How is a normally distributed random variable characterized?

How is a normally distributed random variable characterized?

• A. By its median and interquartile range
• B. By its mean and standard deviation
• C. By its mode and variance
• D. By its skewness and kurtosis

Answer: (B)

A normally distributed random variable is characterized by its mean and standard deviation. The mean indicates the central location of the distribution, while the standard deviation measures the spread or dispersion of the data around the mean. This combination uniquely defines the shape of the normal distribution curve. The mean serves as the peak of the bell-shaped curve, indicating where the majority of the data points cluster. The standard deviation determines the width of this curve; a smaller standard deviation results in a steeper and narrower peak, while a larger standard deviation produces a flatter and wider curve. Because of the properties of the normal distribution, these two parameters are sufficient to describe the entire distribution, illustrating why mean and standard deviation are the correct characterization. Other options do not effectively define a normal distribution: the median and interquartile range focus on different aspects of distribution and do not capture the complete shape; mode and variance provide limited insight into the distribution’s characteristics; while skewness and kurtosis relate to the asymmetry and peakedness of a distribution, respectively, and are not specific enough to define a normal distribution.

A normally distributed random variable is characterized by its mean and standard deviation. The mean indicates the central location of the distribution, while the standard deviation measures the spread or dispersion of the data around the mean. This combination uniquely defines the shape of the normal distribution curve.

The mean serves as the peak of the bell-shaped curve, indicating where the majority of the data points cluster. The standard deviation determines the width of this curve; a smaller standard deviation results in a steeper and narrower peak, while a larger standard deviation produces a flatter and wider curve. Because of the properties of the normal distribution, these two parameters are sufficient to describe the entire distribution, illustrating why mean and standard deviation are the correct characterization.

Other options do not effectively define a normal distribution: the median and interquartile range focus on different aspects of distribution and do not capture the complete shape; mode and variance provide limited insight into the distribution’s characteristics; while skewness and kurtosis relate to the asymmetry and peakedness of a distribution, respectively, and are not specific enough to define a normal distribution.