Why Higher Confidence and Smaller Margin of Error Require Larger Samples

Updated: March 10, 2026
Author: Arpita Goyal

The Cost of Certainty in Statistics is a short comic that explains the relationship between confidence level, margin of error, and sample size. I chose this topic because these ideas are usually introduced through formulas in introductory statistics classes, but students often struggle to conceptually understand them.

The hoop and ball analogy was inspired by a classroom discussion in a critical thinking course that I’m taking this semester. This analogy provides a clear and intuitive way to show how increasing confidence or decreasing margin of error requires a larger sample size. The goal of this project is to make these relationships visually intuitive by changing one variable at a time, allowing students to clearly see how certainty and precision affect how big our data should be.

FINAL DESIGN

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THE PROCESS

Understand (Discover, Interpret, Specify)

DESCRIBE THE CHALLENGE:

• Students are often able to compute or interpret confidence levels and margins of error using formulas and/ or any programming language, but they struggle to intuitively understand the relationship between them. The learning challenge helps develop a conceptual understanding of how changing one of these factors directly affects the others.

CONTEXT AND AUDIENCE:

• The primary audience for this project is undergraduate students enrolled in introductory statistics courses. Students frequently rely on formulas instead of conceptual understanding. They can see that a higher confidence level increases sample size but cannot explain why. This comic solves that problem by focusing on critical thinking rather than just using computational formulas.

POV STATEMENT:

• An undergraduate student in an introductory statistics course needs a clear, memorable and intuitive way to understand how confidence level and margin of error affect sample size so that they can justify their statistical results without relying completely on memorized formulas.

LEARNING OBJECTIVES:

• Explain how increasing/ decreasing the confidence level affects the required sample size.
• Explain how increasing/ decreasing the margin of error affects the required sample size.
• Promote conceptual reasoning over formula memorization.
• Reduce anxiety around understanding complex mathematical concepts.

Plan (Ideate, Sketch, Elaborate)

IDEATION:

• I thought of a few ideas that I could visualize properly in a comic. The ideas were:
  1. Why is the area of circle πr^2?
  2. Understanding the relationship between confidence level, margin of error and sample size using different analogies
  3. How outliers affect mean, median, mode and standard deviation
  
• I chose topic number two and brainstormed a few analogies I could use to explain the concept. Some of them included:
  1. A basketball free throw analogy
  2. A dartboard analogy
  3. A hoop and a ball analogy
  
  I initially decided to use the basketball free throw analogy but it didn’t work because I wasn’t able to clearly show different sizes of a basketball hoop which was essential for the understanding of the concept of margin of error. Then, I decided to use a shopping basket instead.
  
  However, when I tested my comic by asking my peers what they understand, I realized that the problem now is that they interpreted that the capacity of the basket as part of the relationship. This introduced confusion, since capacity is not conceptually tied to margin of error in the intended way.
  
  Therefore, I ultimately chose the hoop and ball analogy. This version allowed the size of the hoop (margin of error), the number of balls (sample size), and the required success rate (confidence level) to be adjusted visually whenever needed.

STORYBOARD OR SCRIPT:

THEORY APPLIED:

This comic was designed using principles from Mayer’s Cognitive Theory of Multimedia Learning, which states that people learn more effectively when information is presented through both words and visuals. The redundancy principle was applied by removing unnecessary visual details. The number of basketballs was simplified and made into a single tracker board. This reduces cognitive load and helps students concentrate on the key statistical relationships. The signaling principle was used by visually emphasizing the most important ideas. The final takeaway statements highlight the core relationships and connect it to classroom learning.

The Contiguity principle was applied by placing related text and visuals close together. For example, the duck representing the sample size is labeled “n”, and the scoreboard displays the numerical values directly beside the relevant categories. This helps learners quickly connect the visual elements with the statistical variables. Finally, the segmenting principle was used by introducing confidence level, margin of error, and sample size in separate panels before showing how they interact. This step-by-step structure helps learners process the relationships gradually.

The Prototype 

PEER FEEDBACK:

My peers appreciated the use of the basketball hoop analogy to explain the relationship between confidence level, margin of error, and sample size. They noted that the visual approach made the concept easier to understand compared to traditional formula-based explanations. The comic was described as easy to follow, visually appealing, and effective at communicating ideas through images rather than large amounts of text.

Several suggestions for improvement were also provided. One reviewer suggested including a formula or a graph for statistical representation to better connect the comic with what students encounter in statistics classes. Another comment noted that some panels initially contained several visual elements, which could make it harder for readers to immediately focus on the key idea. It was also suggested that the key concept should be emphasized more clearly at the end of the comic to strengthen the main takeaway. Additionally, one peer noted that the comic appeared slightly small when displayed on the blog, which made some text difficult to read.

Reflect and Refine

REFLECTION:

One aspect that worked particularly well in this project was the use of the hoop and ball analogy. The analogy made the relationship between confidence level, margin of error, and sample size visually intuitive. Peer feedback confirmed that the analogy helped make the concept easier to understand and more engaging than a traditional formula based explanation. The visual style of the comic was also effective, as peers mentioned that the graphics were clear, visually appealing, and easy to follow.

The peer feedback also highlighted several areas for improvement. Some panels initially contained many visual elements, which made it harder to immediately identify the main idea. In addition, peers suggested emphasizing the key concept more clearly at the end of the comic and including a formula so that the explanation connects more directly with the way the concept is taught in statistics courses. Another comment noted that the comic appeared somewhat small when displayed on the blog, which affected readability.

Based on this feedback, several revisions were made to the prototype. The overall size of the panels was increased to improve readability. Visual clutter was reduced by simplifying the number of basketballs shown and by adding a scoreboard style tracker that displays the sample size, number of successful shots, and the confidence level. This allowed the comic to communicate the same idea without overwhelming the reader with too many visual elements. In addition, the key relationships were highlighted more clearly in the final panels, and the sample size formula was added to connect the visual explanation to the mathematical concept used in statistics classes.

While the comic uses a visual story and analogy to make a mathematical concept easier to understand, it also follows Mayer’s Cognitive Theory of Multimedia Learning, which suggests that learners process information through both visual and verbal channels. However, there are still some limitations. Although the comic helps build an intuitive understanding of the relationship between the variables, it simplifies the mathematics behind the concept. Students may still need formal lectures, related examples, and formulas to fully understand the statistical relationship. Because of this, the comic works best alongside a traditional lecture and doesn’t replace it.

If I were to improve the project further, I would strengthen the connection between the visual explanation and the mathematical side of the concept. In a future revision, I would consider adding a small graph or an additional panel that shows how sample size changes mathematically as confidence level or margin of error changes. This could help bridge the gap between conceptual understanding and the formal statistical representation used in textbooks.