, h = 20

= `60 + (12 - 10)/(2(12) - 10 - 6) xx 20`

= `60 + 2/8 xx 20`

= 60 + 5 = 65

Thus, the mode is 65.