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Chapter 10: Problem 3

\(\\{15,9,23,8,42,6,35,8\\}\) What is the mode of the data set?

Short Answer

Expert verified
Answer: The mode of the data set is 8.

Step by step solution

01

Organize the data in ascending order

First, we will sort the data in ascending order to see the frequency of each value more clearly. Sorted data set: \(\\{6, 8, 8, 9, 15, 23, 35, 42\\}\)
02

Identify the value with the highest frequency

Now, we can easily identify the highest frequency by looking at how many times each value appears in the sorted data set: - \(6\) occurs \(1\) time - \(8\) occurs \(2\) times (highest frequency) - \(9\) occurs \(1\) time - \(15\) occurs \(1\) time - \(23\) occurs \(1\) time - \(35\) occurs \(1\) time - \(42\) occurs \(1\) time
03

Determine the mode of the data set

Since the value \(8\) appears with the highest frequency (\(2\) times) in the sorted data set, the mode of the data set is \(8\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Frequency in Statistics
In statistics, frequency refers to the number of times a particular value appears in a data set. When studying data, you'll often want to know how common certain numbers are.
  • Frequency is a crucial concept when trying to understand patterns or trends within your data.
  • It helps to identify the most common values and understand how the data is distributed.
  • In our given example with the data set \(\{15,9,23,8,42,6,35,8\}\), the frequency of each value is noted after sorting the data.
By counting each occurrence, you observe which numbers are repeated. This provides insight into the most common values, or the "mode," of the data set.
Understanding frequency is foundational in descriptive statistics, where getting a sense of the data's structure is key.
Organizing Data
Organizing data is a critical step in statistical analysis. It involves arranging data in a systematic order so patterns can be easily recognized.
  • One common way to organize data is by sorting it in ascending or descending order.
  • This makes it easier to analyze and allows you to quickly identify key characteristics, like the mode.
Looking at our exercise example, we first rearranged the data set \(\{15,9,23,8,42,6,35,8\}\) in ascending order to \(\{6, 8, 8, 9, 15, 23, 35, 42\}\).
This sorting enables us to clearly see that the number eight occurs twice, making it simpler to identify it as the mode.
When organizing data, clarity and ease of analysis are the main goals, and it forms the basis for more complex analytical tasks.
Descriptive Statistics
Descriptive statistics provides a summary of a data set through numbers and graphical representation. Its purpose is to highlight patterns without making predictions based on data.
  • The mode, which we identified as 8 for our exercise, is one type of descriptive statistic.
  • Other components include the mean (average), median (middle value), and range (difference between the highest and lowest values).
  • Using descriptive statistics allows you to simplify large amounts of data into more understandable terms.
These statistics help in making quick sense of data by breaking it down into understandable formats and highlighting essential trends or frequencies.
Descriptive statistics are often the first step in any data analysis because they provide pivotal insights and a solid groundwork for further statistical exploration.

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