How does bin width change the story?
You will be able to: Build a histogram with explicit boundaries and distinguish it from a bar chart.
How does bin width change the story?
A club’s 12 travel times range from 2 to 14 minutes. Grouping times makes a compact picture, but wide groups can hide clusters.
A useful starting point: How can a graph keep individual values visible? →
Words and symbols before equations
- Bin
- A numerical interval used to group observations.
- Histogram
- Adjacent interval bars showing quantitative frequencies.
- Bin width
- Difference between a bin’s endpoints.
- Boundary convention
- A rule deciding which bin contains an endpoint.
What this picture assumes
The same 12 travel times are used at every width. Equal-width bins include the left endpoint and exclude the right. Count heights begin at zero.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- Bin width 4 minutes. Counts 4, 5, 1, 2 sum to 12; individual times remain 2, 2, 3, 3, 4, 4, 5, 5, 6, 10, 12, 14.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
In this lab, a bin [0,5) includes 0 and 4.9 but excludes 5. The next bin includes 5. That convention assigns each observation exactly once.
With equal-width bins, height is count and bars touch. With unequal widths, counts as heights distort area; use frequency density if comparing areas. Our investigation uses equal widths only.
Changing width changes the visual detail but not the observations. Histograms hide exact within-bin values, so do not reconstruct an exact mean from midpoint estimates without labeling the estimate.
A worked example, step by step
Group times 2,3,5,5,8,9,11,14 minutes into [0,5), [5,10), [10,15).
- Use width 5 minutes and the left-inclusive, right-exclusive convention.
- The first bin contains 2,3, giving count 2.
- The second contains 5,5,8,9, giving 4; the third contains 11,14, giving 2.
- Draw adjacent bars of heights 2,4,2 and label both axes; counts total 8.
A histogram’s horizontal axis is ordered numerical intervals; categorical bars represent labels.
In [5,10), does 10 belong?
Compare with an explanation
No. It belongs in the next left-inclusive bin.
Predict. Change one thing. Explain.
Change bin width from 2 to 4 to 8 minutes. Identify a detail that becomes harder to see and verify the count total remains 12.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Bin width 4 minutes. Counts 4, 5, 1, 2 sum to 12; individual times remain 2, 2, 3, 3, 4, 4, 5, 5, 6, 10, 12, 14.
| Interval (minutes) | Count |
|---|---|
| [0, 4) | 4 |
| [4, 8) | 5 |
| [8, 12) | 1 |
| [12, 16) | 2 |
The same 12 travel times are used at every width. Equal-width bins include the left endpoint and exclude the right. Count heights begin at zero.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the data values, graph scales, summary statistics or study-design conditions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionPlace 1,4,4,6,7,9 into width-3 bins beginning at 0. Interpret the graph’s limitation.
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Compare with the answer and four-point rubric
- 1 point: Use [0,3),[3,6),[6,9),[9,12).
- 1 point: Counts are 1,2,2,1.
- 1 point: Total count is 6; 9 belongs in the final bin.
- 1 point: The bars do not reveal all individual values within each interval.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What is bin width?
The numerical length of a bin.
RECALL 2What happens to n when bins change?
It stays fixed.
RECALL 3What do histogram bars represent?
Frequencies or relative frequencies over numerical intervals.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How does bin width change the story?
- Each observation enters exactly one bin.
- Equal widths permit count-height comparison.
- Changing bins changes appearance, not raw data.
Remember: A histogram’s horizontal axis is ordered numerical intervals; categorical bars represent labels.
Conditions: The same 12 travel times are used at every width. Equal-width bins include the left endpoint and exclude the right. Count heights begin at zero.
Refresh Kid · AP Statistics Unit 1 · Objectives 1.5.A · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 1.5, objectives 1.5.A. Framework effective Fall 2026, checked September 17, 2026. Unit 1 includes one-variable data and data collection; it is part of the revised five-unit course.
Examples and datasets are synthetic, independently authored teaching material. Quartile calculations state a median-of-halves convention. Outlier screens identify values to investigate, not data to discard. Random selection and random assignment have different inferential roles. These lessons introduce design and descriptive reasoning; formal inference comes in later units.
The Organic Chemistry Tutor companion title and destination were checked; the full video was not reviewed. Khan Academy’s destination was checked, but its lesson content was not fully readable by the research tool. OpenStax provides optional reference reading. No provider scripts, questions or graphics were copied. Refresh Kid is not affiliated with these providers.
GitHub’s 3D website collection informed optional spatial inspection. Our original sampling model uses self-hosted Three.js with its MIT license. It shows labeled units in four groups; camera rotation does not change the sampling procedure. Quantitative graphs remain 2D to avoid perspective distortion. Complete labeled diagrams, selected IDs and explanations remain available without 3D.
Independent teacher review and observation of students remain pending. Technical checks do not certify statistical accuracy, accessibility or learning effectiveness. This is a review edition.
Released AP Statistics questions and scoring guides are optional. Older exams use the earlier framework, so check alignment before selecting parts. All practice on this page is original, not official AP material.
Learn → Explore → Practice → Review is informed by the IES learning guide. This implementation has not yet been evaluated with learners.
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