How do you turn a measurement interval into an area?
You will be able to: Standardize normal bounds and calculate left, right and interval probabilities.
How do you turn a measurement interval into an area?
If package mass is normal with μ=100 grams and σ=5 grams, a 110-gram package is two standard deviations above the mean.
A useful starting point: When does the bell curve describe a distribution? →
Words and symbols before equations
- z-score
- z=(x−μ)/σ, position in standard-deviation units.
- Φ(z)
- Standard normal cumulative probability P(Z≤z).
- Left tail
- Area below a cutoff.
- Right tail
- Area above a cutoff.
What this picture assumes
A theoretical normal model has mean 100 g and the selected positive standard deviation. Probability is area, not curve height. The display spans 70–130 g on fixed axes; calculations include the full infinite tails. Reversed bounds give an empty between-event.
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 100 g; SD 5 g. Between 95 and 110 g: probability 0.818595 (81.9%).
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
First identify the model, sketch its center and shade the requested region. Translate each measurement bound using z=(x−μ)/σ.
For the filling model, P(X≤110)=Φ(2)≈0.97725. The right tail is 1−Φ(2)≈0.02275. A between probability subtracts lower cumulative area from upper cumulative area.
For a continuous normal model, including a single endpoint does not change the area. A larger z has a larger normal percentile; raw measurements from different scales should be standardized before comparing relative standing. Without normality, z alone does not determine a percentile.
A worked example, step by step
Under a normal model μ=100 grams, σ=5 grams, find P(95<X<110).
- Shade the region between 95 and 110 on the mass axis.
- Standardize: lower z=(95−100)/5=−1; upper z=(110−100)/5=2.
- Use Φ(2)−Φ(−1)≈0.97725−0.15866.
- The probability is about 0.81859, or 81.86% of fills under the model.
Check which tail is requested. A cumulative table usually gives area to the left, not the area beyond the cutoff.
For a continuous normal model, is P(X=100) positive?
Compare with an explanation
No. A single point has zero width and therefore zero area; rounded recorded values represent intervals instead.
Predict. Change one thing. Explain.
Switch between left, right and between regions. Keep the same bounds and check that complementary tails add to 1. Explain why a point’s height is not its probability.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Mean 100 g; SD 5 g. Between 95 and 110 g: probability 0.818595 (81.9%).
| Value | Standardized z | Area below |
|---|---|---|
| 95 g | -1 | 0.158655 |
| 110 g | 2 | 0.97725 |
A theoretical normal model has mean 100 g and the selected positive standard deviation. Probability is area, not curve height. The display spans 70–130 g on fixed axes; calculations include the full infinite tails. Reversed bounds give an empty between-event.
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 questionX is normal with μ=40,σ=8. Find P(32<X<48) and compare 56’s relative position with a value at z=1 in another normal population.
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Compare with the answer and four-point rubric
- 1 point: Bounds standardize to −1 and 1.
- 1 point: Between area is about 0.68269.
- 1 point: For 56, z=(56−40)/8=2.
- 1 point: z=2 is at a higher normal percentile than z=1, regardless of the raw scales.
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 Φ(z) give?
Standard normal area at or below z.
RECALL 2How do you calculate a between area?
Subtract the lower cumulative area from the upper.
RECALL 3Does including a normal endpoint change probability?
No; a single point has zero probability in the continuous model.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you turn a measurement interval into an area?
- P(X≤b)=Φ((b−μ)/σ).
- P(X>b)=1−Φ((b−μ)/σ).
- P(a<X<b)=Φ(z_b)−Φ(z_a).
Remember: Check which tail is requested. A cumulative table usually gives area to the left, not the area beyond the cutoff.
Conditions: A theoretical normal model has mean 100 g and the selected positive standard deviation. Probability is area, not curve height. The display spans 70–130 g on fixed axes; calculations include the full infinite tails. Reversed bounds give an empty between-event.
Refresh Kid · AP Statistics Unit 2 · Objectives 2.11.D, 2.11.F · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 2.11, objectives 2.11.D, 2.11.F. 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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