Problem 10 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The mean score of the data items in the given frequency distribution is approximately 4.13.

Step 1: List down the Score and Frequency

Extract the given scores (\(x\)) and their respective frequencies (\(f\)). Thus, \(x = [1, 2, 3, 4, 5, 6, 7]\) and \(f = [2, 4, 5, 7, 6, 4, 3]\).

Step 2: Calculate the product of each score and frequency

Multiply each score with its corresponding frequency. This gives: \(x \cdot f = [1*2, 2*4, 3*5, 4*7, 5*6, 6*4, 7*3]\) or \(x \cdot f = [2, 8, 15, 28, 30, 24, 21]\).

Step 3: Calculate the sum of the products

Add all the products from step 2. The sum is \(2 + 8 + 15 + 28 + 30 + 24 + 21 = 128\).

Step 4: Calculate the sum of frequencies

Add all the frequency values. The sum is \(2 + 4 + 5 + 7 + 6 + 4 + 3 = 31\).

Step 5: Calculate the mean

The mean is given by the formula: \[mean = \frac{{\text{{sum of products}}}}{{\text{{sum of frequencies}}}}\] Thus, the mean is \(\frac{128}{31} \approx 4.13\).

Key Concepts

Frequency Distribution

When working with data, organizing it into a frequency distribution helps in understanding how often each data point occurs. This approach is particularly useful when dealing with large datasets. A frequency distribution presents data in a tabular format using two columns:

• Score - Represents the value or category of the data.
• Frequency - Indicates the number of times the score occurs.

In our example, each score from 1 to 7 is recorded along with its frequency, such as a score of 1 appearing 2 times and a score of 4 appearing 7 times. This structured layout allows for easy identification of patterns and trends within data. Furthermore, frequency distributions simplify further statistical analysis, such as calculating the mean or other measures of central tendency.

Mathematical Problem Solving

Solving problems mathematically involves breaking down a complex challenge into manageable steps. This methodology not only aids in understanding the problem better but also helps in arriving at accurate solutions.
In the context of our exercise, the goal was to find the mean using a defined series of actions:

• List and organize the data.
• Apply basic arithmetic operations such as multiplication and addition.
• Use a formula to derive the final result (mean).

By using these steps, complex data becomes manageable and understandable, enhancing one's problem-solving capabilities and fostering a deeper appreciation for mathematical applications in data analysis.

Data Analysis

Data analysis involves interpreting and deriving meaningful insights from raw data. With a frequency distribution, it becomes straightforward to not only identify but quantify how often each score occurs. This is fundamental in statistical analysis, where the focus is often on summarizing data and identifying patterns.
In our case, after calculating the products of each score and its frequency, we sum these products to prepare for the calculation of the mean—a step which enables us to ascertain an average measure of the dataset.
Analyzing frequency data will not only tell us about the central tendency (as seen in the mean calculation), but also about the variability or spread, should we require it.

Step by Step Solution

Following a step-by-step solution ensures clarity and reduces the risk of errors when performing calculations.
In our example, the solution was systematically approached through:

• Identification of individual scores and their frequencies.
• Calculation of the product of each score and its frequency.
• Addition of all products to find the total.
• Summation of all frequencies.
• Final computation of the mean using the formula \( \frac{\text{sum of products}}{\text{sum of frequencies}} \).

Each step builds upon the previous one, ensuring that by the time you compute the mean, each vital part of the process has been meticulously addressed. For students, internalizing this process enhances their ability to tackle similar mathematical challenges with confidence.