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,
Mode = 3(median) – 2(mean)
= 3 × 61.67 – 2 × 62.4
= 185.01 – 124.8
= 60.21
Hence, Mean = 62.4, Median = 61.67 and Mode = 60.21