The normal distribution is completely described by these two parameters. Multiple choice question.
The population mean and the population standard deviation.
The population mode and the population mean.
The population covariance and the population range.
The population mode and the population size.
The correct answer and explanation is:
Correct Answer:
The population mean and the population standard deviation.
Explanation (300 words):
The normal distribution, also called the Gaussian distribution, is a symmetric, bell-shaped curve that is entirely defined by two parameters:
- The population mean (μ)
- The population standard deviation (σ)
1. Population Mean (μ)
The mean is the center of the distribution. It represents the average value of all data points in the population. In a normal distribution, the mean determines the location of the peak of the bell curve and is also equal to the median and mode, due to the symmetry of the distribution.
2. Population Standard Deviation (σ)
The standard deviation describes the spread or dispersion of the data. A small σ means the data points are clustered closely around the mean, resulting in a narrow bell shape. A large σ implies that the data are more spread out, resulting in a flatter, wider curve.
Why the Other Choices Are Incorrect:
- The population mode and the population mean:
The mode is not a required parameter for defining a normal distribution. In fact, in a normal distribution, the mode is the same as the mean and median, so it adds no new information. - The population covariance and the population range:
Covariance measures the relationship between two variables, not one. Range (max – min) does not define the shape of the distribution. - The population mode and the population size:
Population size affects precision in sampling but does not define the shape of a distribution. Mode alone cannot determine the structure of the normal curve.
Conclusion:
Only the mean (μ) and standard deviation (σ) are needed to fully describe a normal distribution, including its center and variability.