How variable is a chance outcome?
You will be able to: Calculate probability-weighted variance and standard deviation.
How variable is a chance outcome?
Two reward systems can average 2 points but feel very different: one always pays 2, while another pays 0 or 4 with equal probability.
A useful starting point: Can an average be an impossible single outcome? →
Words and symbols before equations
- Variance σ²
- Probability-weighted average squared distance from μ.
- Standard deviation σ
- Square root of the variance.
- Deviation
- A possible value minus the distribution mean.
- Distribution parameter
- A fixed property of the probability model.
What this picture assumes
X is a hypothetical reward in points taking values 0, 2 or 5. P(X=2)=0.30 is fixed; P(X=0)=0.70−P(X=5). Probabilities remain nonnegative and sum to 1.
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.
- Mean 1.6 points; SD 1.908 points. P(X ≤ 2) = 0.8. The mean is a long-run average and need not be a possible single reward.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For the constant 2-point reward, every deviation is zero. For the 0-or-4 reward, μ=2 and both deviations have magnitude 2, so σ²=4 and σ=2 points.
Use σ²=Σ(x−μ)²P(X=x), then take the square root. This formula summarizes a complete probability model; it is not the sample variance formula using n−1.
A standard deviation gives a typical scale of variation around the mean, not a guarantee that every outcome falls within that distance. Variance has squared units; standard deviation restores the original unit.
A worked example, step by step
X is 0 or 4 points with probabilities 0.75 and 0.25. Find μ, variance and standard deviation.
- μ=0×0.75+4×0.25=1 point.
- Squared deviations are (0−1)²=1 and (4−1)²=9.
- Variance is 1×0.75+9×0.25=3 points².
- σ=√3≈1.732 points, a scale of long-run outcome deviation from the 1-point mean.
Do not divide a probability-weighted variance by n−1. Its weights already sum to 1.
What is σ for a variable that always equals 7?
Compare with an explanation
0, since every outcome equals the mean.
Predict. Change one thing. Explain.
Change the high-bonus probability and read the weighted squared-deviation table. Check that the mean used in every deviation changes with the distribution.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Mean 1.6 points; SD 1.908 points. P(X ≤ 2) = 0.8. The mean is a long-run average and need not be a possible single reward.
| x (points) | P(X=x) | F(x)=P(X≤x) | x × P(X=x) | (x−μ)² × P(X=x) |
|---|---|---|---|---|
| 0 | 0.5 | 0.5 | 0 | 1.28 |
| 2 | 0.3 | 0.8 | 0.6 | 0.048 |
| 5 | 0.2 | 1 | 1 | 2.312 |
| Summary | Value |
|---|---|
| Mean μ (points) | 1.6 |
| Variance (points²) | 3.64 |
| Standard deviation (points) | 1.908 |
| P(X ≤ 2) | 0.8 |
X is a hypothetical reward in points taking values 0, 2 or 5. P(X=2)=0.30 is fixed; P(X=0)=0.70−P(X=5). Probabilities remain nonnegative and sum to 1.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the probability values, reference groups, graph scales or model assumptions. 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 questionA variable is 2 or 6 minutes, each with probability 1/2. Calculate and interpret its mean and spread.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: μ=4 minutes.
- 1 point: Both squared deviations are 4 minutes².
- 1 point: Variance=4 and standard deviation=2 minutes.
- 1 point: The long-run mean is 4 minutes with typical deviation scale 2 minutes; no single outcome equals 4.
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 opposite signs canceling.
RECALL 2Why take a square root?
To return to the original measurement units.
RECALL 3Does the model formula use n−1?
No; it weights all possible outcomes by their probabilities.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How variable is a chance outcome?
- σ²=Σ(x−μ)²P(X=x).
- σ=√σ².
- Use squared units for variance and original units for σ.
Remember: Do not divide a probability-weighted variance by n−1. Its weights already sum to 1.
Conditions: X is a hypothetical reward in points taking values 0, 2 or 5. P(X=2)=0.30 is fixed; P(X=0)=0.70−P(X=5). Probabilities remain nonnegative and sum to 1.
Refresh Kid · AP Statistics Unit 2 · Objectives 2.9.A, 2.9.B · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 2.9, objectives 2.9.A, 2.9.B. Framework effective Fall 2026, checked September 17, 2026. Unit 2 includes probability, random variables, probability models and introductory sampling distributions; it is part of the revised five-unit course.
Examples and datasets are synthetic, independently authored teaching material. Assumptions about independence, replacement and equal likelihood are stated before calculations. Simulation estimates fluctuate. Discrete probability is summed; continuous probability is area. Sampling distributions and randomization distributions use different repetition mechanisms. 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 three-toss outcome cube uses self-hosted Three.js with its MIT license. Eight corners represent eight equally likely sequences of three fair independent tosses. Conditioning removes ineligible sequences; camera rotation never changes probabilities. Quantitative graphs remain 2D to avoid perspective distortion. Complete labeled diagrams, outcome lists 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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