Given four normal curves with the same mean ($\mu$) but different spreads, which one of the following curves has the smallest standard deviation ($\sigma$)?

• A. The curve that is the flattest and most spread out
• B. The curve that is symmetric but has the widest tails
• C. The curve that is shifted farthest to the right
• D. The curve that is the most narrow and peaked around the mean

Answer: (D)

Verified step by step guidance

1. Recall that the standard deviation of a normal distribution measures the spread or dispersion of the data around the mean.
2. Understand that a smaller standard deviation means the data is more tightly clustered around the mean, resulting in a curve that is narrower and more peaked.
3. Recognize that a larger standard deviation produces a curve that is flatter and more spread out, with wider tails.
4. Note that shifting the curve to the right or left (changing the mean) does not affect the standard deviation, as standard deviation depends only on the spread, not the location.
5. Therefore, among the given options, the curve that is the most narrow and peaked around the mean corresponds to the smallest standard deviation.