🎯 Introduction to the Normal Distribution
The normal distribution, also known as the Gaussian distribution or bell curve, is arguably the most important probability distribution in statistics. It is named after the German mathematician Carl Friedrich Gauss (1777-1855).
Why is it so important?
The normal distribution appears naturally in countless phenomena:
- Human height in a population
- Measurement errors in laboratories
- Standardized test scores (IQ, SAT, etc.)
- Blood pressure
- Weight of manufactured products
📊 Characteristics of the Normal Distribution
The mathematical formula
f(x) = (1 / σ√(2π)) × e-½((x-μ)/σ)²
Where:
- μ (mu) = the mean (center of the curve)
- σ (sigma) = the standard deviation (measure of spread)
- e = Euler's constant (≈ 2.718)
- π = pi (≈ 3.14159)
The 68-95-99.7 Empirical Rule
This fundamental rule allows you to quickly interpret your data:
📌 68% of data falls within ±1 standard deviation of the mean
📌 95% of data falls within ±2 standard deviations of the mean
📌 99.7% of data falls within ±3 standard deviations of the mean
🔬 Practical Applications
Quality Control
In manufacturing, the normal distribution is used to define control limits.
Finance and Investment
Stock returns are often modeled using the normal distribution, enabling Value at Risk (VaR) calculations.