Question

Find the mean, median and mode of the following frequency distribution:

Answer 1

Given

Class $0-10$ $10-20$ $20-30$ $30-40$ $40-50$

Frequency $5$ $27$ $58$ $20$ $10$

(i) Mean

Class Mid point($x_i$) Frequency($f_i$) $f_i{x}_i$

$0-10$ $5$ $5$ $25$

$10-20$ $15$ $27$ $405$

$20-30$ $25$ $58$ $1450$

$30-40$ $35$ $20$ $700$

$40-50$ $45$ $10$ $450$

Here

$\sum f_i=120$

$\sum f_i{x}_i=3030$

Mean $\left(\overline{x}\right)=\frac{\sum f_i{x}_i}{\sum f_i}=\frac{3030}{120}$

$25.25$

(ii) Median

Class Frequency Cumulative frequency

$0-10$ $5$ $5$

$10-20$ $21$ $32\left(cf\right)$

$20-30$(l) $58$(f) $90$

$30-40$ $20$ $110$

$40-50$ $10$ $120$

(iii) Mode

Class Frequency

$0-10$ $5$

$10-20$ $27$ fo

$20-30$(l) $58$f1

$30-40$ $20f2$

$40-50$ $10$

$N=120\Rightarrow \frac{N}{2}=60$

so median class is $20-30$

$l=20$, class width$=h=10$

Median $=l+\left(\frac{\frac{N}{2}-cf}{f}\right)\times h$

$=20+\frac{60-32}{58}\times 10$

$=20+\frac{28}{58}\times 10=20+\frac{280}{58}$

$=20+4.83$

$=24.83$

Mean$=25.25$

So model class is $20-30$

$\therefore$ Mode$=l+\left(\frac{f_1-f_o}{2f_1-f_o-f_2}\right)\times h$

$=20+\frac{58-27}{2\times 58-27-20}\times 10$

$=20+\frac{310}{69}$

$=20+4.493$

$=24.493$