Question

The data in Exercises \(1-3\) represent different ways to classify a group of 100 students in a statistics class. Construct a bar chart and pie chart to describe each set of data. $$\begin{array}{l|c}\text { Status } & \text { Frequency } \\\\\hline \text { Freshman } & 32 \\\\\text { Sophomore } & 34 \\\\\text { Junior } & 17 \\\\\text { Senior } & 9 \\\\\text { Grad Student } & 8\end{array}$$

Short answer

Answer: 17%

Step-by-step solution

(Step 1: Creating the categories and calculating percentages)

(First, list the five classification categories: Freshman, Sophomore, Junior, Senior, and Grad Student. Next, calculate the percentage of each category within the total number of students (100). Using the frequency count provided for each category, divide this frequency by the total number of students and multiply by 100 to find the percentage. Freshman: \(32/100 * 100 = 32\%\) Sophomore: \(34/100 * 100 = 34\%\) Junior: \(17/100 * 100 = 17\%\) Senior: \(9/100 * 100 = 9\%\) Grad Student: \(8/100 * 100 = 8\%\))

(Step 2: Constructing the Bar Chart)

(A bar chart should represent each classification category as a bar, with the height of the bar corresponding to the frequency of students in that category. The x-axis should have the five classification categories (Freshman, Sophomore, Junior, Senior, and Grad Student), and the y-axis should indicate the frequency (0, 10, 20, 30, 40). Create a bar for each category and label the height with its corresponding frequency number. Also, clearly label the axes.)

(Step 3: Constructing the Pie Chart)

(A pie chart represents each classification category as a 'slice' of a whole pie. To create the pie chart, use the percentages calculated in Step 1. Draw a circle and divide it into five sections (or 'slices') corresponding to the five student classifications. The size of each slice should be representative of its percentage from Step 1. Label each slice with its classification category and the corresponding percentage: Freshman - 32% Sophomore - 34% Junior - 17% Senior - 9% Grad Student - 8%)

Key concepts

Constructing Bar Charts

Visualizing the distribution of data is a fundamental aspect of statistics, and bar charts are one of the most common methods used. A bar chart represents categorical data with rectangular bars where the length of each bar is proportional to the value it represents. In our textbook exercise, we began by categorizing the data: Freshman, Sophomore, Junior, Senior, and Grad Student. These categories formed the basis of the x-axis.

Next, we determined the height of each bar by using the frequency of students in each category. For instance, the Freshman category had 32 students, so the height of the Freshman bar was set to represent this frequency. It's crucial when constructing a bar chart to keep the bars uniformly spaced and to have a clear label for each category along the x-axis. Additionally, the y-axis should be appropriately labeled with a scale that fits the frequency of the data, in this case from 0 to at least 34 to accommodate the Sophomore frequency which is the highest value. Labeling each bar with its frequency enhances the chart's readability.

Constructing Pie Charts

A pie chart is a circular visual representation of data that illustrates numerical proportions as slices of a pie. Each slice's size is representative of the contribution that category makes to the whole. To construct a pie chart based on our exercise, we first calculated the percentage of the total for each student classification.

For example, Freshmen make up 32% of all students. With these percentages, we set about dividing a circle into slices that correspond to these values. It's important to make sure each slice accurately reflects the calculated percentage, so a protractor or a pie chart tool can be invaluable for accuracy. After dividing the pie, labeling each slice with both the category and its percentage ensures that the information is easily understandable. A pie chart enables viewers to quickly grasp the relative weights of the parts to the whole, making this an effective tool for visually summarizing proportional data.

Categorical Data Analysis

Categorical data is qualitative and is grouped into non-numeric categories or groups. Categorical data analysis is the process of summarizing, describing, and making inferences based on the patterns and frequencies of these categories. In the context of the given exercise where students are categorized by academic status, we performed a categorical data analysis by first enumerating how many students fall under each category and then assessing the relative frequencies of these categories.

Tools like bar charts and pie charts are particularly suitable for this type of analysis as they visually emphasize the differences between categories. When analyzing the constructed visuals, one could interpret the majority or minority groups and assess trends or patterns, such as whether underclassmen or upperclassmen are more prevalent in a statistics class. This analysis forms the basis for making predictions or informed decisions related to the data at hand.

Frequency Distribution

A frequency distribution lists each category of data and the number of occurrences for each category. In statistical analysis, it serves as a basic summary of the data that allows us to see quickly where the values lay within a dataset. The exercise involved creating a frequency distribution for the number of students in each class status. Once we had our categories—Freshman, Sophomore, Junior, Senior, and Grad Student—we then listed out how many students were in each category to create the frequency distribution table.

Understanding the frequency distribution is key before creating any visual representation as it provides the raw numbers we need to construct our charts. For example, knowing there were 34 sophomores informs us that this category will have the largest representation on our bar chart and pie chart. Frequency distributions are especially helpful when comparing different datasets or looking for patterns within a single dataset.