Chapter 2.2, Problem 4E

Number of College Faculty The number of faculty listed for a sample of private colleges that offer only bachelor’s degrees is listed below. Use these data to construct a frequency distribution with 7 classes, a histogram, a frequency polygon, and an ogive. Discuss the shape of this distribution. What proportion of schools have 180 or more faculty?

To determine

A grouped frequency distribution with 7 classes and draw a histogram, frequency polygon, and ogive; the explanation of these shapes and calculate the proportions of schools that have 180 or more faculty.

Answer to Problem 4E

Output using EXCEL software is given below:

Histogram:

Frequency polygon:

Ogive:

The proportion of schools that has 180 or more faculty is 0.448.

Explanation of Solution

Given info:

The data shows the number of college faculty listed in the table is given below:

165	221	218	206	138	135	224	204
70	210	207	154	155	82	120	116
176	162	225	214	93	389	77	135
221	161	128	310

Calculation:

The class boundaries for any class are given by:

$\begin{array}{l}\text{Upper}\text{class}\text{boundary}=\text{Upper}\text{class}\text{limit+}\frac{\text{d}}{2}\\ \text{Lower}\text{class}\text{boundary=Lower}\text{class}\text{limit-}\frac{\text{d}}{2}\end{array}$

Where, $\text{d}$ is the gap between the upper limit of any class and the lower limit of the succeeding class.

The grouped frequency distribution is as follows:

Class limit	Class boundaries	Tally	Frequency
70-115	69.5-115.5	$\left|\right|\left|\right|$	4
116-161	115.5-161.5	$\cancel{\left|\right|\left|\right|}\left|\right|\left|\right|$	9
162-207	161.5-207.5	$\left|\right||$	3
208-253	207.5-253.5	$\cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}$	10
254-299	253.5-299.5		0
300-345	299.5-345.5	|	1
346-391	345.5-391.5	|	1
Total			28

The midpoint of class boundaries is obtained by adding lower and upper limit and dividing by 2.

The relative frequency is the ration of a class frequency to the total frequency. Cumulative relative frequency can also defined as the sum of all previous frequencies up to the current point.

Class boundaries	Mid point	Frequency	Cumulative frequency
69.5-115.5	92.5	4	4
115.5-161.5	138.5	9	13
161.5-207.5	184.5	3	16
207.5-253.5	230.5	10	26
253.5-299.5	276.5	0	26
299.5-345.5	322.5	1	27
345.5-391.5	368.5	1	28

The histogram is a graph that displays the data by using contiguous vertical bars of various heights to represent the frequencies of the classes.

Upper Limit	Frequency
69.5	0
115.5	4
161.5	9
207.5	3
253.5	10
299.5	0
345.5	1
391.5	1

Histogram:

Software procedure:

Step-by-step procedure to construct the histogram using EXCEL software is given below:

• Press [Ctrl]-N for a new workbook.
• Enter the data in column A, one number per cell.
• Enter the upper boundaries into column B.
• From the toolbar, select the Data tab, then select Data Analysis.
• In Data Analysis, select Histogram and click [OK].
• In the Histogram dialog box, select relative column in the Input Range box and select upper limit column in the Bin Range box.
• Select New Worksheet Ply and Chart Output. Click [OK].

The distribution is slightly symmetric as the histogram shows that majority of the data value fall around the mean of the data. And the frequency is maximum at 253.5 and minimum at 391.5.

Frequency polygon:

Software procedure:

Frequency polygons are a graphical device for understanding the shapes of distributions. They serve the same purpose as histograms, but are especially helpful for comparing sets of data.

Step-by-step procedure to construct the frequency polygon using EXCEL software is given below:

• Press [CTRL]-N for a new notebook.
• Enter the midpoints of the data into column A and the frequencies into column B including labels.
• Press and hold the left mouse button, and drag over the Frequencies (including the label) from column B.
• Select the Insert tab from the toolbar and the Line Chart option.
• Select the 2-D line chart type.

Ogive:

Software procedure:

Step-by-step procedure to construct the ogive using EXCEL software is given below:

• Create an ogive, use the upper class boundaries (horizontal axis) and cumulative frequencies (vertical axis) from the frequency distribution.
• Type the upper class boundaries (including a class with frequency 0 before the lowest class to anchor the graph to the horizontal axis) and
• Corresponding cumulative frequencies into adjacent columns of an Excel worksheet.
• Press and hold the left mouse button, and drag over the Cumulative Frequencies from column B.
• Select Line Chart, then the 2-D Line option.

The points plotted are the upper class limit and the corresponding cumulative frequency.

The observations are based on number of faculty is greater or equal to 180 given as:

224

204

210

207

225

214

389

221

310

The sum of the above observations is 2204.

The total observations for number of college faculty are given as:

165	221	218	206	138	135	224	204
70	210	207	154	155	82	120	116
176	162	225	214	93	389	77	135
221	161	128	310

The sum of all above observations is 4916.

The proportion is calculated by dividing 2204 with 4916, that is,

$\begin{array}{c}\text{Proportion}=\frac{2204}{4916}\\ =0.448\end{array}$

The value 0.448 is the proportion of schools that has 180 or more faculty