A normal distribution is a bell-shaped curve with a majority of the cases falling within the
Blank
______ of the range.

Question

A normal distribution is a bell-shaped curve with a majority of the cases falling within the 
Blank
______ of the range.

BlackTom AI · Accepted Answer

A normal distribution is symmetric about its mean, so the majority of cases are concentrated in the middle of the range.

1. Relevant knowledge points and concepts
- Normal (Gaussian) distribution: bell-shaped, symmetric around the mean; mean = median = mode.
- Empirical (68–95–99.7) rule: about 68% of observations lie within ±1 standard deviation (σ) of the mean, about 95% within ±2σ, and about 99.7% within ±3σ.
- "Majority" means more than 50% of observations.

2. Step-by-step reasoning
- Because the curve is symmetric, the center (mean) is where data are densest.
- Use the empirical rule to quantify "majority": within ±1σ of the mean is considered the central/middle portion.
- Since 68% > 50%, a majority of observations are in that central region.

3. Calculations and intermediate steps
- Total percentage within ±1σ = 68% (empirical rule).
- Percentage outside ±1σ = 100% − 68% = 32%, split equally into two tails: 16% in the left tail beyond −1σ and 16% in the right tail beyond +1σ.
- Therefore central (middle) portion contains a clear majority (68%).

4. Why the correct answer is correct
- The definition and empirical rule show that most observations cluster at the center of the bell curve; numerically, 68% (a majority) lie in the middle region, so "middle" correctly describes where the majority fall.

5. Why other options are incorrect
- "Extreme left" and "left": tails on either side contain relatively small percentages (e.g., 16% beyond −1σ), so they are not where the majority lie.
- "Right": similarly, the right tail holds a minority, not the majority.
- Only "middle" fits both the descriptive shape and the quantitative empirical rule.

6. Conclusion
The correct answer is "middle," which matches the provided answer.

A normal distribution is a bell-shaped curve with a majority of the cases falling within the Blank ______ of the range.单项选择题

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