Question

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

Class	0-10	10-20	20-30	30-40	40-50	50-60	60-70
Frequency	5	10	18	30	20	12	5

Answer 1

We have,

(i) Mean:
Let the assumed mean, A, be 35; h = 10
$\mathrm{Mean},\overline{\mathrm{x}}=\mathrm{A}+\left\{\mathrm{h}\times \frac{\sum \left({\mathrm{f}}_{\mathrm{i}}\times {\mathrm{u}}_{\mathrm{i}}\right)}{\sum {\mathrm{f}}_{\mathrm{i}}}\right\}$
$=35+\left\{10\times \frac{6}{100}\right\}=35+0.6=35.6$
(ii) Median:

$\frac{N}{2}=\frac{100}{2}=50$
Cumulative frequency just greater than 50 is 63.
Corresponding class interval is 30 - 40.
Thus, median class is 30 - 40.
Cumulative frequency just before this class = 33.
So, l = 30, f = 30 , $\frac{N}{2}=50$, h = 10 and cf = 33
$\mathrm{Median},M_e=l+\left\{h\times \frac{\left({\displaystyle \frac{N}{2}}-cf\right)}{f}\right\}$
$=30+\left\{10\times \left(\frac{50-33}{30}\right)\right\}=30+\left(10\times \frac{17}{30}\right)=30+\frac{17}{3}=30+5.67=35.67$
(iii) Mode:
Mode = (3 × median) - (2 × mean)
= (3 × 35.67) - (2 × 35.6)
= (107.01 - 71.2) = 35.81

Hence, Mean = 35.6, Median = 35.67 and Mode = 35.81