If annual returns are approximately normal, about what percent of outcomes lie between the mean and +1 standard deviation (equivalently, between the mean and –1 standard deviation)?

Question

BlackTom AI · Accepted Answer

Approximately 68.27% of values lie within ±1 standard deviation for a normal distribution, so the proportion between the mean and +1 SD is half of that: about 34.13%.

1) Relevant knowledge points and concepts
- The normal (Gaussian) distribution is symmetric about its mean.
- The empirical rule: about 68.27% of data lie within ±1 standard deviation of the mean.
- Using the standard normal cumulative distribution function Φ(z), Φ(0)=0.5 and Φ(1)≈0.8413447.

2) Step-by-step reasoning
- The fraction of outcomes between the mean (z = 0) and +1 SD (z = 1) equals Φ(1) − Φ(0).
- Φ(1) is the cumulative probability up to one standard deviation above the mean; Φ(0)=0.5 is the cumulative at the mean.

3) Calculations with intermediate steps
- Φ(1) ≈ 0.841344746
- Φ(0) = 0.5
- Difference: 0.841344746 − 0.5 = 0.341344746
- As a percentage: 0.341344746 × 100% ≈ 34.1345% ≈ 34.1%

4) Why the correct answer is correct
- The normal distribution’s symmetry and the tabulated standard normal CDF give the exact fraction between 0 and +1 SD as ≈34.13%, which rounds to ≈34.1%. This directly matches the provided correct choice.

5) Why the other options are incorrect
- 47.5%: roughly half of the ~95% within ±2 SD (95%/2 ≈ 47.5%), but that refers to mean to ±2 SD, not mean to +1 SD; it’s not the correct interval.
- 68.0%: this is the total proportion within ±1 SD (both sides combined), not the one-sided share.
- ≈15.9%: equals 1 − Φ(1) ≈ 0.15866, the proportion above +1 SD, not the proportion between the mean and +1 SD.

6) Conclusion
The correct answer is ≈34.1%, which matches the provided answer.

If annual returns are approximately normal, about what percent of outcomes lie between the mean and +1 standard deviation (equivalently, between the mean and –1 standard deviation)?单项选择题

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