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Normal distribution

The Normal Distribution, also known as the Gaussian Distribution, is one of the most important and widely used probability distributions in statistics and machine learning. Here are some key characteristics and information about the Normal Distribution:

  1. Shape: The Normal Distribution is symmetric and bell-shaped. It has a single peak at its mean value.


  2. Mean and Median: The mean (average) and median (middle value) of a Normal Distribution are equal. This is typically denoted as μ (mu).


  3. Standard Deviation: The spread or dispersion of data in a Normal Distribution is determined by the standard deviation (σ). A larger standard deviation indicates greater spread.


  4. Probability Density Function (PDF): The probability density function of a Normal Distribution is given by the famous bell-shaped curve formula:

    Normal Distribution PDF

    • The peak of the curve is at the mean (μ).
    • The spread of the curve is determined by the standard deviation (σ).

  5. 68-95-99.7 Rule: In a Normal Distribution:

    • Approximately 68% of the data falls within one standard deviation of the mean (μ ± σ).
    • Approximately 95% falls within two standard deviations (μ ± 2σ).
    • About 99.7% falls within three standard deviations (μ ± 3σ).


  6. Z-Score: The Z-score measures how far away a data point is from the mean in terms of standard deviations. The formula is: Z=X−μσ, where X is the data point.


  7. Use Cases:

    • Many natural phenomena, such as heights, weights, and test scores, follow a Normal Distribution.
    • It is commonly used in hypothesis testing, confidence intervals, and statistical modeling.
    • In machine learning, it's used in algorithms like Gaussian Naive Bayes and for data preprocessing in methods like feature scaling.


  8. Symmetry: The Normal Distribution is symmetric around its mean, meaning that the probabilities of values above the mean are mirrored by the probabilities of values below the mean.


  9. Central Limit Theorem: This theorem states that the sampling distribution of the sample mean becomes approximately normally distributed as the sample size increases, even if the population distribution is not normal. This is fundamental in statistical inference.

In summary, the Normal Distribution is a fundamental concept in statistics and machine learning due to its prevalence in natural phenomena and its mathematical properties, making it a powerful tool for data analysis and modeling.

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