Problem 6 - College Algebra with Modeling and Visualization

Question

Sort the list of numbers from smallest to largest and display the result in a table. (a) Determine the maximum and minimum values. (b) Calculate the mean and median. Round each result to the nearest hundredth when appropriate. $$ -1.25,4.75,-3.5,1.5,2.5,4.75,1.5 $$

Step-by-Step Solution

Answer

Max: 4.75, Min: -3.5, Mean: 1.46, Median: 1.5.

Step 1: Sort the List

Arrange the numbers from smallest to largest. To do this, list the numbers in ascending order: \[ -3.5, -1.25, 1.5, 1.5, 2.5, 4.75, 4.75 \]

Step 2: Display in a Table

Record the sorted numbers in a table format: | Sorted Numbers | |----------------| | -3.5 | | -1.25 | | 1.5 | | 1.5 | | 2.5 | | 4.75 | | 4.75 |

Step 3: Determine the Maximum and Minimum

Identify the smallest and largest numbers in the sorted list. - Minimum value: \( -3.5 \)- Maximum value: \( 4.75 \)

Step 4: Calculate the Mean

The mean is calculated by dividing the sum of the numbers by the number of numbers.First, find the sum: \[ -3.5 + (-1.25) + 1.5 + 1.5 + 2.5 + 4.75 + 4.75 = 10.25 \]Then, divide by the number of values (7):\[\text{Mean} = \frac{10.25}{7} = 1.46 \, (\text{rounded to two decimal places})\]

Step 5: Calculate the Median

The median is the number in the middle of a sorted list. Since there are 7 numbers, the median is the fourth number.- Median value: \( 1.5 \)

Key Concepts

Sorting

Sorting is a fundamental concept in statistics that involves arranging data in a specific order. In this exercise, numbers are sorted from smallest to largest. This is known as ascending order. Sorting data helps simplify analysis and comparison, especially in statistical computations.
For instance, consider the list:

• -1.25
• 4.75
• -3.5
• 1.5
• 2.5
• 4.75
• 1.5

After sorting the numbers in ascending order, your new list becomes:

• -3.5
• -1.25
• 1.5
• 1.5
• 2.5
• 4.75
• 4.75

Sorting is the first step in many statistical procedures as it helps identify patterns and simplifies further data analysis.

Mean calculation

Calculating the mean, often referred to as the average, helps summarize a set of values with a single representative number. To find the mean, add all the data points together and then divide by the total number of points.
For the sorted list:

• -3.5
• -1.25
• 1.5
• 1.5
• 2.5
• 4.75
• 4.75

First, calculate the sum:\[-3.5 + (-1.25) + 1.5 + 1.5 + 2.5 + 4.75 + 4.75 = 10.25\]Then, divide by the number of values, which is 7:\[\text{Mean} = \frac{10.25}{7} = 1.46\]This process gives you the mean value of the data, rounding to the nearest hundredth. The mean provides an overall summary of the dataset, indicating the central point around which the data values cluster.

Median determination

The median is the middle value of a data set that’s organized in ascending or descending order. Finding the median helps identify the center of the dataset without being influenced by extreme values.
If the dataset is sorted:

• -3.5
• -1.25
• 1.5
• 1.5
• 2.5
• 4.75
• 4.75

For a list with an odd number of terms, such as the 7 in this example, the median is the fourth number, which is 1.5.

• The position can be found using: \( \frac{n+1}{2} \) where \( n \) is the number of values.

This results in a median position of \( \frac{7+1}{2} = 4 \). The median provides a sense of balance in the dataset, demonstrating that approximately half of the numbers are below and half are above this point.

Maximum and minimum values

Identifying the maximum and minimum values in a dataset reveals the range and limits of the data. These values provide insights into the spread and variability of the dataset.
In the given sorted list:

• -3.5
• -1.25
• 1.5
• 1.5
• 2.5
• 4.75
• 4.75

The minimum value, identified as the smallest number, is -3.5. The maximum value, the largest number, is 4.75. These values define the boundaries of the dataset and help understand the extent to which the data ranges. Knowing the maximum and minimum can also assist in detecting outliers or unusual variations in the dataset.