How far are values from their mean?
You will be able to: Calculate sample standard deviation and interpret its units.
How far are values from their mean?
Two teams both average 5 minutes per task. One records 4,5,6 and the other 1,5,9. Their centers match, but their consistency does not.
A useful starting point: What does the middle half tell you? →
Words and symbols before equations
- Deviation
- A value minus the mean, xᵢ−x̄.
- Variance s²
- Sum of squared deviations divided by n−1 for a sample.
- Standard deviation s
- Square root of the sample variance.
- n−1
- Degrees of freedom after estimating one mean from n observations.
What this picture assumes
Start with 2,4,6 minutes and stretch deviations about mean 4 by the nonnegative multiplier. Sample standard deviation uses n−1. Negative displayed times at large multipliers are algebraic illustrations, not physical waits.
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=3; mean 4, median 4, sample s 2, IQR 4, 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
Signed deviations sum to zero, so averaging them directly cannot measure spread. Squaring makes both positive and negative deviations contribute.
For a sample, s=√[Σ(xᵢ−x̄)²/(n−1)], requiring n≥2. Once the mean is fixed, the last deviation is determined by the others. The n−1 adjustment makes sample variance an unbiased estimator under usual random-sampling assumptions; it does not make s itself exactly unbiased.
Take a square root to return to the original unit. A standard deviation of 4 minutes indicates a typical scale of distance from the mean, not that every time lies within 4 minutes. Extreme values strongly influence it.
A worked example, step by step
Calculate sample standard deviation for 2,4,6 minutes.
- The mean is (2+4+6)/3=4 minutes.
- Deviations are −2,0,2 minutes; their squares sum to 8 minutes².
- Sample variance is 8/(3−1)=4 minutes².
- Sample standard deviation is √4=2 minutes. A calculator’s sample Sx corresponds to s; population σx uses a different denominator.
Use n−1 for sample standard deviation. Variance has squared units; standard deviation has original units.
If every measurement is 8 seconds, what is s?
Compare with an explanation
0 seconds, because every deviation from the mean is zero.
Predict. Change one thing. Explain.
Increase the spread multiplier while keeping the mean fixed. Predict what happens to s and s² before checking.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
n=3; mean 4, median 4, sample s 2, IQR 4, range 4 minutes.
| Summary | Value |
|---|---|
| Data | 2, 4, 6 |
| Mean | 4 |
| Median | 4 |
| Sample variance | 4 squared units |
| Sample s | 2 |
| Q₁ / Q₃ | 2 / 6 |
Start with 2,4,6 minutes and stretch deviations about mean 4 by the nonnegative multiplier. Sample standard deviation uses n−1. Negative displayed times at large multipliers are algebraic illustrations, not physical waits.
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 3,6,9 meters, calculate x̄, squared deviations, s² and s.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: x̄=6 meters.
- 1 point: Squared deviations are 9,0,9, totaling 18 meters².
- 1 point: s²=18/(3−1)=9 meters².
- 1 point: s=3 meters, a scale of deviation from the mean.
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 1Why square deviations?
To prevent positive and negative deviations canceling.
RECALL 2Which denominator does sample variance use?
n−1.
RECALL 3What units does standard deviation use?
The same units as the observations.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How far are values from their mean?
- s=√[Σ(xᵢ−x̄)²/(n−1)], n≥2.
- s² has squared units.
- s=0 exactly when all sample values are equal.
Remember: Use n−1 for sample standard deviation. Variance has squared units; standard deviation has original units.
Conditions: Start with 2,4,6 minutes and stretch deviations about mean 4 by the nonnegative multiplier. Sample standard deviation uses n−1. Negative displayed times at large multipliers are algebraic illustrations, not physical waits.
Refresh Kid · AP Statistics Unit 1 · Objectives 1.7.B · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 1.7, objectives 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.
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