As the standard deviation $\sigma$ decreases, what happens to the graph of the normal distribution curve?

• A. The curve becomes taller and narrower around the mean.
• B. The area under the curve increases.
• C. The curve becomes shorter and wider.
• D. The mean of the curve shifts to the right.

Answer: (A)

Verified step by step guidance

1. Recall that the normal distribution curve is defined by the mean (\mu) and the standard deviation (\sigma). The standard deviation controls the spread or dispersion of the data around the mean.
2. Understand that the total area under the normal distribution curve is always equal to 1, representing the total probability.
3. When the standard deviation decreases, the data points are less spread out, meaning the distribution is more concentrated around the mean.
4. This concentration causes the curve to become taller (higher peak) and narrower (less spread out) because the same total area (1) is compressed into a smaller range of values.
5. Therefore, the shape of the curve changes such that it becomes taller and narrower around the mean, while the mean itself does not shift.