Representing a Categorical Variable with Tables Chapter Notes | AP Statistics - Grade 9 PDF Download

Introduction to Organizing Data

Raw data can be overwhelming and difficult to interpret without proper organization. Statistics provides tools to structure and analyze data effectively. The process often begins with organizing data into tables, followed by creating visual representations like graphs to uncover patterns and insights. For categorical variables, tables are a foundational step, but graphical displays like bar graphs and pie charts offer a more intuitive way to visualize trends and distributions.

Frequency Tables

Imagine conducting a survey in an AP Statistics class to assess how stressful students find their academic workload, with response options of "very stressful," "somewhat stressful," or "not stressful." After collecting 30 responses, how do we make sense of the data?
A frequency table is an effective way to summarize this data. It lists each category and counts the number of responses in each one. For example, a frequency table for qualitative data, such as stress levels, displays categories in one column and their corresponding counts in another. Tallies (e.g., ||||) can help count responses during data collection.
In this case, the variable "stress level" is an ordinal variable because the categories have a natural order (very > somewhat > not stressful), though the differences between categories aren't precisely measurable. The table below illustrates a hypothetical frequency table for the survey:
The sum of frequencies always equals the total number of responses, in this case, 30.

Relative Frequency Tables

To gain deeper insights, we can extend frequency tables by calculating relative frequencies and percentages. A relative frequency is computed by dividing the frequency of a category by the total number of responses. The percentage is then found by multiplying the relative frequency by 100.

• Relative Frequency = Frequency of a category / Total frequency
• Percentage = Relative Frequency × 100

A relative frequency table replaces counts with proportions or percentages. For example, using the stress survey data, if 10 students reported "very stressful," the relative frequency is 10/30 = 0.333, or 33.3%. The table might look like this:
From this, we can conclude that 33.3% of students find their workload very stressful, and combining "very" and "somewhat" stressful categories, 80% (33.3% + 46.7%) report at least some stress. The sum of relative frequencies should equal 1.00, and percentages should total 100%, though rounding may cause slight variations.

Example: Frequency and Relative Frequency Tables

Consider a dataset: 2, 3, 3, 5, 5, 5, 7, 7, 9. A frequency table counts how often each value appears:
A relative frequency table expresses these counts as proportions:
These tables help identify patterns, such as the value 5 being the most frequent, and are essential for summarizing data.

Key Terms

• Bar Graph: A bar graph uses bars to represent different categories, with each bar’s height or length indicating the frequency or value of that category. It’s an effective way to compare categories and identify trends in categorical data.
• Ordinal Variable: An ordinal variable is a categorical variable with a defined order among its categories, but the differences between categories aren’t measurable. For example, stress levels (very, somewhat, none) can be ranked but not quantified precisely.
• Pie Chart: A pie chart is a circular graph divided into slices, where each slice represents a category’s proportion of the total. It’s useful for visualizing how categories contribute to the whole dataset.