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:

Frequency distribution of number of letters in the surnames

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:

Height of rectangle for frequency distribution of letters of names

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.

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Created with Highcharts 4.2.5length(in mm)Number of leavesLength (in mm) Vs Number of leaves24681012142468101214Length (in mm) Vs Number of leaves117.5-126.5126.5-135.5135.5-144.5144.5-153.5153.5-162.5162.5-171.5171.5-180.5051015153.5-162.5Length (in mm) Vs Number of leaves: 10

Solution (II):

The maximum number of surnames lies in interval of 6-8 and it has 44 surnames.