The normal-model rule
For a roughly normal, bell-shaped distribution, about 68%, 95% and 99.7% lie within one, two and three SD of the mean.
Under a normal model, approximately 68%, 95% and 99.7% lie within one, two and three SD of the mean. These are model probabilities, not guarantees for a finite sample. A symmetric distribution can be bimodal and unsuitable for this approximation. One observed proportion near 95% does not prove normality.
For mean 65.9 and SD 10.9, the intervals are 55.0-76.8 for one SD, 44.1-87.7 for two, and 33.2-98.6 for three. Say "approximately" and state the roughly normal assumption. Do not automatically apply this interpretation to strongly right-skewed minutes.
Sources: W2 PDF pp. 2-60; L2 Sections 1-8. The personalised Lab 2 has 240 synthetic students, unlike the 1,200-student lecture dataset. Do not mix their numbers.
Mean, median and skewness are clues; the four plots show what they miss. Changing bins changes the picture, not the data.
The 68–95–99.7 rule is compared with observed proportions. It is an approximate model statement, not a guaranteed sample result.
Practice problems
Supplementary study questions on the lecture concepts — not verbatim exam questions. Each answer counts toward spaced review.
1. Conceptual understanding
Definitions and distinctions.
When is the 68–95–99.7 rule appropriate?
For mean 65.9 and SD 10.9, what is the two-SD interval?
2. Application
Apply the concept to a case or calculation.
Under a normal model, about what share lies within one SD of the mean?
A normal model has mean 50 and SD 4. What is the three-SD interval?
3. Interpretation and limitations
Evaluate assumptions and justify conclusions.
A small roughly normal sample has 92% inside two SD. Is the rule necessarily broken?
Which feature makes automatic use of the normal rule questionable?
Explain it in writing
Self-checked, not marked. Write a few sentences, then compare with the example answer. Your writing is saved as you type.
Show an example answer
No. Other distributions or small samples can produce a similar proportion. Check the whole shape rather than using one proportion as proof.
Check: did you answer the question, use evidence or working, and avoid claiming more than the evidence supports?
Show an example answer
Describe the actual distribution with plots, suitable summaries and observed proportions. You can calculate SD without claiming normal-model percentages.
Check: did you answer the question, use evidence or working, and avoid claiming more than the evidence supports?