11-1 Representing Data

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Before creating any graph, you must identify the type of univariate data (data tracking exactly one variable) you are working with. Data falls into two main categories:

• Categorical Data (Qualitative): Deals with descriptions, names, or labels. It categorizes items but cannot be meaningfully added, subtracted, or averaged.
• Quantitative Data (Numerical): Deals with measurable quantities. These are true numbers where mathematical operations (like finding a mean or median) make logical sense.

Examples

• Categorical Data: Eye color (brown, blue, green), favorite sport, or grade level.
• Quantitative Data: Height in inches (62, 68, 71), number of pets (0, 2, 5), or test scores.
• Disguised Categories: Sometimes categorical data uses numbers as labels, like zip codes or sports jersey numbers. You cannot meaningfully calculate an "average zip code," which proves it is categorical, not quantitative.

Explanation

The key to classifying variables is asking yourself: "Does it make sense to do math with these answers?" If you ask 10 people their favorite pet, you cannot calculate the "average pet"—you can only find the mode (the most popular category). Understanding whether your data is a descriptive label or a measurable quantity is the crucial first step because it completely dictates which type of graph you are allowed to use.

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A dot plot displays the frequency of quantitative data along a number line. It is ideal for smaller data sets.

To construct a dot plot, order the data from least to greatest and place exactly one dot directly above its value on the number line. When values repeat, the dots must be stacked perfectly vertically.
Dot plots allow you to easily identify the shape of the data, including:

• Peaks: The value(s) with the tallest stack of dots (the mode).
• Clusters: Groups of data points gathered closely together.
• Gaps: Empty intervals on the number line.
• Outliers: Individual values that stand far away from the rest of the data.

Examples

• Constructing and Stacking: Data set is {7, 3, 5, 3, 7, 5, 3}. Order it first: 3, 3, 3, 5, 5, 7, 7. Place three dots above the 3, two dots above the 5, and two dots above the 7.
• Stacking Error: A student plots the value 4 twice but places the second dot slightly to the right instead of directly above the first. This makes it look like a new data value near 4.5 exists. Dots must be stacked straight up.
• Analyzing Features: Looking at a dot plot of "pets owned," you see a peak at 1, a cluster from 0 to 2, a gap at 4, and an outlier at 7.

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A bar graph is used specifically for categorical data. It uses rectangular bars separated by gaps to represent the frequency of each category.
A valid bar graph requires a title, labeled axes, and a properly scaled vertical axis. Crucially, the vertical axis (y-axis) showing the frequency MUST start at 0. Starting the axis at a number greater than 0 visually distorts the proportions of the bars.

Examples

• Standard Bar Graph: A horizontal bar graph displaying favorite colors: Red (12 students), Blue (18 students), Green (8 students). The longer the bar, the more popular the choice. The categories are separated by gaps.
• The Zero-Axis Distortion: A graph shows votes for two candidates: Candidate A (52 votes) and Candidate B (48 votes). The true difference is very small. If the vertical axis starts at 40 instead of 0, Candidate A's bar will be 12 units tall and Candidate B's bar will be 8 units tall. This visually makes Candidate A look 50% more popular, misleading the reader.

Explanation

Bar charts are fantastic for comparing discrete groups, which is why there are visible gaps between the bars—the gaps signal that the categories don't bleed into one another. However, you must be a critical reader of graphs! The human eye naturally compares the total height of the bars. If a graph cuts off the bottom by starting the y-axis at a number like 50 instead of 0, it artificially stretches small differences to look like massive gaps.