gen-define-the-normal-distribution-and-give-one-short-worked-exa-c306

Define the normal distribution and give one short worked example.

# Normal Distribution ## Definition The **normal distribution** (or Gaussian distribution) is a continuous probability distribution characterized by its bell-shaped curve. It is defined by two parameters: - **μ** (mu): the mean (center of the distribution) - **σ** (sigma): the standard deviation (spread of the distribution) The probability density function is: $$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$ **Key properties:** - Symmetric about the mean - Mean = Median = Mode - About 68% of data falls within μ ± σ - About 95% of data falls within μ ± 2σ --- ## Worked Example **Problem:** IQ scores are normally distributed with mean μ = 100 and standard deviation σ = 15. What percentage of people have IQ scores between 85 and 115? **Solution:** 1. Notice that 85 = 100 - 15 and 115 = 100 + 15 2. So we're looking for values within **one standard deviation** of the mean 3. By the empirical rule, approximately **68%** of values fall within μ ± σ **Answer:** About 68% of people have IQ scores between 85 and 115.