Problem 30 - Basic Statistics for Business and Economics

Question

The numbers of shareholders for a selected group of large companies (in thousands) are: $$ \begin{array}{|lclc|} \hline & \begin{array}{c} \text { Number of } \\ \text { Shareholders } \\ \text { (thousands) } \end{array} & \text { Company } & \begin{array}{c} \text { Number of } \\ \text { Shareholders } \\ \text { (thousands) } \end{array} \\ \text { Company } & 144 & \text { Standard Oil (Indiana) } & 173 \\ \text { Southwest Airlines } & 177 & \text { Home Depot } & 195 \\ \text { General Public Utilities } & 266 & \text { Detroit Edison } & 220 \\ \text { Occidental Petroleum } & 133 & \text { Eastman Kodak } & 251 \\ \text { Middle South Utilities } & 209 & \text { Dow Chemical } & 137 \\ \text { DaimlerChrysler } & 264 & \text { Pennsylvania Power } & 150 \\ \text { Standard Oil of California } & 160 & \text { American Electric Power } & 262 \\ \text { Bethlehem Steel } & 143 & \text { Ohio Edison } & 158 \\ \text { Long Island Lighting } & 246 & \text { Transamerica Corporation } & 162 \\ \text { RCA } & 151 & \text { Columbia Gas System } & 165 \\ \text { Greyhound Corporation } & 239 & \text { International Telephone \& } & \\\ \text { Pacific Gas \& Electric } & 204 & \text { Telegraph } & 223 \\ \text { Niagara Mohawk Power } & 204 & \text { Union Electric } & 158 \\ \text { E.l: du Pont de Nemours } & 195 & \text { Virginia Electric and Power } & 162 \\ \text { Morris Knudsen Corporation } & 176 & \text { Public Service Electric \& Gas } & 225 \\ \text { Union Carbide } & 175 & \text { Consumers Power } & 161 \\ \text { BankAmerica } & 200 & & \\ \text { Northeast Utilities } & & & & \\ \hline \end{array} $$ The numbers of shareholders are to be organized into a frequency distribution and several graphs drawn to portray the distribution. a. Using seven classes and a lower limit of \(130,\) construct a frequency distribution. b. Portray the distribution as a frequency polygon. c. Portray the distribution in a cumulative frequency polygon. d. According to the polygon, three out of four \((75\) percent) of the companies have how many shareholders or less? e. Write a brief analysis of the number of shareholders based on the frequency distribution and graphs.

Step-by-Step Solution

Answer

15 companies have 209 shareholders or less.

Step 1: Organize the Data into Intervals

Given the shareholder counts ranging from 133 to 266, we must organize this data into 7 intervals, starting with a lower limit of 130. Given the range and desired class number, choose a class interval width of approximately 20. This gives intervals such as: 130-149, 150-169, 170-189, 190-209, 210-229, 230-249, 250-269.

Step 2: Count the Frequency for Each Interval

Count the number of companies whose shareholder number falls into each interval. For example, count how many companies have shareholders between 130-149, and so forth for each interval.

Step 3: Construct the Frequency Distribution Table

Create a table listing each interval and the corresponding frequency of companies within those intervals. For instance: 130-149 contains 3 companies, 150-169 contains 5 companies, and continue similarly for each class.

Step 4: Draw the Frequency Polygon

Plot the midpoints of each class interval against their frequencies. Join the points with straight lines to form the frequency polygon. For the midpoint of an interval e.g., 130-149, it would be (130+149)/2 = 139.5.

Step 5: Draw the Cumulative Frequency Polygon

Calculate cumulative frequencies and plot them against the upper limits of each interval. The cumulative frequency for each interval is the sum of its frequency and those of all previous intervals. Plot these points and connect them with a smooth line.

Step 6: Determine 75th Percentile from Polygon

Identify the point on the cumulative frequency polygon that represents 75% of the companies. Look for the shareholder number at which the cumulative frequency first reaches or exceeds 75% of the total companies (20), which would be 0.75 * 20 = 15 companies.

Step 7: Analyze Distribution and Graphs

Assess the frequency distribution and graphs to understand the distribution of shareholder numbers. Note any patterns such as central tendencies and spread, highlighting which classes are most and least frequent and their implications.

Key Concepts

Cumulative Frequency

Cumulative frequency is a cumulative count of frequencies within a data set as you move through the class intervals. Each cumulative frequency value shows how many data points lie below or within a specific range. This is calculated by adding each frequency from a frequency distribution table to the sum of the previous frequencies, creating a running total.

For example, if the frequencies for your class intervals are as follows:

• 130-149 has a frequency of 3
• 150-169 has a frequency of 5
• 170-189 has a frequency of 2

The cumulative frequencies will be:

• 130-149: 3
• 150-169: 3+5=8
• 170-189: 8+2=10

The cumulative frequency is used primarily when you need to determine how many data points fall below a certain value or when creating a cumulative frequency polygon.

Understanding cumulative frequency is vital for interpreting the spread and concentration of data within different intervals, aiding in more comprehensive data analysis.

Frequency Polygon

A frequency polygon is a graphical representation of the distribution of a dataset. It is similar to a histogram but uses points connected by straight lines instead of bars. This graph represents frequencies in a more visually appealing and insightful way by displaying them as a continuous line. To construct a frequency polygon, you plot points at the midpoint of each class interval, where the frequency of that interval is shown.

Let's break down the process:

• Determine the midpoints of the class intervals. For example, the midpoint of 130-149 is \( \frac{130 + 149}{2} = 139.5 \) .
• Plot a point for each class where the x-coordinate is the class midpoint and the y-coordinate is the frequency.
• Connect all the points with straight lines.

Frequency polygons are helpful because they allow you to compare different distributions easily. Because the lines provide a clearer perspective on trends over different intervals, they are excellent for analyzing fluctuations and patterns across the dataset.

Additionally, they can be used to identify peaks (modes) and understand the data's symmetry or skewness.

Class Interval

Class intervals divide a range of data into different segments, each defined by specific lower and upper limits. They are crucial for organizing large sets of data into a structured format that simplifies data interpretation, such as in histograms or frequency distributions.

To determine class intervals, follow these steps:

• Find the range of the data by subtracting the smallest value from the largest value.
• Decide on the number of intervals you want to create.
• Calculate the interval width by dividing the range by the number of intervals.
• Assign start and end points to each class interval based on your calculations.

For instance, suppose the shareholder data ranges from a minimum of 133 to a maximum of 266, and you choose 7 intervals, you might use 130-149 through 250-269 based on an interval width of around 20.

Utilizing class intervals not only aids in creating frequency tables and graphs like frequency polygons but also makes complex datasets easier to understand at a glance.

Data Analysis

Data analysis involves interpreting collected data to uncover useful insights and trends. For a dataset of shareholder numbers, organizing this data effectively enables us to make informed conclusions.

In our example, examining the frequency distribution and polygons can illuminate the following aspects:

• Central Tendency: Where most of the data points cluster, such as the middle intervals having higher frequencies.
• Variability: How spread out the data points are, or the distribution's range.
• Percentiles: Identifying positions within the dataset, such as determining how many companies have fewer than a specific number of shareholders.

Using methods such as frequency distribution tables, frequency polygons, and cumulative frequency polygons, you can visually and numerically analyze data more deeply, allowing for better identification of patterns and making predictions or inferences.

A comprehensive understanding of data analysis techniques will improve your ability to make strategic decisions based on the data your analysis provides.