Describing Data with Tables and Graphs

Organizing Qualitative Data

When data is collected from surveys or experiments, it must be organized to be useful for analysis. Unorganized data is called raw data. Organizing qualitative (categorical) data helps reveal patterns and supports further statistical analysis.

• Qualitative data refers to non-numeric information that describes categories or qualities (e.g., favorite day of the week, education level).

• Common ways to organize qualitative data include frequency tables, bar graphs, and pie charts.

Ways to Organize Data

1. Frequency Tables: Summarize how often each category occurs.

2. Bar Graphs: Visualize the frequency or relative frequency of categories.

3. Pie Charts: Show the proportion of each category as a sector of a circle.

Organizing Qualitative Data in Tables

A frequency distribution lists each category and the number of observations (frequency) in each. A relative frequency distribution lists each category with its relative frequency (proportion or percent of total observations).

• Relative frequency is calculated as:

• Relative frequencies should sum to 1 (or 100%).

Example: Frequency and Relative Frequency Distribution

Suppose a survey asks, "What is the best day of the week?" The responses are tallied in a frequency table, and relative frequencies are calculated for each day.

• Frequency distribution: Lists each day and the number of responses for that day.

• Relative frequency distribution: Lists each day and the proportion of responses for that day.

Constructing Bar Graphs

A bar graph is a visual representation of categorical data. Each category is represented by a bar, and the height of the bar corresponds to the frequency or relative frequency of the category.

• Categories are labeled on one axis (usually horizontal), and frequencies or relative frequencies on the other (usually vertical).

• Bars are of equal width and separated by spaces to emphasize that the data is categorical.

Example: Frequency and Relative Frequency Bar Graph

Using the best day of the week data, a bar graph can be constructed to display the frequency or relative frequency for each day.

Pareto Charts

A Pareto chart is a bar graph where categories are ordered from highest to lowest frequency or relative frequency. This helps identify the most common categories quickly.

• Pareto charts are useful for highlighting the most significant categories in a dataset.

Side-by-Side Bar Graphs

Side-by-side bar graphs are used to compare two or more groups across the same categories. For example, comparing educational attainment by gender or nativity.

• Relative frequencies are preferred for comparison, especially if group sizes differ.

Example: Educational Attainment by Gender

The following table summarizes educational attainment for males and females aged 25 and older:

To compare educational attainment by gender, construct a side-by-side bar graph using the relative frequencies for each category.

Constructing Pie Charts

A pie chart is a circular graph divided into sectors, where each sector represents a category. The area of each sector is proportional to the frequency or relative frequency of the category.

• Pie charts are useful for showing the proportion of each category in the whole dataset.

Example: Pie Chart of Best Day of the Week

Using the best day of the week data, a pie chart can be constructed to visually display the proportion of responses for each day.