Question

Find the mean, median and mode of the following data: [CBSE 2013]

Class	0−50	50−100	100−150	150−200	200−250	250−300	300−350
Frequency	2	3	5	6	5	3	1

Answer 1

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

Class	Frequency (fi)	Class mark (xi)	fixi
0−50	2	25	50
50−100	3	75	225
100−150	5	125	625
150−200	6	175	1050
200−250	5	225	1125
250−300	3	275	825
300−350	1	325	325
Total	∑fi = 25		∑fixi = 4225

$$\begin{gathered} \mathrm{Mean}=\frac{{\displaystyle \sum _i}{f}_i{x}_i}{{\displaystyle \sum _i}{f}_i} \\ =\frac{4225}{25} \\ =169 \end{gathered}$$

Thus, mean of the given data is 169.

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

Class	Frequency (fi)	Cumulative frequency (cf)
0−50	2	2
50−100	3	5
100−150	5	10
150−200	6	16
200−250	5	21
250−300	3	24
300−350	1	25
Total	N = ∑fi = 25

Now, N = 25 $\Rightarrow \frac{N}{2}=12.5$.

The cumulative frequency just greater than 12.5 is 16, and the corresponding class is 150−200.

Thus, the median class is 150−200.

∴ l = 150, h = 50, N = 25, f = 6 and cf = 10.

Now,

$$\begin{gathered} \mathrm{Median}=l+\left(\frac{{\displaystyle \frac{N}{2}}-cf}{f}\right)\times h \\ =150+\left(\frac{12.5-10}{6}\right)\times 50 \\ =150+20.83 \\ =170.83 \end{gathered}$$

Thus, the median is 170.83.

We know that,
Mode = 3(median) − 2(mean)
= 3 × 170.83 − 2 × 169
= 512.49 − 338
= 174.49

Hence, Mean = 169, Median = 170.83 and Mode = 174.49