# NCERT Class 9 Solutions: Statistics (Chapter 14) Exercise 14.3-Part 4

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Q-9 100 surnames were randomly picked up from a local telephone directory and a frequency distribution of the number of letters of the English alphabet in the surnames was found as follows:

 Number of letters Number of surnames 1-4 6 4-6 30 6-8 44 8-12 16 12-20 4
1. Draw a histogram to depict the given information.

2. Write the class interval in which the maximum number of surname lies.

Solution (I):

The problem with this data is that the class intervals are of varying width. The principle behind histograms is that the area of each bar represents the fraction of a frequency (probability) distribution within each bin (class, interval). Therefore, as the bar in a histogram is spread across bins, it is squished in height (that is, area remains the same). Here we choose the bin size to be 2 letters. So we divide the frequency (number of surnames names) into bins of 2 letters.

For example, number of surnames with 1 – 4 letters is 6 however since 4 – 1 = 3, this frequency is not spread across 2 letters but 3. The height in frequency representation in histogram is the proportion of the number of surnames per 2 letters interval, that is .

We calculate the histogram heights for other intervals as follows:

 Number of letters Frequency (Number of surnames) Width of class Height of rectangle 1-4 6 3 4-6 30 2 6-8 44 2 8-12 16 4 12-20 4 8

Now plot the histogram

On x-axis represent the class intervals of the number of letters and on y-axis the proportion of the number of surnames per 2 letters interval.