The Normal Distribution

The ubiquitous 'bell curve' that forms the bedrock of modern statistics.

The Bell Curve

The Normal (or Gaussian) distribution is arguably the most important probability distribution in statistics. It's defined by its mean (μ\mu) and standard deviation (σ\sigma), and its symmetric, bell-shaped curve is instantly recognizable.

Many natural phenomena, from heights and weights to measurement errors, tend to follow a normal distribution. In finance, it's the standard (though often flawed) assumption for modeling asset returns.

The Formula
The probability density function (PDF) is:
f(xμ,σ2)=1σ2πe12(xμσ)2f(x | \mu, \sigma^2) = \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2 }
  • μ\mu is the mean (center of the peak).
  • σ\sigma is the standard deviation (controls the spread).
Interactive Normal Distribution
Adjust the mean and standard deviation to see how they affect the shape of the curve.