When does the bell curve describe a distribution?
You will be able to: Describe a normal model and use the empirical rule with its conditions.
When does the bell curve describe a distribution?
A filling machine’s output is modeled as normal with mean 100 grams and standard deviation 5 grams. Most fills cluster near 100, with progressively fewer far away.
A useful starting point: What should repeated success counts look like? →
Words and symbols before equations
- Continuous random variable
- Can take any value in an interval.
- Normal distribution
- A symmetric, unimodal continuous bell-shaped probability model.
- Density
- Curve height whose area over an interval gives probability.
- Standard normal Z
- Normal distribution with mean 0 and standard deviation 1.
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
A normal model is specified by μ and a positive σ. Moving μ shifts the center; increasing σ spreads the curve and lowers its peak so total area remains 1.
For a normal model, about 68%,95%,99.7% lie within one,two,three standard deviations of μ. For the filling model these intervals are 95–105,90–110 and 85–115 grams.
These percentages require a roughly normal shape; they are not universal rules for skewed or bimodal data. A continuous model assigns zero probability to one exact mathematical value, although a rounded measurement covers an interval. Real-world normal models may also need limits if impossible values receive appreciable probability.
A worked example, step by step
A normal model has μ=50 minutes and σ=4 minutes. Estimate the share between 42 and 58, and the share above 58.
- The endpoints are 50−2×4 and 50+2×4.
- The empirical rule places about 95% within two standard deviations.
- About 5% lies outside that interval.
- Symmetry places about 2.5% above 58 minutes; these are empirical-rule approximations.
Curve height is not a probability. The empirical rule requires a normal or approximately normal model.
Does every dataset have 95% within two standard deviations?
Compare with an explanation
No. That empirical-rule percentage applies to a normal-shaped model, not arbitrary distributions.
Predict. Change one thing. Explain.
Change σ while keeping μ fixed. Explain why the curve becomes wider and lower, then shade the interval μ±2σ and compare its area with about 95%.
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 questionFor a normal model with μ=80 points and σ=6 points, give the 68% and 99.7% intervals and the approximate percentage below 62.
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Compare with the answer and four-point rubric
- 1 point: Within one σ: 74 to 86 points, about 68%.
- 1 point: Within three σ: 62 to 98 points, about 99.7%.
- 1 point: About 0.3% lies outside the three-σ interval.
- 1 point: About 0.15% is below 62 by symmetry; these are approximate model percentages.
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 area under a density mean?
Probability over the horizontal interval.
RECALL 2What changes when σ increases?
The curve spreads out and its peak lowers.
RECALL 3When should the empirical rule be used?
When a normal or approximately normal model is justified.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
When does the bell curve describe a distribution?
- Normal model: mean μ, standard deviation σ>0.
- Within 1σ,2σ,3σ: about 68%,95%,99.7%.
- Total density area=1; standard normal has μ=0,σ=1.
Remember: Curve height is not a probability. The empirical rule requires a normal or approximately normal model.
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.A, 2.11.B, 2.11.C · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 2.11, objectives 2.11.A, 2.11.B, 2.11.C. 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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