What does the middle half tell you?
You will be able to: Calculate quartiles and IQR and interpret a percentile with a stated convention.
What does the middle half tell you?
Eight wait times are 2,4,4,6,8,10,10,12 minutes. The middle half is a narrower interval than the full 2-to-12-minute range.
A useful starting point: Which center should represent the data? →
Words and symbols before equations
- Q₁
- First quartile, a lower-quarter location in ordered data.
- Q₃
- Third quartile, an upper-quarter location.
- Interquartile range IQR
- Q₃−Q₁, the spread of the middle half.
- Percentile
- A value locating a stated percentage at or below it in a distribution.
What this picture assumes
Data: 2,4,4,6,8,10,10 and the adjustable maximum. Quartiles use medians of the halves. Modified boxplot flags strictly beyond 1.5-IQR fences. The two-s screen uses sample s here and may differ.
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.
- n=8; mean 9.25, median 7, sample s 8.876, IQR 6, range 28 minutes. Fences -5, 19; IQR outliers: 30. Two-s screen: 30. 87.5% at or below 10 minutes.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
We find quartiles as medians of the lower and upper halves; for odd n we exclude the overall median from both halves. Software conventions may differ for small datasets, so state the convention.
Here the median is 7, Q₁=(4+4)/2=4 and Q₃=(10+10)/2=10. Thus IQR=6 minutes. Range is 10 minutes, which uses the extremes instead.
A 90th-percentile time has about 90% of times at or below it; it is not 90% of the largest time. For an observed value our lab reports the exact percentage at or below, including ties. Small samples and ties can prevent exact target percentages.
A worked example, step by step
For ordered scores 1,3,5,7,9,11, find median, quartiles and IQR. What percent are at or below 7?
- The median is (5+7)/2=6.
- Lower half 1,3,5 has Q₁=3; upper half 7,9,11 has Q₃=9.
- IQR=9−3=6 score units.
- Four of six scores are at or below 7, so the empirical share is about 66.7%.
A percentile is a relative position, not a percentage correct. Quartile conventions can differ on small samples.
Does the 80th percentile mean 80% correct?
Compare with an explanation
No. It locates a score relative to the distribution, not relative to the number of test items.
Predict. Change one thing. Explain.
Change the largest observation and the percentile threshold. Compare the IQR, range and share at or below the threshold; explain ties.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
n=8; mean 9.25, median 7, sample s 8.876, IQR 6, range 28 minutes. Fences -5, 19; IQR outliers: 30. Two-s screen: 30. 87.5% at or below 10 minutes.
| Summary | Value |
|---|---|
| Data | 2, 4, 4, 6, 8, 10, 10, 30 |
| Mean | 9.25 |
| Median | 7 |
| Sample variance | 78.786 squared units |
| Sample s | 8.876 |
| Q₁ / Q₃ | 4 / 10 |
Data: 2,4,4,6,8,10,10 and the adjustable maximum. Quartiles use medians of the halves. Modified boxplot flags strictly beyond 1.5-IQR fences. The two-s screen uses sample s here and may differ.
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 questionFind the quartiles and IQR for 2,4,6,8,10,12,14,16. Interpret the share at or below 10.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Median is 9.
- 1 point: Q₁=(4+6)/2=5 and Q₃=(12+14)/2=13.
- 1 point: IQR=8 units.
- 1 point: 5/8=62.5% of the observations are at or below 10.
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 does IQR measure?
The spread between the lower and upper quartiles.
RECALL 2Why state a quartile convention?
Small-sample definitions can produce different values.
RECALL 3How are ties counted here?
Every value at or below the threshold is counted.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What does the middle half tell you?
- IQR=Q₃−Q₁.
- State the quartile convention.
- At-or-below percentage = 100 × count(xᵢ≤x)/n.
Remember: A percentile is a relative position, not a percentage correct. Quartile conventions can differ on small samples.
Conditions: Data: 2,4,4,6,8,10,10 and the adjustable maximum. Quartiles use medians of the halves. Modified boxplot flags strictly beyond 1.5-IQR fences. The two-s screen uses sample s here and may differ.
Refresh Kid · AP Statistics Unit 1 · Objectives 1.7.A, 1.7.B · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 1.7, objectives 1.7.A, 1.7.B. 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.
Want to work through this with a tutor?
Bring your question about What does the middle half tell you? Your explanation and answers remain free to access.