Question

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 mode of the data set? A. 28.5 B. 31.5 C. 86 D. There is no mode.

Short answer

D. There is no mode.

Step-by-step solution

Define the Mode

The mode of a data set is the number or numbers that appear most frequently. If no number repeats, the data set has no mode.

Organize Data

List all the numbers in the data set: \[\{1, 3, 14, 28, 2, 18, 27, 86, 34, 45, 44, 36, 21, 11, 51, 23, 37, 52, 29, 41, 33, 19, 24, 38, 15, 87\}\]

Identify Frequency of Each Number

Count how many times each number appears in the data set. 1: 1 time3: 1 time14: 1 time28: 1 time2: 1 time18: 1 time27: 1 time86: 1 time34: 1 time45: 1 time44: 1 time36: 1 time21: 1 time11: 1 time51: 1 time23: 1 time37: 1 time52: 1 time29: 1 time41: 1 time33: 1 time19: 1 time24: 1 time38: 1 time15: 1 time87: 1 time

Determine the Mode

Since each number appears only once, there is no number that appears more frequently than others.

Key concepts

headline of the respective core concept

To understand and identify the mode of a data set, the first step is to organize the data effectively. Organizing data helps you see patterns and outliers more easily.
Begin by listing all the numbers in the data set in a logical order. This could be numerical order, from smallest to largest, or simply noting each entry clearly. In our exercise, we have the dataset: \[ \{1, 3, 14, 28, 2, 18, 27, 86, 34, 45, 44, 36, 21, 11, 51, 23, 37, 52, 29, 41, 33, 19, 24, 38, 15, 87\} \]
By organizing the data set in this manner, we make it easier to move onto the next step, which involves counting the frequency of each number.

headline of the respective core concept

With the data neatly organized, the next step is to count how frequently each number appears in the dataset. Counting frequency tells us how often each number occurs and helps us in identifying modes.
To count the frequency, go through the list of numbers one by one and keep a tally of how many times each number appears. For our dataset, each number appears exactly once:

• 1: 1 time
• 3: 1 time
• 14: 1 time
• 28: 1 time
• 2: 1 time
• 18: 1 time
• 27: 1 time
• 86: 1 time
• 34: 1 time
• 45: 1 time
• 44: 1 time
• 36: 1 time
• 21: 1 time
• 11: 1 time
• 51: 1 time
• 23: 1 time
• 37: 1 time
• 52: 1 time
• 29: 1 time
• 41: 1 time
• 33: 1 time
• 19: 1 time
• 24: 1 time
• 38: 1 time
• 15: 1 time
• 87: 1 time

This counting confirms that all numbers in the dataset occur exactly once.

headline of the respective core concept

Identifying the mode becomes straightforward once we have the frequencies of all numbers. The mode is the number that appears most frequently in a dataset. However, if no number repeats more than others, then the data set has no mode.
In our dataset, since each number appears only once, no number is more frequent than another. Therefore, our conclusion is simple: there is no mode in this data set.
This example highlights that not all datasets will have a mode and reinforces the importance of counting frequencies accurately to draw valid conclusions on modes.