The Normal Distribution and the 68-95-99.7 Rule
The normal distribution (also called the Gaussian distribution or bell curve) is the most important distribution in statistics. It is symmetric around its mean μ, with spread determined by standard deviation σ. Its shape — a perfect bell — arises naturally whenever a large number of independent random factors additively combine. Height, IQ scores, measurement errors, and many natural phenomena follow approximately normal distributions. The 68-95-99.7 rule (empirical rule) describes what percentage of data falls within each standard deviation of the mean: approximately 68% of data falls within μ ± 1σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ. For adult male heights in the US (μ = 70 inches, σ = 3 inches): 68% of men are between 67–73 inches; 95% are between 64–76 inches; 99.7% are between 61–79 inches. These percentages are exact for a perfect normal distribution, and useful approximations for approximately normal data. The area under the entire bell curve equals exactly 1 (100% of probability). The tails extend infinitely but become vanishingly small beyond 3σ — values beyond 3σ are statistically rare events.