Question

100 surnames were randomly picked up from a local telephone directory and a frequency distribution of the number of letters in the English alphabet in the surnames was found as follows:(i) Draw a histogram to depict the given information. (ii) Write the class interval in which the maximum number of surnames lie.

\begin{tabular}{|c|c|} \hline Number of letters & Number of surnames \\ \hline \( 1-4 \) & 6 \\ \( 4-6 \) & 30 \\ \( 6-8 \) & 44 \\ \( 8-12 \) & 16 \\ \( 12-20 \) & 4 \\ \hline \end{tabular}

Answer 1

Step 1: Plot a histogram to depict the given information

Since the class sizes vary to make the histogram we have to calculate the weighted frequency for each rectangle as per its width

Weighted frequency=$\frac{\mathrm{minimum}\mathrm{class}\mathrm{size}}{\mathrm{class}\mathrm{size}\mathrm{of}\mathrm{the}\mathrm{interval}}\times \mathrm{Frequency}$

Length of the rectangle=$\frac{\mathrm{minimum}\mathrm{width}}{\mathrm{Width}\mathrm{of}\mathrm{the}\mathrm{rectangle}}\times \mathrm{Frequency}$

Minimum class size $=6-4=2$

The modified table showing the weighted frequency as per the size of the class intervals is as follows.

The width of the class intervals in the given data is varying in nature.

Also, The area of a rectangle is proportional to the frequencies in the histogram.

Now, the proportion of children per year can be calculated as shown in the table below.

Number of letters	Number of surnames	Width of class	Length of rectangle
$1-4$	$6$	$3$	$\frac{6}{3}\times 2=4$
$4-6$	$30$	$2$	$\frac{30}{2}\times 2=30$
$6-8$	$44$	$2$	$\frac{44}{2}\times 2=44$
$8-12$	$16$	$4$	$\frac{16}{4}\times 2=8$
$12-20$	$4$	$8$	$\frac{4}{8}\times 2=1$

By taking the number of letters on $\mathbf{x}$-axis and the proportion of the number of surnames per 2 letters interval on $\mathbf{y}$-axis and choosing an appropriate scale ( 1 unit$\mathbf{=}\mathbf{4}$ students for $\mathbf{y}$axis), the histogram can be constructed as follows.

Step2 : Write the class interval in which the maximum number of surnames lie.

6-8 is the class interval in which the maximum number of surnames lie.