Question

A larger standard deviation for a normal distribution with an unchanged mean indicates that the distribution becomes:

• A. flatter and wider
• B. more skewed to the left
• C. narrower and more peaked
• D. Change in standard deviation does not change the shape

Hint:

The standard deviation controls the width of the normal curve. A larger $\sigma$ stretches the curve horizontally (making it wider) and pushes it down vertically (making it flatter) to keep the total area equal to 1.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The shape of a normal distribution curve is determined by its two parameters: the mean ($\mu$), which determines the center location, and the standard deviation ($\sigma$), which determines the spread.

Step 2: Detailed Explanation:
The standard deviation ($\sigma$) measures the dispersion or variability of the data points around the mean.
Since the total area under any probability density curve must remain equal to 1, any change in the spread of the curve must be accompanied by a change in its height to maintain this constant unit area.
If the standard deviation increases ($\sigma \uparrow$) while the mean remains constant:
- The data points become more spread out over a wider range.
- The peak of the normal curve decreases in height to compensate for the wider base.
Consequently, the normal curve becomes flatter (platykurtic) and wider.
Conversely, a smaller standard deviation would make the curve narrower and more peaked.

Step 3: Final Answer:
The distribution becomes flatter and wider.