Problem 53 - Algebra 2

Question

Find the median of each set of numbers. \(\\{2.5,7.8,5.5,2.3,6.2,7.8\\}\)

Step-by-Step Solution

Answer

The median is 5.85.

Step 1: Arrange Numbers in Ascending Order

First, we need to order the numbers from smallest to largest. So, the set \( \{2.5, 7.8, 5.5, 2.3, 6.2, 7.8\} \) becomes \( \{2.3, 2.5, 5.5, 6.2, 7.8, 7.8\} \).

Step 2: Identify the Middle Numbers

Since there are 6 numbers, which is an even amount, we take the middle two numbers. In our set \( \{2.3, 2.5, 5.5, 6.2, 7.8, 7.8\} \), the middle numbers are 5.5 and 6.2.

Step 3: Calculate the Median

To find the median, average the two middle numbers: \( \frac{5.5 + 6.2}{2} = 5.85 \).

Key Concepts

Arranging Numbers

To properly find the median of any set of numbers, the first essential step is to arrange the numbers in order. This means lining them up from the smallest to the largest value. Let’s say you have a set of numbers: \( \{2.5, 7.8, 5.5, 2.3, 6.2, 7.8\} \). Rendering them in ascending order would give you \( \{2.3, 2.5, 5.5, 6.2, 7.8, 7.8\} \).
Arranging numbers helps in identifying the position of numbers in a dataset. It is a crucial step to accurately determine the median, which represents the middle value of a dataset. Without sorting, it's impossible to know the true "middle." This step is especially important if numbers are provided in a random order.

Even Number Set

In statistics, the term 'even number set' refers to a collection of numbers where the total count is an even quantity. This is significant because the method for calculating the median varies based on whether the dataset contains an odd or even number of elements.
For an even number set such as \( \{2.3, 2.5, 5.5, 6.2, 7.8, 7.8\} \), there are 6 numbers in total. Therefore, the median cannot simply be the center number. Instead, it requires finding the average of the two middle numbers after arranging the numbers in order. This helps in capturing the central tendency of the dataset more accurately when the count is even.

Calculating Median

Once the numbers are arranged and you know the set either has an odd or an even count, you can calculate the median. For an even set like our example, the median is located between the two middle numbers. In our ordered set \( \{2.3, 2.5, 5.5, 6.2, 7.8, 7.8\} \), the two middle numbers are 5.5 and 6.2.

To find the median, you simply calculate the average of these two numbers:

• Add the two middle numbers together: \( 5.5 + 6.2 = 11.7 \)
• Divide the sum by 2 to get the median: \( \frac{11.7}{2} = 5.85 \)

Thus, the median of this set is 5.85. Calculating the median for an even set ensures a more accurate representation of the dataset's central point.