Problem 73 - Prealgebra

Question

Find the median and the range for each set of numbers. $$15,18,21,24,29$$

Step-by-Step Solution

Answer

The median is 21, and the range is 14.

Step 1: Arrange the Numbers

The numbers given are already arranged in ascending order: \(15, 18, 21, 24, 29\). Thus, no sorting is necessary.

Step 2: Find the Median

Since the set contains an odd number of elements (five numbers), the median is the middle number. Here, the third number is 21, so the median is \(21\).

Step 3: Find the Range

The range is the difference between the largest and smallest numbers in the set. The smallest number is 15, and the largest is 29. Thus, the range is \(29 - 15 = 14\).

Key Concepts

Understanding Range

The range is a measure of how spread out a set of numbers is. It's calculated simply by taking the difference between the highest and lowest numbers in a set. This gives us a sense of the dispersion within the data. For instance, in the set \(15, 18, 21, 24, 29\), the largest number is \(29\) and the smallest is \(15\). Therefore, the range is \(29 - 15 = 14\).

This means that there is a 14-unit gap between the smallest and largest values, signifying the total distribution width of the numbers provided. The range is useful for understanding the variability of the data. A larger range indicates more variability, while a smaller range suggests the data values are closer together.

What is Ascending Order?

Placing numbers in ascending order means arranging them from the smallest to the largest. This order is essential for easily identifying the median of a set or checking the spread between numbers. In our given set of numbers, they have already been arranged: \(15, 18, 21, 24, 29\).

Arranging numbers in ascending order is often the first step when dealing with statistics, as it provides a clear sequence.

• The first number will be the smallest.
• The last number will be the largest.
• This helps in calculating other statistics like the median or range effortlessly.

Defining an Odd Number Set

An odd number set, in terms of data, refers to the number of elements in a collection when it is not divisible evenly by two. This is quite significant when finding the median. With an odd number of items, there is always one middle number, which simplifies median calculation.

For example, in the series \(15, 18, 21, 24, 29\), we have five numbers, making it an odd number set. Thus, we can easily pick the third number, 21, as the median. When sets have an even number of elements, the median would be the average of the two middle numbers instead.

Identifying the Middle Number

Finding the middle number is key when calculating the median, especially when working with odd-numbered sets.

In a list ordered in ascending sequence: - This means finding the number that lies at the center of the sequence.For odd sets like \(15, 18, 21, 24, 29\), the middle number is straightforward. Here, 21 is situated in the center, making it the median.

Conversely, if you had an even-numbered set, like \(10, 15, 20, 25\), with no single middle number, you'd calculate the median by averaging the two central numbers, 15 and 20. The median then becomes \((15 + 20) / 2 = 17.5\).