Which center should represent the data?
You will be able to: Calculate mean and median and justify a choice when values are skewed.
Which center should represent the data?
Four lunch waits are 3, 5, 5 and 7 minutes. If the final wait becomes 27, the average rises sharply, but the middle of the ordered list stays at 5.
A useful starting point: How do you describe a distribution in context? →
Words and symbols before equations
- Mean x̄
- Sum of sample values divided by n.
- Median
- Middle of the sorted values, or average of the two middle values.
- Resistant
- Not strongly affected by a small number of extreme observations.
- Σ
- The summation symbol, meaning add the indicated values.
What this picture assumes
Four synthetic waits: 3, 5, 5 and the adjustable final wait. Axis remains 0–30 minutes. The median stays 5 over this control range.
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=4; mean 5, median 5, sample s 1.633, IQR 2, range 4 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
The mean is an equal-share or balance-point summary: total waiting time divided among the observations. Every value contributes, so extremes pull it.
The median depends on position in the sorted list. In this example it remains (5+5)/2=5 minutes as the last value increases.
For skewed data or outliers, median and IQR often describe a typical observation better. Mean and standard deviation are useful for relatively symmetric data without strong outliers. The choice should match the question; total resource use may still make a mean useful.
| Feature | Mean | Median |
|---|---|---|
| Uses | Every numerical value | Ordered positions |
| Extreme values | Can strongly shift it | Usually more resistant |
| Helpful context | Equal-share average or total | Typical value in skewed data |
A worked example, step by step
Find and compare the centers for waits 2,4,4,6,24 minutes.
- The data are already sorted and n=5.
- The total is 40 minutes, so x̄=40/5=8 minutes.
- The middle value is 4 minutes, the median.
- For a typical wait, 4 minutes is less influenced by the 24-minute extreme. Report the long wait separately instead of hiding it.
Resistant does not mean the median can never change; changing enough observations or their order positions can change it.
Must mean and median be equal?
Compare with an explanation
No. They can differ, especially with asymmetric data or extreme values.
Predict. Change one thing. Explain.
Increase the fourth wait from 7 to 27. Predict both centers first, then explain their different reactions.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
n=4; mean 5, median 5, sample s 1.633, IQR 2, range 4 minutes.
| Summary | Value |
|---|---|
| Data | 3, 5, 5, 7 |
| Mean | 5 |
| Median | 5 |
| Sample variance | 2.667 squared units |
| Sample s | 1.633 |
| Q₁ / Q₃ | 4 / 6 |
Four synthetic waits: 3, 5, 5 and the adjustable final wait. Axis remains 0–30 minutes. The median stays 5 over this control range.
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 questionFor 4,5,5,6,30 dollars, compute both centers and defend a summary of a typical purchase.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Sum 50, n=5, mean $10.
- 1 point: The median is $5.
- 1 point: The high $30 purchase pulls the mean upward.
- 1 point: Use the median for a typical purchase while reporting the unusually large purchase and relevant spread.
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 1How is the mean calculated?
Add all values and divide by their count.
RECALL 2Why sort before a median?
The median uses order positions.
RECALL 3Which center resists one extreme value?
Usually the median, because it is based on position.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Which center should represent the data?
- x̄=Σxᵢ/n.
- Sort before finding the median.
- Justify center choice using shape, unusual values and context.
Remember: Resistant does not mean the median can never change; changing enough observations or their order positions can change it.
Conditions: Four synthetic waits: 3, 5, 5 and the adjustable final wait. Axis remains 0–30 minutes. The median stays 5 over this control range.
Refresh Kid · AP Statistics Unit 1 · Objectives 1.7.A, 1.7.F · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 1.7, objectives 1.7.A, 1.7.F. 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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