Problem 51 - Algebra 2

Question

Find the median of each set of numbers. \(\\{9,3,7,5,6,3,7,9\\}\)

Step-by-Step Solution

Answer

The median is 6.5.

Step 1: Organize the Data

To find the median, first arrange the numbers in the set from smallest to largest. Here, the given numbers are \(9, 3, 7, 5, 6, 3, 7, 9\). Organizing them in ascending order, we get: \(3, 3, 5, 6, 7, 7, 9, 9\).

Step 2: Identify the Middle Positions

Since there are 8 numbers, the middle positions are the 4th and 5th numbers in the ordered set. In the sequence \(3, 3, 5, 6, 7, 7, 9, 9\), the 4th number is \(6\) and the 5th number is \(7\).

Step 3: Find the Median

The median of an even set of numbers is calculated by taking the average of the two middle numbers. Therefore, the median is \(\frac{6 + 7}{2}\).

Step 4: Calculate the Median

Perform the arithmetic to find the median: \(\frac{6 + 7}{2} = \frac{13}{2} = 6.5\).

Key Concepts

Organizing Data

Before you calculate the median, you need to organize the numbers in your data set. This involves sorting the elements in ascending order, which means arranging them from the smallest number to the biggest. For example, with the set \(\{9, 3, 7, 5, 6, 3, 7, 9\}\), start by identifying the smallest number and work your way up to the largest. After organizing, your sequence should be \(3, 3, 5, 6, 7, 7, 9, 9\). This step is crucial because an unordered set won't provide accurate results when finding the median. Remember to count the numbers in your set to ensure none are missing after sorting.

Median Calculation

Once your data set is organized, finding the median becomes straightforward. The median represents the middle value in a set of numbers. For odd-numbered sets, it's as simple as picking the middle number. However, in even-numbered sets, such as \(\{3, 3, 5, 6, 7, 7, 9, 9\}\), there are two middle numbers. You must determine these positions to calculate the median correctly. In this example, the numbers in the 4th and 5th positions are \(6\) and \(7\). These positions are in the middle of the ordered set.

Average of Numbers

When dealing with even-numbered sets, you'll often need to calculate the average of the two middle numbers to find the median. To find the average, sum the two numbers and divide by 2. In our previous example, the two middle numbers, \(6\) and \(7\), have a sum of \(6 + 7 = 13\). Divide this sum by 2 to get the average: \[ \frac{13}{2} = 6.5 \]. This result, \(6.5\), is the median of your data set. This step ensures that you account for all values in your calculation.

Arithmetic Operations

Arithmetic operations are the basic math processes of addition, subtraction, multiplication, and division. These operations help simplify complex steps like finding the median. For example, to find the median of certain numbers in our data set, you use addition and division. Add the two middle values, \(6 + 7\), to get \(13\). Then divide the result, \(13\), by \(2\) to find the median: \[ \frac{13}{2} = 6.5 \].Understanding these basic arithmetic operations can make solving many mathematical problems more manageable. They are fundamental for various calculations beyond just finding the median.