Chapter 2, Problem 2.2.8RE

Find the relative frequency for the frequency distribution for the data in Exercise 6.

Use 6 classes. The data for this exercise will be used for Exercises 8, 10, and 12.

Source: World Almanac and Book of Facts.

To determine

The relative frequency of the data given.

Answer to Problem 2.2.8RE

A relative frequency table shown below,

Class	Frequency $f$	Relative frequency
50.5-59.5	5	$\frac{5}{41}=0.122$
59.5-68.5	12	$\frac{12}{41}=0.293$
68.5-77.5	11	$\frac{11}{41}=0.268$
77.5-86.5	8	$\frac{8}{41}=0.195$
86.5-95.5	3	$\frac{3}{41}=0.073$
95.5-104.5	2	$\frac{2}{41}=0.049$
Total	41

Explanation of Solution

Given info:

Refer to Exercise $6$.

The data represent the ages of the Vice Presidents of the United States at the time of their death. The number of classes is $6$.

Calculation:

Step-by-step procedure to construct frequency distribution is as follows:

Here, the number of classes is 6.

Here, the minimum number is $51$ and maximum number is $98$. Thus, the range is

$98-51=47$.

Class width is obtained by dividing the range by the number of classes.

Thus,

$\begin{array}{c}\text{Class}\text{width}=\frac{\text{Range}}{\text{Number}\text{of}\text{calsses}}\\ =\frac{47}{6}\\ =7.8\\ \approx 8\left(\text{rounded}\text{up}\text{for}\text{convenient}\right)\end{array}$

For making $6$ classes for given data take class width 9 for convenient.

Lower class limits:

• The class limits are obtained by taking the minimum value or less than minimum value as lower limit for the first class. For second class, the lower limit is obtained by adding the class width to the previous class lower limit. That is, $51+9=60$. Similarly, the lower limits for the remaining classes can be obtained.

Upper class limits:

• For the first class, the upper limit is one less than the lower limit of the second class that is, $60-0.5=59.5$. The other upper limits are obtained by adding the upper limit of the preceding class and the class width.

Thus, the lower class and upper class limits are as follows:

Lower limit	Upper limit
50.5	$60-0.5=59.5$
$50.5+9=59.5$	$59.5+9=68.5$
$59.5+9=68.5$	$68.5+9=77.5$
$68.5+9=77.5$	$77.5+9=86.5$
$77.5+9=86.5$	$86.5+9=95.5$
$86.5+9=95.5$	$95.5+9=104.5$

• Make a tally mark for each entry in the corresponding class and continue for all number of children in the data.
• The number of tally marks in each class represents the frequency, f of that class.

Thus, the frequency distribution table the distribution of declaration of independence is as follows,

Class	Tally	Frequency $f$
50.5-59.5	$\cancel{\left|\right|\left|\right|}$	5
59.5-68.5	$\cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}\left|\right|$	12
68.5-77.5	$\cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}|$	11
77.5-86.5	$\cancel{\left|\right|\left|\right|}\left|\right||$	8
86.5-95.5	$\left|\right||$	3
95.5-104.5	$\left|\right|$	2

Relative frequency:

The portion of the data that falls in the corresponding class is relative frequency of that class.

$\begin{array}{c}\text{Relative}\text{frequency}=\frac{\text{Class}\text{frequency}}{\text{Sample}\text{size}}\\ =\frac{f}{n}\end{array}$

Thus, the relative frequency for each class is tabulated below,

Class	Frequency $f$	Relative frequency
50.5-59.5	5	$\frac{5}{41}=0.122$
59.5-68.5	12	$\frac{12}{41}=0.293$
68.5-77.5	11	$\frac{11}{41}=0.268$
77.5-86.5	8	$\frac{8}{41}=0.195$
86.5-95.5	3	$\frac{3}{41}=0.073$
95.5-104.5	2	$\frac{2}{41}=0.049$
Total	41

From the frequency table it is observed that maximum relative frequency is for class $59.5-68.5$ and also relative frequency is maximum for that class.