2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs

Graphical Representations of Data

Data visualization turns raw numbers into pictures you can actually read. Graphs like stemplots, line graphs, and bar graphs each reveal different things about a dataset: its shape, its center, how spread out it is, or how categories compare. Choosing the right graph depends on what kind of data you have and what you're trying to show.

Data Visualization and Statistical Analysis

Graphs make patterns visible that are hard to spot in a table of numbers. A well-chosen graph can quickly reveal trends, clusters, gaps, and outliers in your data. In statistics, this matters because seeing the data is often the first step before doing any calculations.

• Quantitative data (numbers with meaningful order and magnitude) calls for stemplots or line graphs
• Categorical data (groups or labels, like "red," "blue," "green") calls for bar graphs
• A good graph highlights the story in the data without distorting it

Stem-and-Leaf Graphs (Stemplots)

A stem-and-leaf graph (or stemplot) displays the distribution of a small quantitative dataset while preserving every individual data value. That's its big advantage over most other graphs: you can recover the original numbers from it.

Each data value gets split into two parts:

• The stem is the leading digit(s)
• The leaf is the trailing digit

For example, the value 47 has a stem of 4 and a leaf of 7. The value 153 might have a stem of 15 and a leaf of 3.

How to construct a stemplot:

1. Separate each data value into a stem and a leaf.
2. List all stems vertically in ascending order. Include stems that have no leaves so there are no gaps.
3. Write each leaf horizontally next to its stem, in increasing order from left to right.

What to look for when reading a stemplot:

• Shape: Is the distribution symmetric, skewed right (tail stretches toward higher values), or skewed left (tail stretches toward lower values)?
• Center: You can find the approximate median by counting to the middle value in the ordered data.
• Spread: The range is the maximum value minus the minimum value.
• Outliers: Look for data points that sit far away from the rest of the distribution.

Line Graphs (Frequency Polygons)

A line graph (sometimes called a frequency polygon) shows how data values are distributed across intervals. It plots frequency on the y-axis against data values on the x-axis, then connects the points with straight line segments.

How to construct a line graph:

1. Determine the range of data values and divide it into evenly spaced intervals (classes).
2. Count how many data points fall within each interval (the frequency).
3. Plot a point at the midpoint of each interval, at the height of that interval's frequency.
4. Connect the plotted points with line segments.

How to interpret a line graph:

• Shape: Just like stemplots, look for symmetry or skewness.
• Center: The approximate median corresponds to the x-value where about half the total area under the graph falls on each side.
• Spread: Identified by the minimum and maximum x-values where the data appears.
• Peaks: A high point on the graph shows where data values are most concentrated.

Line graphs are especially useful for comparing two distributions on the same axes, since overlapping lines are easier to read than overlapping bars.

Bar Graphs for Categorical Data

A bar graph displays the distribution of categorical data. Each category gets its own bar, and the bar's height shows either the frequency (count) or the relative frequency (proportion/percentage) for that category.

How to construct a bar graph:

1. List the categories on the x-axis with equal spacing between bars.
2. Calculate the frequency or proportion for each category.
3. Draw bars with heights matching the frequency or proportion. Bars should be the same width and should not touch (the gaps signal that the categories are distinct, not continuous).
4. Label both axes and include a title.

How to interpret a bar graph:

• Compare bar heights to see which categories are more or less common.
• The mode is the category with the tallest bar.
• The order of categories on the x-axis is arbitrary (unlike a histogram, where order matters), so you can rearrange them for clarity.