Problem 27 - Basic College Mathematics

Question

Highest Bridges. The following list gives the distance, in meters, from the road surface to the water or ground below for the 20 highest bridges in the world. Create a stem-and-leaf plot to represent the data. Hint: Let each leaf represent the digit in the tens place. $$ 570,500,360,340,370,320,490,390,320,430, $$ 330,360,400,330,370,460,390,320,380,340

Step-by-Step Solution

Answer

320: 2 2 2330: 3 3340: 4 4360: 6 6370: 7 7380: 8390: 9 9400: 0430: 3460: 6490: 9500: 0570: 7

Step 1 - Organize the Data

Start by listing the data in ascending order. This will make it easier to create the stem-and-leaf plot.320, 320, 320, 330, 330, 340, 340, 360, 360, 370, 370, 380, 390, 390, 400, 430, 460, 490, 500, 570

Step 2 - Determine the Stems and Leaves

Each stem will represent the hundreds place, and each leaf will represent the tens place. Identify the stems and the corresponding leaves from the data.Stems: 320-570Leaves: 2, 3, 4, 5, 6, 7, 8, 9

Step 3 - Create the Stem-and-Leaf Plot

Now, organize the data into the stem-and-leaf plot format. List the stems in the left column, and write the leaves next to the corresponding stem.320: 2 2 2330: 3 3340: 4 4360: 6 6370: 7 7380: 8390: 9 9400: 0430: 3460: 6490: 9500: 0570: 7

Key Concepts

data representation

Data representation helps us understand and interpret sets of numbers in an easy and organized way. One common method is using a stem-and-leaf plot. Stem-and-leaf plots break down data into stems (the broad category) and leaves (the specific details). For instance, if we look at bridge heights, the stems can be in hundreds (like 300, 400) and the leaves in tens (like 20, 30). This helps us to quickly see the frequency and distribution of values. For example, in the given exercise:
320, 320, 320, 330, 330, 340, 340, 360, 360, 370, 370, 380, 390, 390, 400, 430, 460, 490, 500, 570
It's clear that the 300-399 range has more entries than the 500-599 range.

statistics

Statistics is all about collecting, analyzing, interpreting, and presenting data. The stem-and-leaf plot is a basic statistical tool. It not only provides a quick way to visualize data points, but it also retains the original data. For example, looking at our stem-and-leaf plot, it's easy to determine that the most common height range for our bridges is between 320 and 390 meters. This concise way of presenting data helps in quickly spotting trends, medians, and outliers without complex computations.

mathematical graphs

Mathematical graphs like stem-and-leaf plots convert numerical data into a visual format. This visual representation aids in better comprehension and comparison of data sets. While creating such plots:

• Think of the stems as the main trunk of the graph representing grouped data.
• Leaves are the branches showing the finer details within each group.

Using our exercise, for instance:
320: 2 2 2
330: 3 3
This layout immediately informs us about the distribution frequency and helps in the interpretation of mathematical theories or statistical information.

organizational skills

Organizing data effectively is crucial in any mathematical task. Stem-and-leaf plots are an excellent way to practice and improve organizational skills. They require careful arranging of data points in ascending order and correctly assigning them to stems and leaves without error. This meticulous approach to organizing data:

• Reduces the risk of mistakes.
• Aids in clearer interpretation.

In our given exercise, arranging the bridge heights in order made it simpler to construct the plot and assured that no data point was skipped or duplicated. Skills like this are invaluable in both academic and real-world data analysis scenarios.