Why the Normal Distribution Shapes Everything: From Bell Curves to 3‑Sigma Rule

1 Normal Distribution

The normal (Gaussian) distribution, also called the bell curve, was first derived by A. De Moivre and later by C.F. Gauss while studying measurement errors. It is a fundamental probability distribution widely used in mathematics, physics, engineering, and statistics.

1.1 Probability Density Function

The probability density function (PDF) of the normal distribution is:

For the standard normal distribution, the mean μ = 0 and the standard deviation σ = 1.

1.2 The 3‑Sigma Rule

The 3‑sigma principle states that approximately 68.27% of values lie within (μ‑σ, μ+σ), 95.45% within (μ‑2σ, μ+2σ), and 99.73% within (μ‑3σ, μ+3σ). Thus, almost all observations fall inside the μ ± 3σ interval, with less than 0.3% outside.

In statistics, confidence intervals for the normal distribution are derived by reversing the probability calculation; for example, a 95% confidence interval for a standard normal distribution corresponds to μ ± 1.96σ.

2 Bivariate Normal Distribution

The probability density function of a multivariate normal distribution extends the univariate case.

The figure below shows a three‑dimensional view of a bivariate normal distribution.

A contour plot can also illustrate its two‑dimensional density.

References

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