Table of Contents
What Is the Normal Distribution?
The normal distribution (also called Gaussian distribution or bell curve) is the most important probability distribution in statistics. It is characterized by its bell shape and is defined by two parameters: the mean (μ) and standard deviation (σ).
Many natural phenomena approximate the normal distribution: heights, test scores, measurement errors, and more. The Central Limit Theorem states that the mean of many independent random variables tends toward a normal distribution, regardless of the original distribution.
Formulas
Empirical Rule (68-95-99.7)
| Range | Percentage | Standard Normal |
|---|---|---|
| μ ± 1σ | 68.27% | -1 to 1 |
| μ ± 2σ | 95.45% | -2 to 2 |
| μ ± 3σ | 99.73% | -3 to 3 |
Frequently Asked Questions
What is the standard normal distribution?
The standard normal distribution has mean 0 and standard deviation 1. Any normal distribution can be converted to it using the z-score formula: z = (x - μ)/σ. This standardization allows using a single table for all normal distributions.
What is the z-score?
The z-score tells you how many standard deviations a value is from the mean. A z-score of 2 means the value is 2 standard deviations above the mean. Negative z-scores indicate values below the mean.
Is my data normally distributed?
You can check with visual methods (histogram, Q-Q plot) or statistical tests (Shapiro-Wilk, Anderson-Darling, Kolmogorov-Smirnov). No real-world data is perfectly normal, but many datasets are close enough for normal-based methods to work well.