Question

The mode of the data \(6,4,3,6,4,3,4,6,5\) and \(x\) can be (1) only 5 (2) both 4 and 6 (3) both 3 and 6 (4) 3,4 or 6

Short answer

Answer: The possible modes are both 3 and 6, or 3, 4, or 6 depending on the value of x.

Step-by-step solution

Count the frequency of each number in the data

In the given data: \(6,4,3,6,4,3,4,6,5\), we will count the frequency of each number. - Number 3 occurs 2 times. - Number 4 occurs 3 times. - Number 5 occurs 1 time. - Number 6 occurs 3 times. Now, we will check the different possibilities for x.

Determine the possible frequency of x in the data

There are four options to consider: 1. If the mode is only 5, x must be equal to 5 and increase the frequency of 5 to be the highest. In that case, frequency of 5 would be at least 4, which would not make sense since 4 and 6 already have a higher frequency. 2. If the mode is both 4 and 6, the frequency of x should be such that the frequency of 3, 4, and 6 all become equal. In this case, the frequency of x should be 0. Since x is a part of the data, this option is not possible. 3. If the mode is both 3 and 6, x must be equal to 3, and the frequency of 3 would be 3. 4. If the mode is 3, 4 or 6, x could be 3, 4, or 6 making their frequencies 3, 4, or 4, respectively.

Check which options are valid

After analyzing the cases for each option: - Option 1 is not valid – 5 cannot be a mode due to the higher frequency of 4 and 6. - Option 2 is not valid – since x is part of the data, the frequency of x cannot be 0. - Option 3 is valid – x can be 3, leaving both 3 and 6 as modes. - Option 4 is valid – x can be 3, 4, or 6 to make either of them the mode. So the answer is (3) both 3 and 6, and (4) 3, 4 or 6.

Key concepts

Mode

The mode in a set of data is the number that appears most frequently. It's one of the measures of central tendency, alongside the mean and median. Understanding the mode is crucial as it shows you the most common number or numbers in a data set.
In the given exercise, you have the data set: 6, 4, 3, 6, 4, 3, 4, 6, 5, and the variable \(x\). By determining which number appears most frequently, you identify the mode.

• If \(x\) is such that three different numbers (like 3, 4, and 6) can all be the mode at the same time depending on \(x\)'s value, it indicates how the mode can shift based on the data composition.
• When two numbers can both be the mode, it's called a "bimodal" distribution.
• A data set with more than two modes is "multimodal."

This exercise helps illustrate that the mode depends on the frequency of each number, and slight changes in data can result in different modes.

Frequency Distribution

Frequency distribution is a way to organize data to show how often each value occurs. By examining a frequency distribution, you can get a more detailed understanding of the data's characteristics.
In the original exercise, you can count how many times each number appears in the data set:

• 3 occurs twice
• 4 occurs three times
• 5 occurs once
• 6 occurs three times

Frequency distribution is a valuable part of data analysis. It helps identify the most common values in a data set, like the mode. By calculating these frequencies, you're setting the stage for further analysis like determining the mode or even looking for patterns.
It provides a structured way to assess data, especially when it comes with elements such as \(x\) that might change the outcome.

Data Analysis

Data analysis is the process of evaluating data using various techniques to uncover valuable information. For any data set, like the one in the exercise, data analysis allows one to draw meaningful conclusions.
In this case, after clarifying the frequency of each number, you interpreted how \(x\) changes potential modes. Data analysis involves:

• Counting frequencies to determine how many times each number appears, which helps highlight key statistics like the mode.
• Evaluating potential values of \(x\) and observing their impact on the data set.
• Assessing which options make sense based on numerical reasoning.

By thoroughly analyzing these observations, it's possible to see how \(x\) affects possible modes and ensure no logical inconsistencies occur.

Mathematical Reasoning

Mathematical reasoning involves logical thinking in order to solve problems, and in the exercise, it is key to understanding how the mode was determined.
In the context of the problem, mathematical reasoning requires you to:

• Consider what each number's frequency should be for a number to be the mode.
• Evaluate each potential value for \(x\) to understand how it changes data frequencies.
• Apply logic to weigh options and determine which modes are possible without inconsistencies.

For instance, determining that \(x = 3\) can make both 3 and 6 have the highest frequency shows mathematical reasoning at work.
It emphasizes approaching challenges methodically and applying trustful techniques to reach valid conclusions.