Problem 17 - Thinking Mathematically

Question

In Exercises 13-20, find the median for each group of data items. \(100,40,70,40,60\)

Step-by-Step Solution

Answer

The median value of the given data set is 60.

Step 1: Arrange in Numerical Order

First, arrange the given values in ascending order: \(40, 40, 60, 70, 100\)

Step 2: Identify the Position of the Median

There are 5 data points, an odd number, hence the median will be at the 3rd position, as it is in the middle when data is arranged in ascending order.

Step 3: Extract the Median

Hence, the median value is the 3rd value in the ordered list, which is 60.

Key Concepts

Central Tendency

Central tendency is a statistical measure that identifies a single value as representative of an entire data set. It provides an insight into the "center" or typical value of the data points. This concept is all about having a summary that speaks for the overarching group of numbers, making data interpretation simpler and more meaningful.

There are three main indicators of central tendency: mean, median, and mode. Each represents a different aspect of the data's central value:

• Mean: The average of all data points.
• Median: The middle value when data is arranged in numerical order.
• Mode: The most frequently occurring data point.

For this particular exercise, we are focusing on the median because it is less sensitive to extreme values or outliers compared to the mean. When tasked with identifying a central tendency, choosing the right measure (like median in this case) can highlight meaningful patterns that might otherwise be hidden.

Numerical Order

Numerical order refers to arranging data from the smallest to the largest value. This step is crucial for finding the median, because without ordering the data, it would be impossible to accurately identify the middle value.

Let's take the given data set for the exercise:

• First, arrange the data: 100, 40, 70, 40, 60.
• The numbers are then sorted to become: 40, 40, 60, 70, 100.

This sequential arrangement helps easily pinpoint the center of the data set, necessary for determining the median. Without proper ordering, evaluations and analyses might lead to incorrect conclusions. In many statistical tasks, particularly those using central tendency, this orderly arrangement is a foundational step.

Odd Number Data Set

An odd number data set simply means that the total count of data points is an odd number. Knowing whether a data set contains an odd number of values is important for finding the median because it determines the position of the median directly.

For any data set with an odd number of points, the median will always be the middle point so it can easily divide the data into two equal halves.

Consider our example with the data set: 40, 40, 60, 70, 100. This set has 5 numbers:

• Count the numbers: 1, 2, 3, 4, 5.
• The median, being the middle number, is the 3rd value when counted, which is 60.

If the data set had an even amount of data points, the median would be the average of the two middle values. But with an odd data set, we can quickly locate and identify the central position without further computations. This simplicity makes median calculations straightforward with odd-numbered data.