Problem 9 - 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. \(\\{31.7,22.3,12.3,26.8,24.9,23.0\\}\) (Areas of largest freshwater lakes in thousands of square miles) (Source: U.S. National Oceanic and Atmospheric Administration.)

Step-by-Step Solution

Answer

Mean is 23.5; median is 23.95. Largest area, 31.7, is Lake Superior.

Step 1: Plot Numbers on a Number Line

First, arrange the given numbers in ascending order: 12.3, 22.3, 23.0, 24.9, 26.8, 31.7. Next, draw a horizontal number line. Mark each number from the set on this line, ensuring they are proportionally spaced based on their values. This visual representation helps in understanding their distribution.

Step 2: Calculate the Mean

To find the mean, sum all numbers in the set and then divide by the number of elements in the set. Using the numbers \{31.7, 22.3, 12.3, 26.8, 24.9, 23.0\}, first calculate the sum: \[31.7 + 22.3 + 12.3 + 26.8 + 24.9 + 23.0 = 141.0\]Now, divide the sum by the total number of data points (6):\[\text{Mean} = \frac{141.0}{6} = 23.5\]

Step 3: Calculate the Median

The median is the middle number in an ordered data set. Since we have an even number of elements (6), the median will be the average of the third and fourth numbers when ordered. In the sorted set \{12.3, 22.3, 23.0, 24.9, 26.8, 31.7\}, the third number is 23.0 and the fourth is 24.9:\[\text{Median} = \frac{23.0 + 24.9}{2} = 23.95\]

Step 4: Interpret the Mean and Median

The mean value of 23.5 thousand square miles is the average area size for the largest freshwater lakes according to the data provided, while the median (23.95) indicates the midpoint when the lake areas are ordered from smallest to largest. Since the median is slightly higher than the mean, it suggests a slightly skewed distribution with larger lakes.

Step 5: Identify the Geographic Feature of Largest Number

The largest number in the set is 31.7, which represents Lake Superior's area in thousands of square miles, making it the largest freshwater lake by surface area.

Key Concepts

Mean

The mean is a fundamental concept in descriptive statistics. It is simply the average of a set of numbers. To calculate the mean, you add up all the numbers in your data set and then divide by how many numbers there are. This gives you the typical value, or the central point of the data. For instance, using the set \(\{31.7, 22.3, 12.3, 26.8, 24.9, 23.0\}\):

• First, calculate the sum: \(31.7 + 22.3 + 12.3 + 26.8 + 24.9 + 23.0 = 141.0\).
• Next, divide the total by the number of values (6): \(\text{Mean} = \frac{141.0}{6} = 23.5\).

In this example, the mean is 23.5 thousand square miles. This tells us that, on average, the largest freshwater lakes in our data set have a size of 23.5 thousand square miles.
Understanding the mean helps us summarize a large data set with a single number. However, it is always good to check how close numbers are to each other as well. That’s where the median comes in handy.

Median

The median is another central measure in statistics, offering a different perspective from the mean by highlighting the middle value in a data set. The benefit of the median is that it isn’t affected by very high or very low values, which can skew the mean.

• First, organize the numbers in ascending order, for instance: \(\{12.3, 22.3, 23.0, 24.9, 26.8, 31.7\}\).
• Since the data set has an even number of entries, the median will be the average of the third and fourth values: \(\frac{23.0 + 24.9}{2} = 23.95\).

Thus, the median for this set is 23.95 thousand square miles. This median value represents the middle point of our dataset. Since it is slightly higher than the mean, it suggests there are some larger than average lakes pulling the mean down slightly.
Comparing both the mean and the median offers insights into the nature of the data set, such as skewness or symmetry.

Number Line

A number line is a visual tool in mathematics that helps to plot and understand the relative position of numbers. Here's how it works:

• First, arrange the numbers in ascending order, like this: \(\{12.3, 22.3, 23.0, 24.9, 26.8, 31.7\}\).
• Then, draw a horizontal line and mark equal intervals to represent these numbers proportionally.
• Place each point on this line according to its value.

This visual method allows you to quickly understand the distribution and spread of the data. You can see gaps between values, check for any clusterings, and identify outliers more easily. A number line helps in showing relationships within a set of numbers and is a great way to interpret numerical data, making the concepts of mean and median more intuitive!