Frequency & Relative Frequency Tables and Histograms
Learning Objectives
- Construct and understand frequency tables for a set of data
- Construct and understand relative frequency tables for a set of data
- Create and interpret histograms
Why This Matters
Every time a news outlet publishes a histogram of home prices, income levels, or COVID case counts, someone chose how wide to make the bars. Zillow's price histogram for your zip code can look like a tight cluster of similar homes or a broad spectrum of wildly different price points -- same data, different bin width, different headline. Learning to see that choice is the difference between reading a graph and reading what the graph is hiding.
How to Use This Simulation
- Drag the bin-width slider and watch the histogram and frequency table rebuild in real time -- same data, different groupings.
- Toggle the Y-axis between Frequency and Relative Frequency to see counts versus proportions.
- Switch between preset datasets to see how different data shapes respond to bin-width changes.
- Open the Build a Table tab to practice constructing a frequency table by hand.
| Bin | Frequency (f) | Relative Frequency (f/n) |
|---|
Practice: Build a Frequency Table
A campus survey recorded daily app screen time (in hours) for 20 students. The data is shown below. Using a bin width of 2 hours, count the frequency for each bin, then calculate the relative frequencies.
| Bin (hours) | Your Frequency (f) | Relative Frequency (f/n) |
|---|---|---|
| [1, 3) | -- | |
| [3, 5) | -- | |
| [5, 7) | -- | |
| [7, 9] | -- | |
| Total | -- | -- |
What's Happening
Quick Check
You create a histogram of 50 students' exam scores using a bin width of 10 points. The histogram shows two clear peaks -- one around 60-70 and another around 85-95. Your study partner recreates the histogram with a bin width of 30 points and sees only one broad hump centered around 75. They say: "Your histogram must have errors. Mine shows the scores follow a single-peak pattern, so the data can't be bimodal." Which response best addresses your study partner's reasoning?
Try This
Load the "Campus Meal Wait" preset. With the bin width set to 2 minutes, read the histogram and write down the frequency for each bin by hand. Then calculate the relative frequency for each bin (divide each frequency by n). Verify your table matches the simulation's table below the histogram. Which bin has the highest frequency? What is its relative frequency?
Load the "Spotify Listening" preset. Build frequency tables at two bin widths: 10 minutes and 40 minutes. For each, sketch the histogram shape (symmetric, left-skewed, right-skewed, bimodal). Compare the two histograms: does the skew direction stay the same at both bin widths? Does the location of the tallest bar change? Identify one feature that stays the same regardless of bin width and one feature that changes. Verify using the simulation.
You work at a city transit agency. The "Commute Times" dataset represents 40 surveyed riders. Your manager wants a histogram for a public report to city council arguing that most commuters have short, efficient trips. Explore the dataset using the bin-width slider. What bin width would make the data look like most commuters have short trips? What bin width would reveal that a large group actually has long transit rides? Recommend a bin width that honestly represents the data's structure and defend your choice in two sentences, addressing both readability for a non-technical audience and accuracy in representing the data's true shape.
Instructor Notes
Teaching Notes
This simulation is most effective when you start with the "Commute Times" preset at its default bin width of 5. Ask students to describe the shape they see (bimodal -- two peaks). Then ask them to predict what will happen when they drag the bin-width slider to 15 or 20. Most students will expect the shape to "smooth out" but keep its two peaks. When the peaks merge into a single hump, the surprise creates a powerful anchor for why bin width is a researcher choice with real consequences.
The "Build a Table" tab addresses Objective 1 directly. Have students complete it before exploring the Histogram Explorer so they appreciate what the automation is doing for them. The relative frequency column introduces the sum-to-1 constraint that reappears in Sim 10 (probability distributions) and Sim 15 (continuous density functions).
Common Student Errors
- Confusing histograms with bar graphs. Histograms display quantitative data with adjacent bars (connected ranges); bar graphs display qualitative categories with separated bars. This distinction is foundational and recurs in Sim 4 and Sim 5.
- Assuming the bin width is determined by the data. Students often think there is one "right" bin width, when in fact it is a researcher decision. The slider makes this tangible.
- Forgetting that relative frequencies must sum to 1. Students sometimes treat relative frequencies as standalone numbers without checking the constraint.
- Misinterpreting the y-axis. Students may confuse frequency (count) with relative frequency (proportion), especially when comparing datasets of different sizes.
Discussion Questions
- If two news outlets show histograms of the same city's home prices but one uses $50,000 bins and the other uses $200,000 bins, which would make the market look more affordable? Why?
- Can you think of a situation where a very narrow bin width (many bars) is more useful than a moderate one? What about the reverse?
- Why do relative frequencies matter when comparing two datasets of different sizes? What would go wrong if you compared raw frequencies instead?
Exam Connection
Typical exam questions give a dataset and ask students to construct a frequency table with a specified bin width, compute relative frequencies, and interpret or sketch the histogram. The "Build a Table" tab directly practices this format. Some exam items present two histograms with different bin widths and ask whether the underlying data could be the same -- the slider interaction prepares students to reason about this.