Normal Distribution

Normal Distribution

Definition

• A continuous probability distribution that is symmetric and bell-shaped
• Also known as the Gaussian distribution or bell curve
• Describes how variables are distributed in a population

Key Features

• Mean (μ): the average value of the distribution
• Standard Deviation (σ): a measure of the spread or dispersion of the data
• Median and mode are equal to the mean
• 68-95-99.7 Rule: approximately 68% of data falls within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean

Properties

• The normal distribution is symmetric around the mean
• The curve is continuous and smooth
• The distribution is asymptotic, meaning it approaches but never touches the x-axis
• The normal distribution is a limiting case of the binomial distribution

Z-Scores

• A measure of how many standard deviations an observation is away from the mean
• Calculated using the formula: z = (x - μ) / σ
• Used to compare observations from different normal distributions
• Z-scores can be used to find the percentile or probability of an observation

Applications

• Modeling real-valued variables in natural and social sciences
• Analyzing data that is normally distributed or can be transformed to normality
• Used in statistical inference, hypothesis testing, and confidence intervals
• Applications in finance, engineering, medicine, and social sciences