Problem 76 - Prealgebra

Question

Find the median and the range for each set of numbers. $$53,61,67,75$$

Step-by-Step Solution

Answer

The median is 64, and the range is 22.

Step 1: Order the Numbers

The numbers are already ordered in ascending order: 53, 61, 67, 75.

Step 2: Find the Median

To find the median, which is the middle number, we have an even number of data points (4 numbers), so we take the average of the two middle numbers. These two numbers are 61 and 67.\[\text{Median} = \frac{61 + 67}{2} = \frac{128}{2} = 64\]Thus, the median is 64.

Step 3: Determine the Range

The range is found by subtracting the smallest number from the largest number in the set.\[\text{Range} = 75 - 53 = 22\]Thus, the range is 22.

Key Concepts

Ordering Numbers in Mathematics

Ordering numbers is a fundamental skill in mathematics, helping us understand data by arranging it from smallest to largest or vice versa. This process is crucial when calculating the median, as it allows us to identify the central part of a data set.
Ensuring numbers are in the correct order can:

• Reveal trends or patterns within a data set.
• Make subsequent calculations, like finding medians and ranges, more straightforward.
• Assist in comparing different data sets for analysis.

For example, in the set \( 53, 61, 67, 75 \), the numbers are already in ascending order. If the numbers were unordered, such as \( 75, 53, 67, 61 \), you would rearrange them to be \( 53, 61, 67, 75 \). This ordered list is essential for easily finding other statistics, acting as our roadmap for analyzing data.

Calculating Median

The median is a measure of central tendency, giving us insight into the middle of a data set. It's particularly useful in understanding data that may have outliers or is skewed. Unlike the mean, the median isn't affected by extremely high or low values.
When calculating the median, follow these steps:

• First, ensure the numbers are in ascending order; this makes identifying the center easy.
• If the data set has an odd number of data points, the median is simply the middle number.
• If there are an even number of data points, such as in our example with \( 53, 61, 67, 75 \), we take the average of the two middle numbers. Here, those numbers are \( 61 \) and \( 67 \).
• To find the median, calculate: \[\text{Median} = \frac{61 + 67}{2} = 64\]

Thus, for this data set, the median is \( 64 \), representing the middle value of the ordered numbers.

Calculating Range

The range in mathematics provides a simple measure of the spread within a set of numbers. It shows how much variation there is from the smallest to the largest number. This is a straightforward way to see how spread out or tightly clustered a data set is.
To compute the range:

• Identify the smallest and largest numbers in the set. This step is easier when numbers are ordered.
• Subtract the smallest number from the largest number. This difference reveals the range.

For the data \( 53, 61, 67, 75 \), the range is calculated as:\[\text{Range} = 75 - 53 = 22\]This answer tells us that the numbers \( 53 \) to \( 75 \) span a range of \( 22 \), providing insight into the distribution of values. The range is a valuable statistic that emphasizes the variability in the data set.