Problem 50 - Algebra 2

Question

Find the median of each set of numbers. \(\\{3,2,1,3,4,8,4\\}\)

Step-by-Step Solution

Answer

The median is 3.

Step 1: Understand the Concept of Median

The median is the middle value in a list of numbers. For odd-numbered sets, it's the center number when all numbers are in order. For even-numbered sets, it's the average of the two middle numbers.

Step 2: Arrange the Numbers in Order

Order the set of numbers from smallest to largest: \(1, 2, 3, 3, 4, 4, 8\).

Step 3: Identify the Middle Number

Since the list has 7 numbers (which is odd), find the middle (4th) number. Count to the fourth position: The numbers are \(1, 2, 3, 3, 4, 4, 8\). The middle number is the fourth number which is \(3\).

Key Concepts

Ordered Data Sets

When working with statistics, an important step in finding the median is to first organize your data set. This means arranging the numbers in either ascending or descending order. Ordering your data set allows you to clearly see where the middle value lies. To organize, you should:

• Identify all the numbers in your set.
• List them from the smallest number to the largest.

In our exercise, we had the numbers \(\{3,2,1,3,4,8,4\}\). By arranging them in order, we got \(1, 2, 3, 3, 4, 4, 8\). This ordered arrangement is crucial as it simplifies the process of identifying the median.

Odd-Numbered Sets

An odd-numbered set of data refers to a collection of numbers where the total count of numbers is odd. Essentially, there is an exact middle point among the numbers when they are sorted. This middle number is what we refer to as the median.Odd-numbered sets are easier to deal with when finding the median because:

• There is one clear middle number.
• You don’t need to calculate an average between two numbers as you do with even-numbered sets.

In our example, the set \(\{1, 2, 3, 3, 4, 4, 8\}\) contains 7 numbers, which is odd. This allows for a clear middle point in the sequence.

Middle Value

The middle value or the median in an ordered data set is the number that separates the higher half from the lower half. When you line up all numbers from least to greatest in an odd-numbered set, the middle value is easy to find.To locate the middle value:

• Count towards the center of the data set.
• Find the middle position using the formula: \((n + 1) / 2\), where \(n\) is the total number of values.

For our set \(\{1, 2, 3, 3, 4, 4, 8\}\), there are 7 numbers. The formula gives us \( (7 + 1) / 2 = 4\). Thus, the middle number is the 4th number: \(3\). This make \(3\) the median, representing the middle value of our ordered set.