Problem 8 - Thinking Mathematically

Question

A random sample of 30 male college students is selected. Each student is asked his height (to the nearest inch). The heights are as follows: $$ \begin{aligned} &72,70,68,72,71,71,71,69,73,71,73,75,66,67,75,74,73,71 \text {, } \\ &72,67,72,68,67,71,73,71,72,70,73,70 . \end{aligned} $$ Construct a frequency distribution for the data.

Step-by-Step Solution

Answer

The frequency distribution for the given data would be a table consisting of the heights (66 to 75 inches) in one column, and the frequency of occurrence of each height in another. For example, height 66 inches appears 1 time, 67 inches appears 3 times, 68 inches appears 2 times, etc., up to height 75 inches, which appears 2 times.

Step 1: Organize the Data

Start by arranging the data in ascending order. This will help to identify the various values and their frequencies more accurately.

Step 2: Identify Unique Values

Next, identify the unique values within the dataset. These values correspond to the heights of the male college students and they will form the 'x' (Observation) column in the frequency distribution table.

Step 3: Calculate Frequencies

For each unique height, determine how many times it occurs in the dataset. This count is known as the frequency. Record it in the 'frequency' column relating to the corresponding height.

Step 4: Construct the Frequency Distribution Table

Finally, construct a frequency distribution table with two columns: 'Height' and 'Frequency'. Put the unique heights from step 2 in the 'Height' column, and their corresponding frequencies from step 3 in the 'Frequency' column.

Key Concepts

Data Organization

Before diving into analyzing any data, it is crucial to organize it effectively. Think of data organization as tidying up a messy room. It makes everything easier to find and understand. For this exercise, we have a list of heights from male college students that we need to sort out.
Start by arranging these heights in ascending order. This means listing the heights from the smallest to the largest. Doing so helps us to clearly view the data and recognize patterns or anomalies. This orderly arrangement provides a structured picture of the data, making the next steps in the process much simpler and more accurate.

Unique Values

Once the data is organized, the next step is to identify the unique values. But what are unique values? Simply put, they are the different heights that appear in the data list. Each unique value is distinct and only appears once in the 'Unique Values' list.
Spotting these unique values is important because they form the basis of our frequency distribution table. They will be the 'x' or observation column in the table. So take a careful look at the ordered list, identify the different heights, and jot them down separately.

Calculate Frequencies

After identifying the unique values, we move on to calculating frequencies. Frequency, in this context, refers to how many times a specific height occurs in the data set. It's essentially counting the repetitions of each unique height.
Make sure to go through the organized data carefully, tally how often each unique height appears, and write this count next to the corresponding unique value. This process requires focus as it lays the foundation for accurate data representation in the frequency distribution table.

Frequency Distribution Table

With our list of unique values and their frequencies, we can now construct the frequency distribution table. This table is a powerful way to represent organized data, helping us understand the distribution of different heights.
To create this table, list the unique heights in one column, usually labeled "Height," and the frequency of each height in another column, labeled "Frequency."
- Example:
- **Height**: Each of the unique values from the data.
- **Frequency**: The number of times each unique height appears.
This visual representation makes it easier to see which heights are most common and how the data is spread out. Creating a frequency distribution table not only summarizes the data concisely but also makes it accessible for further statistical analysis.