Question

For questions 5-10, use the data set \(\\{1,3,14,28,2,18\), \(27,86,34,45,44,36,21,11,51,23,37,52,29,41,33\), \(19,24,38,15,87\\}\). What is the median of the data set? A. 27.5 B. 28 C. 28.5 D. 29

Short answer

C. 28.5

Step-by-step solution

- Sort the Data Set

First, arrange the data set in ascending order. The sorted data set is:\[1, 2, 3, 11, 14, 15, 18, 19, 21, 23, 24, 27, 28, 29, 33, 34, 36, 37, 38, 41, 44, 45, 51, 52, 86, 87\]

- Identify the Number of Data Points

Count the total number of data points in the sorted list. There are 26 data points.

- Find the Median Position

The median is the middle value of a data set. For an even number of data points, the median is the average of the two middle numbers. For 26 data points, the two middle numbers are the 13th and 14th values.

- Locate the 13th and 14th Values

In the sorted data set, the 13th value is 28, and the 14th value is 29.

- Calculate the Median

The median is the average of the 13th and 14th values:\[\text{Median} = \frac{28 + 29}{2} = \frac{57}{2} = 28.5\]

Key concepts

Sorting Data

To analyze a data set effectively, it helps to sort the data first. Sorting means arranging data in a specific order—usually ascending or descending. Sorting makes it easier to find values like the median. For example, in the given exercise, the unsorted data set is difficult to analyze. By sorting it into ascending order \[1, 2, 3, 11, 14, 15, 18, 19, 21, 23, 24, 27, 28, 29, 33, 34, 36, 37, 38, 41, 44, 45, 51, 52, 86, 87\], we simplify the process of locating specific values quickly. Sorting the data is often the first step in statistical analysis because it allows us to see the distribution of values and makes further calculations more manageable.

Median of Data Set

The median of a data set is the value that separates the data into two equal halves. To find the median, you must know whether your data set has an odd or even number of values. For an odd number of values, the median is the middle number. For an even number of values, like in this exercise, the median is the average of the two middle numbers.
After sorting the data set, the 13th and 14th values are identified as 28 and 29. By computing the average of these two values: \(\text{Median} = \frac{28+29}{2} = 28.5\), we find that the median of this data set is 28.5. This value indicates the central tendency of the data, providing a sense of the 'middle' among the numbers.

Average Calculation

An average, or mean, is calculated by dividing the sum of all values in a data set by the number of values. Although this exercise focuses on the median, understanding how to compute an average is crucial. For instance, let's say you wanted to find the average of the two middle numbers to determine the median.
Given the numbers 28 and 29: \(\text{Average} = \frac{28+29}{2} = 28.5\). The steps involve summing the two numbers to get 57, then dividing by 2. This simple arithmetic operation helps find the center for even-numbered data sets. Knowing how to calculate averages can also help you understand other central tendencies beyond just the median.

Mathematical Reasoning

Mathematical reasoning involves logically following a sequence of steps to arrive at a solution. In this exercise, we applied reasoning to find the median through a series of steps: sorting the data, identifying the number of points, finding the median position, and calculating the median using the average of the two middle values.
This methodical approach ensures that all necessary steps are considered and nothing is overlooked. It’s an important skill that helps in various types of problem-solving beyond just median calculations. Mathematical reasoning helps you to break down a problem into manageable parts, ensuring accuracy and completeness in your solutions.