20. Mode problems

20. Mode problems - step by step solution procedure and detailed explanation for each problem.

Home page of Arithmetic Problems 

Below are 10 mode problems, each part of arithmetic, with step-by-step solutions and detailed explanations. The mode is the value that appears most frequently in a dataset. These problems range from basic to slightly more complex to provide a comprehensive understanding.
Problem 1: Basic Mode of a Small Dataset
Find the mode of the dataset: 3, 5, 2, 3, 7.
Step-by-Step Solution:
  1. List the numbers: The dataset is 3, 5, 2, 3, 7.
  2. Count the frequency of each number:
    • 2: appears 1 time
    • 3: appears 2 times
    • 5: appears 1 time
    • 7: appears 1 time
  3. Identify the highest frequency: The number 3 appears 2 times, which is the highest.
  4. Determine the mode: The mode is 3.
Detailed Explanation: The mode represents the most frequent value in a dataset. Here, we tally each number’s occurrences. Since 3 appears twice, more than any other number, it is the mode. If no number repeats, there would be no mode, but in this case, 3 clearly stands out.
Answer: Mode = 3
Problem 2: Mode with No Repeated Values
Find the mode of the dataset: 1, 4, 6, 8, 9.
Step-by-Step Solution:
  1. List the numbers: The dataset is 1, 4, 6, 8, 9.
  2. Count the frequency of each number:
    • 1: appears 1 time
    • 4: appears 1 time
    • 6: appears 1 time
    • 8: appears 1 time
    • 9: appears 1 time
  3. Identify the highest frequency: Each number appears exactly once (frequency = 1).
  4. Determine the mode: Since no number repeats, there is no mode.
Detailed Explanation: For a dataset to have a mode, at least one value must appear more frequently than others. Here, every number appears exactly once, so no value stands out as the most frequent. In such cases, we conclude that the dataset has no mode.
Answer: No mode
Problem 3: Bimodal Dataset
Find the mode of the dataset: 2, 4, 4, 6, 6, 8.
Step-by-Step Solution:
  1. List the numbers: The dataset is 2, 4, 4, 6, 6, 8.
  2. Count the frequency of each number:
    • 2: appears 1 time
    • 4: appears 2 times
    • 6: appears 2 times
    • 8: appears 1 time
  3. Identify the highest frequency: Both 4 and 6 appear 2 times, which is the highest.
  4. Determine the mode: The modes are 4 and 6 (bimodal).
Detailed Explanation: A dataset can have more than one mode if multiple values share the highest frequency. Here, both 4 and 6 appear twice, more than any other numbers. This makes the dataset bimodal, with modes 4 and 6. Note that all values with the highest frequency are included as modes.
Answer: Modes = 4, 6
Problem 4: Mode of a Larger Dataset
Find the mode of the dataset: 7, 3, 5, 7, 4, 7, 2, 5.
Step-by-Step Solution:
  1. List the numbers: The dataset is 7, 3, 5, 7, 4, 7, 2, 5.
  2. Count the frequency of each number:
    • 2: appears 1 time
    • 3: appears 1 time
    • 4: appears 1 time
    • 5: appears 2 times
    • 7: appears 3 times
  3. Identify the highest frequency: The number 7 appears 3 times, which is the highest.
  4. Determine the mode: The mode is 7.
Detailed Explanation: With a larger dataset, organizing the counts systematically is key. We can sort or tally the numbers to ensure accuracy. Here, 7 appears three times, more than any other number, making it the mode. The other numbers appear less frequently, so there’s only one mode.
Answer: Mode = 7
Problem 5: Mode with Negative Numbers
Find the mode of the dataset: -2, 3, -2, 5, 1, 3.
Step-by-Step Solution:
  1. List the numbers: The dataset is -2, 3, -2, 5, 1, 3.
  2. Count the frequency of each number:
    • -2: appears 2 times
    • 1: appears 1 time
    • 3: appears 2 times
    • 5: appears 1 time
  3. Identify the highest frequency: Both -2 and 3 appear 2 times, which is the highest.
  4. Determine the mode: The modes are -2 and 3.
Detailed Explanation: Negative numbers are treated the same as positive numbers when finding the mode. The key is to count occurrences accurately. Here, both -2 and 3 appear twice, sharing the highest frequency, so the dataset is bimodal with modes -2 and 3.
Answer: Modes = -2, 3
Problem 6: Mode in a Frequency Table
A frequency table shows the number of books read by students: 0 books (2 students), 1 book (5 students), 2 books (3 students), 3 books (1 student). Find the mode.
Step-by-Step Solution:
  1. Interpret the frequency table:
    • 0 books: 2 students
    • 1 book: 5 students
    • 2 books: 3 students
    • 3 books: 1 student
  2. Identify the highest frequency: The value 1 book has 5 students, which is the highest.
  3. Determine the mode: The mode is 1 book.
Detailed Explanation: A frequency table simplifies mode calculation by directly providing the counts. The mode is the value with the highest frequency. Here, 1 book has the most occurrences (5 students), so it’s the mode. This method is efficient for summarized data.
Answer: Mode = 1 book
Problem 7: Mode with Decimals
Find the mode of the dataset: 2.5, 3.1, 2.5, 4.2, 3.1, 5.0.
Step-by-Step Solution:
  1. List the numbers: The dataset is 2.5, 3.1, 2.5, 4.2, 3.1, 5.0.
  2. Count the frequency of each number:
    • 2.5: appears 2 times
    • 3.1: appears 2 times
    • 4.2: appears 1 time
    • 5.0: appears 1 time
  3. Identify the highest frequency: Both 2.5 and 3.1 appear 2 times, which is the highest.
  4. Determine the mode: The modes are 2.5 and 3.1.
Detailed Explanation: Decimals are handled the same way as integers when finding the mode. Each value’s frequency is counted, and the highest frequency determines the mode. Here, 2.5 and 3.1 both appear twice, making the dataset bimodal. Precision in identifying identical values is crucial with decimals.
Answer: Modes = 2.5, 3.1
Problem 8: Mode in a Real-world Context
The shoe sizes of a group of people are: 7, 8, 6, 7, 9, 7, 8. Find the mode.
Step-by-Step Solution:
  1. List the numbers: The dataset is 7, 8, 6, 7, 9, 7, 8.
  2. Count the frequency of each number:
    • 6: appears 1 time
    • 7: appears 3 times
    • 8: appears 2 times
    • 9: appears 1 time
  3. Identify the highest frequency: The number 7 appears 3 times, which is the highest.
  4. Determine the mode: The mode is 7.
Detailed Explanation: In real-world contexts, the mode often has practical significance (e.g., the most common shoe size for inventory). Here, we count each shoe size’s occurrences. The size 7 appears three times, more than any other, so it’s the mode. This could inform decisions like stocking more size 7 shoes.
Answer: Mode = 7
Problem 9: Mode with All Equal Frequencies
Find the mode of the dataset: 10, 20, 30, 40.
Step-by-Step Solution:
  1. List the numbers: The dataset is 10, 20, 30, 40.
  2. Count the frequency of each number:
    • 10: appears 1 time
    • 20: appears 1 time
    • 30: appears 1 time
    • 40: appears 1 time
  3. Identify the highest frequency: Each number appears exactly once (frequency = 1).
  4. Determine the mode: Since no number repeats, there is no mode.
Detailed Explanation: When all values in a dataset have the same frequency, no value is more frequent than others, so there is no mode. This is similar to Problem 2 but reinforces that the absence of repetition means no mode exists, regardless of dataset size.
Answer: No mode
Problem 10: Mode in a Grouped Frequency Table
A grouped frequency table shows test scores: 0–10 (2 students), 11–20 (4 students), 21–30 (4 students), 31–40 (1 student). Find the mode.
Step-by-Step Solution:
  1. Interpret the frequency table:
    • 0–10: 2 students
    • 11–20: 4 students
    • 21–30: 4 students
    • 31–40: 1 student
  2. Identify the highest frequency: The intervals 11–20 and 21–30 both have 4 students, which is the highest.
  3. Determine the mode: The modal intervals are 11–20 and 21–30.
Detailed Explanation: In grouped data, the mode is the interval with the highest frequency. Here, both the 11–20 and 21–30 intervals have 4 students, the highest frequency, so they are the modal intervals. Unlike ungrouped data, we report the interval(s) rather than a specific value, as individual scores aren’t provided.
Answer: Modal intervals = 11–20, 21–30
Summary
These 10 problems cover various scenarios: single mode, no mode, bimodal datasets, negative numbers, decimals, frequency tables, real-world contexts, and grouped data. Each problem emphasizes counting frequencies and identifying the highest, with clear steps to ensure accuracy. The mode’s simplicity makes it a powerful tool for identifying common values, but careful counting is essential, especially in complex datasets.

 

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