Chapter 2, Problem 2.1.1RE

In Exercises 1 and 2, use the data set, which represents the overall average class sizes for 20 national universities. (Adapted from Public University Honors)

1. Construct a frequency distribution for the data set using five classes. Include class limits, midpoints, boundaries, frequencies, relative frequencies, and cumulative frequencies.

To determine

To construct: A frequency distribution for data set with class limits, midpoints, boundaries, frequencies, relative frequencies and cumulative frequencies.

Answer to Problem 2.1.1RE

The frequency distribution for the data set is as follows:

Class	Mid point	Class boundaries	Frequency $f$	Relative Frequency	Cumulative Frequency
26-31	28.5	25.5-31.5	5	0.25	5
32-37	34.5	31.5-37.5	4	0.20	9
38-43	40.5	37.5-43.5	6	0.30	15
44-49	46.5	43.5-49.5	3	0.15	18
50-55	52.5	49.5-55.5	2	0.10	20
			${\displaystyle \sum f}=20$	${\displaystyle \sum \frac{f}{n}}=1$

Explanation of Solution

Given info:

The data set represents the overall average class sizes for 20 national universities. Also, the number of classes is 5.

Calculation:

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

• The number of classes is 5.
• Here, the minimum class size is 26 and maximum class size is 55. Thus, the range is $55-26=29$.
• Class width is obtained by dividing the range by the number of classes.

Thus,

$\begin{array}{c}\text{Class}\text{width}=\frac{\text{Range}}{\text{Number}\text{of}\text{calsses}}\\ =\frac{29}{5}\\ =5.8\end{array}$

Hence, the class width is 6.

Lower class limits:

• The class limits are obtained by taking the minimum value as lower limit for the first class. For the second class, the lower limit is obtained by adding the class width to the previous class’s lower limit, that is, $26+6=32$. Similarly, the lower limits for the remaining classes can be obtained.

Upper class limits:

• For the first class, the upper limit is one value less than the lower limit of the second class, that is, $32-1=31$. The other upper limits are obtained by adding the upper limit of the preceding class to the class width.

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

Lower limit	Upper limit
26	$32-1=31$
$26+6=32$	$31+6=37$
$32+6=38$	$37+6=43$
$38+6=44$	$43+6=49$
$44+6=50$	$49+6=55$

• Make a tally mark for each score in the corresponding class and continue for all class sizes in the data.
• The number of tally marks in each class represents the frequency (f )of that class.

Thus, the frequency distribution table for music blog reading times is as follows:

Class	Tally	Frequency $f$
26-31	$\cancel{\left|\right|\left|\right|}$	5
32-37	$\left|\right|\left|\right|$	4
38-43	$\cancel{\left|\right|\left|\right|}|$	6
44-49	$\left|\right||$	3
50-55	$\left|\right|$	2
		${\displaystyle \sum f}=20$

Midpoint:

The midpoint is the average of the lower limit and upper limit of a particular class. It is also called as class mark.

$\text{Mid}\text{point}=\frac{\left(\text{Lower}\text{class limit+Upper}\text{class}\text{limit}\right)}{2}$

Thus, the mid points for each class is tabulated below:

Class	Frequency $f$	Mid point
26-31	5	$\frac{26+31}{2}=28.5$
32-37	4	$\frac{32+37}{2}=34.5$
38-43	6	$\frac{38+43}{2}=40.5$
44-49	3	$\frac{44+49}{2}=46.5$
50-55	2	$\frac{50+55}{2}=52.5$
	${\displaystyle \sum f}=20$

Class boundaries:

The numbers that eliminate the difference between the classes is known as class boundaries.

For integers, subtract 0.5 from the lower limit of each class and add 0.5 to the upper limit of each class.

First class lower boundary is $25.5\left(=26-0.5\right)$.

First class upper boundary is $31.5\left(=31+0.5\right)$.

Similarly, for the class boundaries for the remaining classes are tabulated below:

Class	Frequency $f$	Class boundaries
26-31	5	25.5-31.5
32-37	4	31.5-37.5
38-43	6	37.5-43.5
44-49	3	43.5-49.5
50-55	2	49.5-55.5
	${\displaystyle \sum f}=20$

Relative frequency:

The portion of the data that falls in the corresponding class is the 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
26-31	5	$\frac{5}{20}=0.25$
32-37	4	$\frac{4}{20}=0.2$
38-43	6	$\frac{6}{20}=0.3$
44-49	3	$\frac{3}{20}=0.15$
50-55	2	$\frac{2}{20}=0.1$
	${\displaystyle \sum f}=20$	${\displaystyle \sum \frac{f}{n}}=1$

Cumulative frequency:

Cumulative frequency is the sum of all frequencies of the classes. The last class’s cumulative frequency is equal to the sample size ( $n$).

Thus, the cumulative frequency for each class is tabulated below:

Class	Frequency $f$	Cumulative frequency
26-31	5	5
32-37	4	$5+4=9$
38-43	6	$9+6=15$
44-49	3	$15+3=18$
50-55	2	$18+2=20$
	${\displaystyle \sum f}=20$