Question

Determine the value of mode for the following data by using the formula:
Mode = 3 Median − 2 Mean.

Marks	No. of students
Less than 10	5
Less than 20	15
Less than 30	98
Less than 40	242
Less than 50	367
Less than 60	405
Less than 70	425
Less than 80	438
Less than 90	439

Answer 1

Marks	c.f.	f	m	fm
0 – 10	5	5	5	25
10 – 20	15	10	15	150
20 – 30	98	83	25	2075
30 – 40	242	144	35	5040
40 – 50	367	125	45	5625
50 – 60	405	38	55	2090
60 – 70	425	20	65	1300
70 – 80	438	13	75	975
80 – 90	439	1	85	85
		Ʃf = 439		Ʃfm =17365

$\overline{X}=\frac{\mathrm{\Sigma }fm}{\mathrm{\Sigma }f}=\frac{17365}{439}=39.55$
Median class is given by the size of the ${\left(\frac{N}{2}\right)}^{\mathrm{th}}$ item, i.e. the ${\left(\frac{439}{2}\right)}^{\mathrm{th}}$ item, which is the 219.5^th item. This corresponds to the class interval 30−40, so this is the median class.
$$\begin{gathered} \mathrm{Median}=l_1+\frac{{\displaystyle \frac{N}{2}-c.f}}{f}\times i \\ =30+\frac{219.5-98}{144}\times 10 \\ =30+\frac{1215}{144} \\ =30+8.43 \\ =38.43 \end{gathered}$$

Now,
Mode = 3 Median – 2 Mean
= 3 × 38.43 – 2× 39.55
= 115.29 – 79.1
$\therefore$ Mode = 36.19