Question

(b) Calculate the missing frequency from the data if the median is 50.
CI 10-20 20-30 30-40 40-50 50-60 60-70
f 2 8 6 ? 15 10

Answer 1

$$\begin{array}{c:c:c} \begin{matrix} Class \\ Interval\\ CI \end{matrix}& \begin{matrix} Frequency \\ f \end{matrix} &\begin{matrix} Cumulative \\ Frequency \end{matrix} \\ \hline 10-20 & 2 & 2 \\ \hdashline 20-30 & 8 & 10 \\ \hdashline 30-40 & 6 & 16\\ \hdashline 40-50 & f_4 & 16+f_4\\ \hdashline 50-60 & 15 & 31+f_4\\ \hdashline 60-70 & 10 & 41+f_4 \end{array}$$

$$Estimated\ Median=L+{{n \over 2}-B \over G}\times w$$

where:

$L$  is the lower class boundary of the group containing the median

$n$  is the total number of values

$B$  is the cumulative frequency of the groups before the median group

$G$  is the frequency of the median group

$w$ is the group width

We have that $n=2+8+6+f_4+15+10=41+f_4$

$$Median=50$$

The median lies in class interval $CI=50-60$

$L=50, n=41+f_4, B=16+f_4,G=15,w=10$

$$50=50+{{41+f_4 \over 2}-(16+f_4) \over 15}\times 10$$ $${41+f_4 \over 2}-(16+f_4) =0$$

$$41+f_4=32+2f_4$$

$$f_4=9$$

The missing frequency is $9.$