Problem 43 - Thinking Mathematically

Question

Find the mode for the data items in the frequency distribution in \(1.6,3.8,5.0,2.7,4.2,4.2,3.2,4.7,3.6,2.5,2.5\)

Step-by-Step Solution

Answer

The modes of the data set are \(4.2\) and \(2.5\).

Step 1: Compile the data set

To start, identify the numbers in the data set. The numbers are \(1.6,3.8,5.0,2.7,4.2,4.2,3.2,4.7,3.6,2.5,2.5\).

Step 2: Identify frequency of each number

Count the frequency of each number in the data set. After counting, you will find that \(1.6\) and \(3.8\) and \(5.0\) and \(2.7\) and \(3.2\) and \(4.7\) and \(3.6\) each appear once, \(4.2\) appears twice, and \(2.5\) appears twice.

Step 3: Identify the mode

The mode is the number (or numbers) that appear most frequently. In our case, the numbers \(4.2\) and \(2.5\) each appear twice, making them the modes of this data set. This data set is multimodal.

Key Concepts

Understanding Frequency Distribution

Frequency distribution is a simple method to organize data. It helps to see how often each value appears in a data set. This makes it easier to understand what the data is indicating and to identify patterns or trends. A frequency distribution lists each unique data item and counts the number of times each occurs in the data set.
For example, if you have the numbers \(1, 2, 2, 3, 3, 3\), a frequency distribution would look like this:

• \(1\): occurs once
• \(2\): occurs twice
• \(3\): occurs three times

This visualization allows you to quickly see that \(3\) is the most frequent item in the set.

Defining a Data Set

A data set is simply a collection of numbers or values. They are grouped together because they are related or need to be analyzed together. In the context of our example, the data set consists of the numbers \(1.6, 3.8, 5.0, 2.7, 4.2, 4.2, 3.2, 4.7, 3.6, 2.5,\) and \(2.5\).
Data sets can be used in many ways:

• To calculate averages
• To identify trends
• To determine variations

Understanding what data set you have and its purpose is crucial to performing any statistical analysis. This step is foundational because all subsequent steps rely on having clearly defined and accurate data to objectively analyze and interpret its meaning.

What Multimodal Means

Describing a data set as multimodal means that there is more than one mode. The mode is the number that appears most frequently in a data set. Sometimes, multiple numbers can have the same highest frequency.
In the case of the data set \(1.6, 3.8, 5.0, 2.7, 4.2, 4.2, 3.2, 4.7, 3.6, 2.5, 2.5\), both \(4.2\) and \(2.5\) appear twice. Thus, it's a multimodal data set with two modes: \(4.2\) and \(2.5\).

Having multiple modes can indicate that there are different trends or data clusters within the data set. Remember, in a multimodal data set, there can be more than two modes, whenever distinct values share the same frequency.

Counting Frequency Effectively

Counting frequency is the process of tallying how often each value appears in a data set. This step is essential in identifying modes or repeating patterns.
The best approach to counting frequency is:

• List each unique number in the data set.
• Count and note how many times each number occurs.

For example, if you have the numbers \(4.2, 4.2, 2.5, 2.5, 3.0\), you would count:

• \(4.2\): 2 times
• \(2.5\): 2 times
• \(3.0\): 1 time

This careful counting will lead you to identify the modes and precisely describe the data set's characteristics.