Introduction to Graphical Representations

Welcome to one of the most visual parts of AP Statistics! In previous lessons, we looked at categorical data (like eye color or favorite pizza toppings). Now, we are shifting our focus to quantitative variables. These are variables where the data are numbers that represent counts or measurements, such as heights, exam scores, or the number of apps on your phone.

Why do we graph this data? Because a long list of numbers is hard to read! Graphs allow us to see the "story" the data is telling us—where the values cluster, how spread out they are, and if there are any weird surprises. In this chapter, we will master the tools used to turn raw numbers into clear pictures.

1. Dotplots

The dotplot is the simplest way to graph quantitative data. It is perfect for small datasets.

How to Construct a Dotplot:

  1. Draw a horizontal axis (a number line) and label it with the variable name.
  2. Scale the axis based on the minimum and maximum values in your data.
  3. For every data point, place a dot above its corresponding value on the axis.
  4. If a value repeats, stack the dots vertically.

Example: If we measure the number of pets owned by \(5\) students and get the data \( \{0, 1, 1, 2, 4\} \), we would have one dot over \(0\), two stacked dots over \(1\), one over \(2\), and one over \(4\).

Quick Tip: Always use a consistent scale! The distance between \(1\) and \(2\) on your axis should be the same as the distance between \(3\) and \(4\).

2. Stemplots (Stem-and-Leaf Plots)

Stemplots are great because they show the exact data values while still giving you a visual "shape" of the distribution. They work best when you have a small to moderate amount of data.

How to Construct a Stemplot:

  1. Separate each observation into a stem (all but the final digit) and a leaf (the final digit).
  2. Write the stems in a vertical column from smallest to largest and draw a vertical line to their right.
  3. Write each leaf to the right of its stem.
  4. Arrange the leaves in increasing order out from the stem.
  5. Provide a Key that explains what the stem and leaf represent.

The "Key" is Mandatory! On the AP Exam, you will lose points if you forget the key. For example: Key: \(2 | 3\) represents a student who scored \(23\) points.

Don't worry if this seems tricky at first... if your data is too "clumped" together (like everyone scoring in the 60s and 70s), you can split stems. For example, you could have two "6" stems: the first for leaves \(0-4\) and the second for leaves \(5-9\).

3. Histograms

When you have a lot of data, dotplots and stemplots get too crowded. This is where histograms shine. They group nearby values together into "bins" or "intervals."

Key Features of Histograms:

  • The horizontal axis shows the variable (e.g., Weight in kg).
  • The vertical axis shows frequency (count) or relative frequency (percentage).
  • The bars must touch! This is the main visual difference between a histogram (quantitative) and a bar chart (categorical).

How to Construct a Histogram:

  1. Divide the range of data into equal-width intervals (classes).
  2. Count how many individuals fall into each interval.
  3. Label and scale your axes.
  4. Draw bars where the height represents the count (or \(\%\)) of individuals in that interval.

Common Mistake: If a data point falls exactly on the boundary of two bins, the standard convention is to place it in the bin to the right (the higher bin). For example, in bins \(10-20\) and \(20-30\), the value \(20\) usually goes into the \(20-30\) bin.

4. Cumulative Relative Frequency Displays

Sometimes we want to know what percentage of the data falls at or below a certain value. We use a cumulative relative frequency graph (sometimes called an ogive) for this.

  • Cumulative Frequency: The sum of the counts for an interval and all intervals below it.
  • Cumulative Relative Frequency: The cumulative frequency divided by the total number of individuals, usually expressed as a percentage or decimal.

Example: If you are in the \(90\)th percentile for height, a cumulative graph would show a point at your height with a vertical value of \(0.90\) (or \(90\%\)). It means \(90\%\) of people are your height or shorter.

5. Choosing the Right Graph

How do you know which one to use? Here is a simple guide:

  • Dotplot: Best for small datasets where you want to see every individual point.
  • Stemplot: Best for small-to-medium datasets where you want to see the actual digits of the data.
  • Histogram: Best for large datasets where you want to see the overall "shape" of the distribution.
  • Boxplot: These are excellent for seeing the "five-number summary" and identifying outliers. (Note: We will dive deeper into boxplots in Chapter 1.8).

Summary Checklist for Graphs

When constructing any graph in AP Statistics, remember the acronym T.A.L.U. to ensure you don't miss points on the exam:

  • T - Title: Give your graph a descriptive title.
  • A - Axes: Draw your vertical and horizontal axes clearly.
  • L - Labels: Label both axes (e.g., "Number of Pets" and "Frequency").
  • U - Units: Include units if applicable (e.g., "Weight in pounds" or "Time in minutes").

Key Takeaway: Graphical representations are the first step in Analyzing Data (Practice 3). They allow us to visualize the distribution's shape, center, and variability before we start doing complex calculations.

Note: To learn how to describe these graphs using words like "skewed" or "symmetric," check out Chapter 1.6: Descriptions for One Quantitative Variable Distributions.