Problem 11 - Thinking Mathematically

Question

In Exercises 9–12, find the mean for the data items in the given frequency distribution. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Score } \\ \boldsymbol{x} \end{array} & \begin{array}{c} \text { Frequency } \\ \boldsymbol{f} \end{array} \\ \hline 1 & 1 \\ \hline 2 & 1 \\ \hline 3 & 2 \\ \hline 4 & 5 \\ \hline 5 & 7 \\ \hline 6 & 9 \\ \hline 7 & 8 \\ \hline 8 & 6 \\ \hline 9 & 4 \\ \hline 10 & 3 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The mean of the given frequency distribution is approximately 5.28.

Step 1: Calculate the total number of data items

Add all the frequencies together to get the total number of data items. So, \(f = 1 + 1 + 2 + 5 + 7 + 9 + 8 + 6 + 4 + 3 = 46.\)

Step 2: Calculate the sum of the product of each score and its frequency

Multiply each score by its frequency and sum the results. This would be \((1*1) + (2*1) + (3*2) + (4*5) + (5*7) + (6*9) + (7*8) + (8*6) + (9*4) + (10*3) = 243.\)

Step 3: Calculate the mean

The mean is calculated by dividing the sum obtained in step 2 by the total number of data items from step 1. Therefore, the mean \(μ\) is \(μ = 243/46 ≈ 5.28.\)

Key Concepts

Frequency Distribution

Understanding frequency distribution is pivotal in statistics as it helps organize raw data in a structured manner, making it easier to interpret and analyze. In a frequency distribution table, data points, or scores, are listed alongside their corresponding frequencies, which show how often each score appears. For instance, in the given exercise, the scores range from 1 to 10, and their frequencies vary. Each score has a unique frequency, indicating the number of occurrences of that particular score.

Frequency distributions help summarize large datasets by grouping them, allowing easier extraction of insights. By analyzing these tables, one can quickly gather information about the dataset, like the most common scores, or how data points are distributed across a range.

Data Analysis

Data analysis involves examining datasets to draw conclusions, identify patterns, and make informed decisions. By systematically analyzing frequency distribution data, you can uncover details about your dataset that are not immediately obvious. In the context of calculating the mean,

• You identify the total occurrence of all data (by summing up all frequencies), which gives a comprehensive view of the dataset size.
• Determine the weighted impact of each score by calculating a product of score and frequency, which helps determine the overall effect of each data point on the mean.

This structured approach enables you not only to compute averages but also to understand the dataset's shape and central tendency, which are key in statistical analysis.

Step-by-Step Solution

Breaking down a problem into manageable steps is a crucial skill in mathematics. It simplifies the problem-solving process and ensures accuracy and clarity. Here's how the solution for finding the mean from a frequency distribution was structured:

• **Step 1:** Calculate the total number of data items by adding all frequencies. This step defines the sample size and sets the stage for further analysis.
• **Step 2:** Multiply each score by its corresponding frequency to get a weighted total for each score. Then, sum these products to get the total weighted value of the dataset, which is pivotal for calculating the mean.
• **Step 3:** Divide the total weighted value by the total frequency. This step delivers the mean, providing a measure of central tendency.

Following these steps carefully leads you to the precise mathematical solution.

Mathematical Problem-Solving

Mathematical problem-solving is an essential skill, not only for academic endeavors but for real-life applications. It involves logical thinking, pattern recognition, and strategizing to tackle mathematical challenges. When calculating the mean in statistics, you use critical problem-solving skills:

• Decompose the problem into smaller, actionable parts, as seen in breaking down the calculation into manageable steps.
• Operate on data using multiplication and addition to find intermediate results, such as total frequency and sum of weighted scores.
• Apply formulas accurately to find a final solution, ensuring your understanding translates into correct calculations.

This strategic approach not only aids in solving mathematical problems but also hones your ability to approach and resolve various other analytical challenges.