Question

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

$\begin{array}{cc}\text{Monthly profit}& \text{Frequency}\\ \text{Less than 10}& 4\\ \text{Less than 20}& 20\\ \text{Less than 30}& 35\\ \text{Less than 40}& 55\\ \text{Less than 50}& 62\\ \text{Less than 60}& 67\end{array}$

Answer 1

$\begin{array}{cccccc}\text{Class Interval}& \text{Cumulative Frequency}(c.f)& \text{Frequency}\left(f_i\right)& \text{Mid values}\left(x_i\right)& f_i{x}_i& \\ 0-10& 4& 4& 5& 20& \\ 10-20& 25\left(cf\right)& 16& 15& 240& \\ 20-30& 35& 15\left(f_0\right)& 25& 375& \\ 30-40& 55& 20\left(f_1\right)& 35& 700& \text{Modal class}\\ 40-50& 62& 7\left(f_2\right)& 45& 315& \\ 50-60& 67& 5& 55& 275& \\ & & N=\sum f=67& 5& \sum f_i{x}_i=1925& \end{array}$

From the table,

Mean $\left(\overline{x}\right)=\frac{\sum f_i{x}_i}{\sum f_i}=\frac{1925}{67}=28.73$

Mean $\left(\overline{x}\right)=28.73$---(1)
Here, $N=67$ (odd)

Median class $=(\frac{N+1}{2}{)}^{\text{th}}$ class $=\frac{67+1}{2}={34}^{\text{th}}$ class

Hence, Median class $=20-30$

Lower boundary $\left(L\right)=20$

Corresponding frequency $\left(F\right)=15$

Corresponding preceding cumulative frequency $(c.f)=20$

Size of the class interval $\left(h\right)=10$

Median $\left(M\right)=L+\frac{(\frac{N}{2}-c.f)}{F}\times h$

$=20+\frac{(\frac{67}{2}-20)}{15}\times 10$

$=20+\frac{33.5-20}{15}\times 10$

$=20+\frac{13.5}{15}\times 10$

$=20+\frac{135}{15}$

$=20+9$

$=29$

Median $\left(M\right)=29$---(2)

In the table highest frequency is $20$.

So, modal class is $=30–40$.

$l_1=30$

$f_0=15,f_1=20,f_2=7$

Mode $\left(Z\right)=l_1+\left(\frac{f_1-f_0}{2f_1-\left(f_0+f_2\right)}\right)\times h$

$=30+\left(\frac{20-15}{2\left(20\right)-\left(15+7\right)}\right)\times 10$

$=30+\frac{5}{18}\times 10$

$=30+\frac{50}{18}$

$=30+2.77$ (Approximately)

$=32.77$ (Approximately)

Mode $\left(Z\right)=32.77$ (Approximately)---(3)