18. Mean problems

18. Mean problems - step by step solution procedure and detailed explanation for each problem.

Home page of Arithmetic Problems 

Below are 10 mean (arithmetic average) problems, each part of arithmetic, with step-by-step solutions and detailed explanations. The mean is calculated by summing all values in a dataset and dividing by the number of values. These problems range from basic to more complex to provide a comprehensive understanding.
Problem 1: Basic Mean of a Small Dataset
Find the mean of the dataset: 4, 6, 8, 10.
Step-by-Step Solution:
  1. List the numbers: The dataset is 4, 6, 8, 10.
  2. Count the number of values: There are 4 values.
  3. Sum the values: 4 + 6 + 8 + 10 = 28.
  4. Divide the sum by the number of values: 28 ÷ 4 = 7.
  5. State the mean: The mean is 7.
Detailed Explanation: The mean represents the average value of a dataset. We add all numbers to get the total (28) and divide by the count of numbers (4). This balances the dataset, giving a central value. Here, 7 is the mean, indicating the average of the numbers.
Answer: Mean = 7
Problem 2: Mean with Negative Numbers
Find the mean of the dataset: -3, 5, -1, 7.
Step-by-Step Solution:
  1. List the numbers: The dataset is -3, 5, -1, 7.
  2. Count the number of values: There are 4 values.
  3. Sum the values: (-3) + 5 + (-1) + 7 = -3 + 5 - 1 + 7 = 8.
  4. Divide the sum by the number of values: 8 ÷ 4 = 2.
  5. State the mean: The mean is 2.
Detailed Explanation: Negative numbers are added like positive numbers, respecting their signs. The sum (8) divided by the count (4) gives the mean. Negative values can offset positive ones, but here, the positive numbers dominate, resulting in a positive mean of 2.
Answer: Mean = 2
Problem 3: Mean with Decimals
Find the mean of the dataset: 2.5, 3.7, 4.1, 1.9.
Step-by-Step Solution:
  1. List the numbers: The dataset is 2.5, 3.7, 4.1, 1.9.
  2. Count the number of values: There are 4 values.
  3. Sum the values: 2.5 + 3.7 + 4.1 + 1.9 = 12.2.
  4. Divide the sum by the number of values: 12.2 ÷ 4 = 3.05.
  5. State the mean: The mean is 3.05.
Detailed Explanation: Decimals require careful addition to ensure accuracy. The sum (12.2) is divided by the number of values (4), yielding 3.05. This precise calculation is typical when working with measurements or data involving fractions, ensuring the mean reflects the average accurately.
Answer: Mean = 3.05
Problem 4: Mean of a Larger Dataset
Find the mean of the dataset: 12, 15, 18, 20, 25, 30.
Step-by-Step Solution:
  1. List the numbers: The dataset is 12, 15, 18, 20, 25, 30.
  2. Count the number of values: There are 6 values.
  3. Sum the values: 12 + 15 + 18 + 20 + 25 + 30 = 120.
  4. Divide the sum by the number of values: 120 ÷ 6 = 20.
  5. State the mean: The mean is 20.
Detailed Explanation: With a larger dataset, summing accurately is crucial. The total (120) divided by the count (6) gives 20, a whole number, indicating the dataset is balanced around this average. The mean provides a central tendency, useful for summarizing larger datasets.
Answer: Mean = 20
Problem 5: Mean in a Real-World Context
The daily temperatures (°C) over 5 days are: 22, 25, 20, 23, 25. Find the mean temperature.
Step-by-Step Solution:
  1. List the numbers: The dataset is 22, 25, 20, 23, 25.
  2. Count the number of values: There are 5 values.
  3. Sum the values: 22 + 25 + 20 + 23 + 25 = 115.
  4. Divide the sum by the number of values: 115 ÷ 5 = 23.
  5. State the mean: The mean temperature is 23°C.
Detailed Explanation: In real-world contexts, the mean provides a typical value (e.g., average temperature). Summing the temperatures (115) and dividing by the number of days (5) gives 23°C, representing the average weather condition over the period, useful for analysis or forecasting.
Answer: Mean = 23°C
Problem 6: Mean with Repeated Values
Find the mean of the dataset: 3, 3, 4, 5, 5, 5.
Step-by-Step Solution:
  1. List the numbers: The dataset is 3, 3, 4, 5, 5, 5.
  2. Count the number of values: There are 6 values.
  3. Sum the values: 3 + 3 + 4 + 5 + 5 + 5 = 25.
  4. Divide the sum by the number of values: 25 ÷ 6 ≈ 4.1667.
  5. State the mean: The mean is approximately 4.17 (rounded to two decimal places).
Detailed Explanation: Repeated values are counted individually in the sum. The total (25) divided by the count (6) results in a non-integer (4.1667). Rounding to two decimal places (4.17) is common for readability. This shows how the mean balances repeated values.
Answer: Mean ≈ 4.17
Problem 7: Mean from a Frequency Table
A frequency table shows the number of goals scored in games: 0 goals (3 games), 1 goal (4 games), 2 goals (2 games), 3 goals (1 game). Find the mean.
Step-by-Step Solution:
  1. Interpret the frequency table:
    • 0 goals: 3 games
    • 1 goal: 4 games
    • 2 goals: 2 games
    • 3 goals: 1 game
  2. Count the total number of games: 3 + 4 + 2 + 1 = 10 games.
  3. Calculate the total goals: (0 × 3) + (1 × 4) + (2 × 2) + (3 × 1) = 0 + 4 + 4 + 3 = 11.
  4. Divide the total goals by the number of games: 11 ÷ 10 = 1.1.
  5. State the mean: The mean is 1.1 goals per game.
Detailed Explanation: Frequency tables simplify mean calculations by grouping data. Multiply each value by its frequency to get the total sum (11 goals), then divide by the total count (10 games). The result (1.1) indicates the average goals per game, useful for summarizing performance.
Answer: Mean = 1.1 goals per game
Problem 8: Mean with Missing Value
The mean of five numbers is 10. Four of the numbers are 8, 12, 9, and 11. Find the fifth number.
Step-by-Step Solution:
  1. Use the mean formula: Mean = (Sum of values) ÷ (Number of values).
  2. Given mean and count: Mean = 10, number of values = 5.
  3. Calculate the total sum: 10 × 5 = 50.
  4. Sum the known values: 8 + 12 + 9 + 11 = 40.
  5. Find the fifth number: 50 - 40 = 10.
  6. State the fifth number: The fifth number is 10.
Detailed Explanation: The mean gives the total sum when multiplied by the number of values (50). Subtracting the sum of known values (40) leaves the missing value (10). This approach is useful when working backward to find a missing data point in a dataset.
Answer: Fifth number = 10
Problem 9: Mean of a Grouped Frequency Table
A grouped frequency table shows ages: 0–10 years (3 people), 11–20 years (5 people), 21–30 years (2 people). Find the mean age, using the midpoint of each interval.
Step-by-Step Solution:
  1. Determine the midpoints:
    • 0–10: (0 + 10) ÷ 2 = 5
    • 11–20: (11 + 20) ÷ 2 = 15.5
    • 21–30: (21 + 30) ÷ 2 = 25.5
  2. List frequencies:
    • 5: 3 people
    • 15.5: 5 people
    • 25.5: 2 people
  3. Count total people: 3 + 5 + 2 = 10.
  4. Calculate total age: (5 × 3) + (15.5 × 5) + (25.5 × 2) = 15 + 77.5 + 51 = 143.5.
  5. Divide total age by total people: 143.5 ÷ 10 = 14.35.
  6. State the mean: The mean age is 14.35 years.
Detailed Explanation: Grouped data uses interval midpoints as representative values. Multiply each midpoint by its frequency, sum the results (143.5), and divide by the total count (10). The mean (14.35) is an estimate, as exact ages aren’t known, but it’s a standard method for grouped data.
Answer: Mean = 14.35 years
Problem 10: Mean with Outlier
Find the mean of the dataset: 5, 6, 7, 8, 100.
Step-by-Step Solution:
  1. List the numbers: The dataset is 5, 6, 7, 8, 100.
  2. Count the number of values: There are 5 values.
  3. Sum the values: 5 + 6 + 7 + 8 + 100 = 126.
  4. Divide the sum by the number of values: 126 ÷ 5 = 25.2.
  5. State the mean: The mean is 25.2.
Detailed Explanation: Outliers, like 100, significantly affect the mean. The sum (126) is heavily influenced by 100, pulling the mean (25.2) higher than most values. This shows the mean’s sensitivity to extreme values, unlike the median, which would be less affected.
Answer: Mean = 25.2
Summary
These 10 problems cover various scenarios: basic datasets, negative numbers, decimals, larger datasets, real-world contexts, repeated values, frequency tables, missing values, grouped data, and outliers. Each problem follows the same process—sum the values and divide by the count—but requires careful handling of specific conditions (e.g., midpoints for grouped data or outliers). The mean is a powerful measure of central tendency but can be skewed by extreme values, making context important.

 

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