Problem 10 - College Algebra with Modeling and Visualization

Question

The set of numbers contains data about geographic features of the world. (a) Plot the numbers on a number line. (b) Calculate the mean and median for the set of numbers. Interpret your results. (c) Try to identify the geographic feature associated with the largest number in the set. \(\\{19.3,18.5,29.0,7.31,16.1,22.8,20.3\\}\) (Highest eleva- tions of the continents in thousands of feet) (Source: National Geographic Atlas of the World.)

Step-by-Step Solution

Answer

Arrange numbers, plot on number line, then calculate mean ( \approx 19.04 ) and median (19.3). Mount Everest corresponds to the largest number, 29.0.

Step 1: Arrange the Numbers

Start by arranging the numbers in ascending order to make plotting and calculation easier. The numbers are \(7.31, 16.1, 18.5, 19.3, 20.3, 22.8, 29.0\).

Step 2: Plot the Numbers on a Number Line

Draw a number line with a suitable range that covers all the numbers. Mark each number from step 1 onto the line, representing each number as a dot at the corresponding value.

Step 3: Calculate the Mean

The mean is the average of the numbers. Add all the numbers together, then divide by the total number of elements.\[\text{Mean} = \frac{7.31 + 16.1 + 18.5 + 19.3 + 20.3 + 22.8 + 29.0}{7} = \frac{133.31}{7} \approx 19.04\]

Step 4: Calculate the Median

The median is the middle number in an ordered list. As there are 7 numbers, the median is the 4th number in the sorted list: \(19.3\).

Step 5: Interpret the Mean and Median

The mean of approximately \(19.04\) indicates the average elevation is around 19,000 feet. The median of \(19.3\) suggests half the elevations are below 19,300 feet, and half are above. Both metrics indicate a central tendency around 19,000-19,300 feet.

Step 6: Identify the Largest Number

The largest number in the set is \(29.0\). This likely represents the highest elevation, associated with Mount Everest in Asia, the tallest mountain in the world.

Key Concepts

Understanding the Number Line

A number line is a visual representation of numbers laid out in a straight line. It helps in comparing and understanding the relative size of numbers.
In this exercise, we're working with geographical data represented on a number line. Here's how to plot the numbers:

• First, arrange the numbers in increasing order. This helps maintain clarity and ensures accurate plotting.
• Draw a horizontal line and evenly distribute marks along it, representing the range of numbers you've arranged.
• Plot each number as a dot above the corresponding mark on your number line.

This method provides a clear picture of how each data point relates to others, making it easier to visualize distributions and ranges.

Demystifying Mean Calculation

Calculating the mean, also known as the average, gives us a single value that summarizes all the data in a set. The mean is useful for understanding the overall level or central tendency of a dataset.
To calculate the mean:

• Add together all the numbers in your dataset.
• Divide the total by the number of data points you have.

In our example, the mean of the world elevations is approximately 19.04. This suggests that, on average, the elevations of the world are around 19,000 feet. Remember, the mean can be sensitive to extreme values, which might skew this value if there are outliers.

Clarifying Median Calculation

The median is the middle value of an ordered dataset, providing an alternative measure of central tendency that is less affected by outliers or skewed data. Here's how to find the median:

• First, order the numbers from smallest to largest.
• If you have an odd number of elements, the median is the middle number.
• If you have an even number of elements, the median is the average of the two middle numbers.

For our elevations data, the median is 19.3. With the median, half the data points are below this value, and half are above, giving a balanced indication of central tendency without being affected by the extreme high value of 29.0.

Interpreting Statistical Measures

When interpreting the mean and median, it's crucial to understand what these values tell us about the dataset. Both measures offer insights into the central characteristics of the data.
In this context:

• The mean value of approximately 19.04 suggests that the general level of elevations is around this figure.
• The median of 19.3 provides a point where half of the elevations lie below and half above, which can offer a clearer picture when the data may include outliers.

Together, they paint a picture of the distribution's central region, highlighting typical trends. In this case, because they are close, they suggest a fairly balanced dataset without large skewed differences.