Larger values of the standard deviation result in a normal curve that is: $\sigma$

• A. narrower and taller
• B. unchanged in shape
• C. skewed to the right
• D. wider and flatter

Answer: (D)

Verified step by step guidance

1. Recall that the standard deviation (\( \sigma \)) measures the spread or dispersion of data in a normal distribution.
2. Understand that the shape of a normal distribution is defined by its mean (\( \mu \)) and standard deviation (\( \sigma \)). The probability density function is given by: \[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}} \]
3. Note that increasing the standard deviation \( \sigma \) increases the denominator in the fraction, which lowers the peak height of the curve, making it shorter.
4. Also, a larger \( \sigma \) spreads the data out more, causing the curve to become wider because the values are more dispersed around the mean.
5. Therefore, larger values of the standard deviation result in a normal curve that is wider and flatter, not narrower, taller, or skewed.