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Chapter 2: Problem 3

Gulf Coastlines Construct a Pareto chart for the sizes of Gulf coastlines in statute miles for each state. $$ \begin{array}{lc}{\text { State }} & {\text { Coastline }} \\ \hline \text { Alabama } & {53} \\ {\text { Florida }} & {770} \\ {\text { Louisiana }} & {397} \\ {\text { Mississippi }} & {44} \\ {\text { Texas }} & {367}\end{array} $$

Short Answer

Expert verified
Organize, calculate cumulative frequencies, determine percentages, plot in order, then evaluate the chart for key contributors.

Step by step solution

01

Data Organization

Start by organizing the given data in descending order, as this is required for a Pareto chart. The state coastlines in statute miles are: Florida (770), Louisiana (397), Texas (367), Alabama (53), and Mississippi (44).
02

Calculate Cumulative Frequencies

Next, calculate the cumulative frequency for each coastline. Start with Florida at 770 miles, then add Louisiana's 397 miles (770 + 397 = 1167), followed by Texas's 367 miles (1167 + 367 = 1534), Alabama's 53 miles (1534 + 53 = 1587), and finally Mississippi's 44 miles (1587 + 44 = 1631).
03

Determine Percentages

Convert the cumulative frequencies into percentages of the total coastline of 1631 miles. Florida is 47.2% (770/1631*100), Louisiana is 71.5% (1167/1631*100), Texas is 94.0% (1534/1631*100), Alabama is 97.2% (1587/1631*100), and Mississippi is 100% (1631/1631*100).
04

Plotting the Pareto Chart

On the horizontal axis, list the states in the order of largest to smallest coastline: Florida, Louisiana, Texas, Alabama, Mississippi. On the vertical axis, provide two scales: one for literal coastline values and one for the cumulative percentage. Use bars to represent the coastline size for each state, and a line to indicate the cumulative percentage. This visualizes both the actual size and the relative importance of each part.
05

Evaluate the Visual

Assess the Pareto chart to identify the "vital few" which often implies focusing on the biggest contributors (Florida and Louisiana), thus simplifying priorities when necessary.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cumulative Frequency
Cumulative frequency is a crucial concept when organizing data for a Pareto chart. It involves sequentially adding each state's coastline size to the total of all preceding states. This step helps us understand how the total value accumulates as we move through the data points.

In our exercise, we start with the largest coastline, which is Florida's at 770 miles. From there, we add Louisiana's 397 miles, resulting in a cumulative total of 1167 miles. Add Texas's 367 miles next, leading to a cumulative total of 1534 miles. Continue by adding Alabama's 53 miles for a total of 1587 miles, and finally, include Mississippi's 44 miles to reach the total cumulative coastline of 1631 miles.

Cumulative frequency is essential for creating a line graph on the Pareto chart, providing a visual representation of how much each state's coastline contributes to the overall total. This allows viewers to quickly identify which states' coastlines have the most significant impact on the total.
Percentage Calculation
Percentage calculation is the process of converting raw data or cumulative frequencies into percentages, making it easier to visualize and compare parts of a whole.

To calculate the percentage of each state's coastline relative to the total, we use the formula: \[\text{Percentage} = \left(\frac{\text{Part}}{\text{Whole}}\right) \times 100\]For instance, Florida's coastline percentage is calculated as:\[\left(\frac{770}{1631}\right) \times 100 \approx 47.2\%\]Similarly, compute for other states: Louisiana ends up at 71.5%, Texas at 94.0%, Alabama at 97.2%, and Mississippi at 100%.

These calculations help represent each state's contribution to the total coastline as a portion of the whole, making comparisons more intuitive and clear to the observer.
Data Organization
Data organization is the initial and vital step in constructing a Pareto chart. It involves arranging the data in a meaningful order, typically from largest to smallest value. This format helps in correctly plotting the chart and aids in visual analysis.

In this exercise, the state coastlines are sorted from largest to smallest: Florida, Louisiana, Texas, Alabama, and Mississippi. This descending order helps clarify which states have the most significant and least significant coastlines quickly.

Proper organization of data is not just helpful for visualization but also essential for accurate subsequent calculations like cumulative frequency and percentage conversion. Sorting data in this manner ensures a smooth and logical flow into the next steps of Pareto chart construction, enabling an efficient analysis of the dataset.
Data Visualization
Data visualization, particularly through a Pareto chart, transforms numerical data into a graphical representation. This process provides a clear and concise visual summary of the data, highlighting the most impactful elements.

Creating a Pareto chart involves plotting the states' coastlines in order from the largest contributor to the smallest on the horizontal axis. The vertical axis must display two scales: one for actual coastline sizes and another for cumulative percentages.

Using bars, the coastline size of each state is represented, showcasing the literal data value. Concurrently, a line graph runs across these bars reflecting the cumulative percentage. This dual representation allows the viewer to directly see both quantity and cumulative impact.

Ultimately, data visualization through Pareto charts simplifies priorities, making it easier to focus on the states with the most significant coastlines, aiding in decision making and strategic planning.

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Most popular questions from this chapter

Trust in Internet Information A survey was taken on how much trust people place in the information they read on the Internet. Construct a categorical frequency distribution for the data. A = trust in all that they read, \(\mathrm{M}=\) trust in most of what they read, \(\mathrm{H}=\) trust in about one-half of what they read, \(S=\) trust in a small portion of what they read. (Based on information from the UCLA Internet Report.) $$ \begin{array}{llllllllll}{M} & {M} & {M} & {A} & {H} & {M} & {S} & {M} & {H} & {M} \\ {S} & {M} & {M} & {M} & {M} & {A} & {M} & {M} & {A} & {M} \\ {M} & {M} & {H} & {M} & {M} & {M} & {H} & {M} & {H} & {M} \\ {A} & {M} & {M} & {M} & {H} & {M} & {M} & {M} & {M} & {M}\end{array} $$

Years of Experience The data show the number of years of experience the players on the Pittsburgh Steelers football team have at the beginning of the season. Draw and analyze a dot plot for the data. $$ \begin{array}{ccccccccc}{4} & {4} & {2} & {9} & {7} & {3} & {7} & {12} & {6} \\\ {5} & {1} & {4} & {5} & {2} & {7} & {6} & {12} & {3} \\ {12} & {4} & {0} & {4} & {0} & {0} & {0} & {2} & {9} \\ {2} & {6} & {7} & {13} & {4} & {2} & {6} & {9} & {4} \\ {4} & {0} & {3} & {5} & {4} & {2} & {6} & {9} & {4} \\\ {4} & {0} & {3} & {5} & {3} & {11} & {1} & {4} & {2} \\ {3} & {15} & {1} & {6} & {0} & {11} & {3} & {10} & {3}\end{array} $$

Charity Donations A random sample of 30 large companies in the United States shows the amount, in millions of dollars, that each company donated to charity for a specific year. Construct a frequency distribution for the data, using 9 classes. $$ \begin{array}{lllll}{26} & {25} & {19} & {31} & {14} \\ {48} & {35} & {43} & {25} & {46} \\ {17} & {21} & {57} & {58} & {34} \\ {41} & {12} & {27} & {15} & {53} \\ {16} & {63} & {82} & {23} & {52} \\ {56} & {75} & {19} & {26} & {88}\end{array} $$

Average Wind Speeds A sample of 40 large cities was selected, and the average of the wind speeds was computed for each city over one year. Construct a frequency distribution, using 7 classes. $$ \begin{array}{cccc}{12.2} & {9.1} & {11.2} & {9.0} \\ {10.5} & {8.2} & {8.9} & {12.2} \\ {9.5} & {10.2} & {7.1} & {11.0} \\ {6.2} & {7.9} & {8.7} & {8.4} \\\ {8.9} & {8.8} & {7.1} & {10.7} \\ {8.7} & {10.5} & {10.2} & {10.7} \\\ {70.4} & {7.7} & {12.3} & {10.7} \\ {7.7} & {7.8} & {11.8} & {10.5} \\ {9.6} & {9.6} & {8.6} & {10.3}\end{array} $$

Pupils Per Teacher The average number of pupils per teacher in each state is shown. Construct a grouped frequency distribution with 6 classes. Draw a histogram, frequency polygon, and ogive. Analyze the distribution. $$ \begin{array}{ccccc}{16} & {16} & {15} & {12} & {14} \\ {13} & {16} & {14} & {15} & {14} \\ {18} & {18} & {18} & {12} & {15} \\ {15} & {16} & {16} & {15} & {15} \\ {25} & {19} & {15} & {12} & {22} \\ {18} & {14} & {13} & {17} & {9} \\ {13} & {14} & {13} & {16} & {12} \\ {14} & {16} & {10} & {22} & {20} \\\ {12} & {14} & {18} & {15} & {14} \\ {16} & {12} & {12} & {13} & {15}\end{array} $$
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