Which of the following is key to generating a bell curve in statistics?

• A. A normal distribution with any standard deviation value
• B. A uniform distribution with a large $\sigma$ (standard deviation)
• C. A small $\sigma$ (standard deviation), which causes data to cluster tightly around the $\mu$ (mean)
• D. A $\mu =0$ (mean of zero) and a $\sigma =1$ (standard deviation of one) only

Answer: (A)

Verified step by step guidance

1. Understand that a bell curve in statistics refers to the shape of the normal distribution, which is symmetric and centered around the mean.
2. Recognize that the normal distribution is defined by two parameters: the mean (\$\mu\$) and the standard deviation (\$\sigma\$). The mean determines the center of the distribution, and the standard deviation controls the spread.
3. Note that the bell shape is characteristic of the normal distribution regardless of the specific values of the mean and standard deviation; changing these values shifts or stretches the curve but does not change its fundamental bell shape.
4. Contrast this with other distributions, such as the uniform distribution, which does not produce a bell curve because it has equal probability across its range, resulting in a flat shape.
5. Conclude that the key to generating a bell curve is having a normal distribution with any standard deviation value, not necessarily a mean of zero or a specific standard deviation.