Normal and Lognormal Distributions

Solution & Explanation

Understanding Normal and Lognormal Distributions

Normal Distribution

• Definition: A normal distribution, also known as a Gaussian distribution, is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is defined by two parameters: the mean (μ) and the standard deviation (σ).

• Characteristics:

  • Symmetry: The distribution is perfectly symmetrical about the mean, meaning the left and right sides are mirror images.
  • Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all equal.
  • 68-95-99.7 Rule: Approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
  • Range: It covers all real numbers from negative to positive infinity, $(-∞, +∞)$.

• Applications:

  • Used to model variables that have natural variability, such as heights, test scores, and measurement errors.
  • Forms the basis for many statistical tests and procedures.

Lognormal Distribution

• Definition: A lognormal distribution is a probability distribution of a random variable whose logarithm is normally distributed. Unlike the normal distribution, a lognormal distribution is skewed to the right.

• Characteristics:

  • Right-Skewed: The distribution has a long right tail, indicating that it is positively skewed.
  • Non-Negative: All values are positive, ranging from zero to infinity, $(0, +∞)$.
  • Mean and Median: The mean is greater than the median due to the positive skewness.
  • Parameters: Defined by two parameters: the location parameter (μ) and the scale parameter (σ), which are the mean and standard deviation of the logarithm of the variable, respectively.

• Applications:

  • Suitable for modeling data that cannot be negative, such as stock prices, income levels, and biological measurements.
  • Used in fields like finance, environmental studies, and reliability engineering.

Key Differences

• Symmetry: The normal distribution is symmetric, while the lognormal distribution is asymmetric and skewed to the right.
• Range of Values: Normal distribution can take any real number, whereas lognormal distribution is limited to positive values.
• Transformation: The lognormal distribution arises when the logarithm of the variable is normally distributed, making it useful for multiplicative processes.

Conclusion

Understanding the characteristics and differences between normal and lognormal distributions is crucial for selecting the appropriate model for a given dataset. While the normal distribution is ideal for symmetric data around a mean, the lognormal distribution is better suited for data that is positively skewed and cannot take negative values.