91Ó°ÊÓ

Commercial dredging operations in ancient rivers occasionally uncover archaeological artifacts of great importance. One such artifact is Bronze Age spearheads recovered from ancient rivers in Ireland. A recent study gave the following information regarding discoveries of ancient bronze spearheads in Irish rivers. $$ \begin{array}{l|ccccc} \text { River } & \text { Bann } & \text { Blackwater } & \text { Erne } & \text { Shannon } & \text { Barrow } \\ \hline \text { No. of spearheads } & 19 & 8 & 15 & 33 & 14 \end{array} $$ (Based on information from Crossing the Rubicon, Bronze Age Studies 5 , Lorraine Bourke, Department of Archaeology, National University of Ireland, Galway.) (a) Make a Pareto chart for these data. (b) Make a circle graph for these data.

Step by step solution

01

Calculate Total Number of Spearheads

First, sum up the total number of spearheads across all the rivers. We have the following spearhead counts for rivers: Bann = 19, Blackwater = 8, Erne = 15, Shannon = 33, Barrow = 14. Add them up:\[ 19 + 8 + 15 + 33 + 14 = 89 \]Thus, the total number of spearheads is 89.

02

Calculate Percentage for Each River

To create a Pareto chart and a circle graph, calculate the percentage of total spearheads each river contributes.- Bann: \( \frac{19}{89} \times 100 \approx 21.35\% \)- Blackwater: \( \frac{8}{89} \times 100 \approx 8.99\% \)- Erne: \( \frac{15}{89} \times 100 \approx 16.85\% \)- Shannon: \( \frac{33}{89} \times 100 \approx 37.08\% \)- Barrow: \( \frac{14}{89} \times 100 \approx 15.73\% \)

03

Order Data for Pareto Chart

For the Pareto chart, reorder the rivers by the number of spearheads, from highest to lowest: 1. Shannon: 33 spearheads (37.08%) 2. Bann: 19 spearheads (21.35%) 3. Erne: 15 spearheads (16.85%) 4. Barrow: 14 spearheads (15.73%) 5. Blackwater: 8 spearheads (8.99%). These will be displayed in descending order on the Pareto chart.

04

Create the Pareto Chart

A Pareto chart is a bar chart with data ordered by frequency of occurrence. The x-axis lists the rivers: Shannon, Bann, Erne, Barrow, Blackwater. The y-axis shows the number of spearheads. Bars should be drawn in descending order of number of spearheads. Additionally, a line graph representing the cumulative frequency percentage can be added to the chart.

05

Create the Circle Graph (Pie Chart)

For the circle graph, convert each river's percentage into a proportion of the circle's 360 degrees.- Bann: \((21.35/100) \times 360 \approx 76.86\) degrees- Blackwater: \((8.99/100) \times 360 \approx 32.36\) degrees- Erne: \((16.85/100) \times 360 \approx 60.66\) degrees- Shannon: \((37.08/100) \times 360 \approx 133.49\) degrees- Barrow: \((15.73/100) \times 360 \approx 56.63\) degreesCreate the circle graph by drawing a circle, then dividing it into the angles calculated above, labeling each section with the corresponding river.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Pareto Chart

A Pareto Chart is a special type of bar chart invented by Vilfredo Pareto and is used often in the field of statistics to display data in order of significance. The primary purpose is to highlight the most crucial factors in a data set. When creating a Pareto Chart:

• Arrange bars in descending order based on frequency, which makes it visually easier to identify which categories are the most significant.
• The x-axis labels the categories or groups, in this case, the rivers where spearheads were found: Shannon, Bann, Erne, Barrow, Blackwater.
• The y-axis will display the number of spearheads discovered.

A cumulative line graph can be added on top of the bars to represent the cumulative frequency, showing the total percentage that each category and the ones before it contribute to the whole. This cumulative line can further help emphasize the Pareto Principle or "80/20 rule," which indicates that a small number of occurrences often present the majority of the effect.

Circle Graph

A Circle Graph, also known as a Pie Chart, effectively illustrates data in a round, divided format where each slice represents a proportion of the whole. It is a powerful visual tool especially for showing percentages.

• Each slice of the pie represents a category's contribution to the total.
• The graph gets its name because, once filled with sections, it resembles a pie.

To create a circle graph based on the spearhead data:

• Convert the percentage of spearheads found in each river into their respective angles, considering a full circle has 360 degrees.
• Draw the circle and partition it into sections corresponding to these angles, thereby giving each river a portion of the pie.

This visualization allows for an intuitive comparison of each category's share out of the entire dataset, helping identify the rivers with the greatest discoveries.

Percentage Calculation

Understanding how to calculate percentages is vital in data analysis. Percentages enable a relative comparison of a part of a whole in abstract and meaningful ways. Here's how to perform percentage calculations in general terms and specifically for the spearhead example:

• First, determine the total sum of all groups or categories. For the spearhead example, the total was calculated to be 89.
• Next, divide the number of items in each category by the total number to get a fraction representing each part of the whole.
• Lastly, multiply each fraction by 100 to convert it to a percentage.

Through percentage calculation, each river's contributions could be expressed as a percentage of total spearheads, making comparisons more meaningful. This method is not only useful for visualization but also for identifying outliers or main contributors in a dataset.

Cumulative Frequency

Cumulative Frequency is an integral part of data analysis, especially in visual graphs like the Pareto Chart. It represents a running total of frequencies through the data in a sequential manner. By showing cumulative frequency:

• It illustrates not only the individual frequencies but how they contribute to a collective total.
• Greatly aids in understanding the trend or pattern within the data as you move from one category to another.

For instance, when looking at the spearhead data, the cumulative frequency would begin with the first river category and add each subsequent number of spearheads as you proceed across the rivers listed in descending order. This incremental addition creates an integrated look at how much each subsection adds up to the whole.

Descending Order Arrangement

Arranging data in descending order is an essential data organizing technique that often comes into play with Pareto Charts. The biggest advantage:

• This arrangement makes it easier to visualize and prioritize the most significant categories.
• Aligns perfectly with the Pareto Principle, where you can quickly see which few categories contribute most prominently to the total ("the vital few").

For creating a Pareto chart, the data on spearhead discoveries were arranged from biggest to smallest by the number of discovered artifacts. This ordering process involves:

• Listing categories (rivers in our example) with the highest frequency first, followed by those with lesser frequencies.

This descending order allows for efficient analysis and provides readers with a quick visual narrative of who has the leading role in the dataset.