Section 53 Normal Distribution: Introduction

• Relative frequencies as estimates of probabilities

• Plot them as histograms or bar charts

• When we do this with experimental data certain histogram shapes keep recurring, especially when the sample sizes are large.

• The more common shapes have acquired names (Binomial, Poisson, Gaussian or Normal) and have been characterised mathematically.

• These characteristic shapes are referred to as ‘probability distributions’ and the particular response, for example temperature or cloud cover, is called a random variable.

• The Gaussian or Normal distribution is designed for continuous data.

• De Moivre presented this fundamental result, known as the central limit theorem, in 1733. Unfortunately, his work was lost for some time.

• Carl Friedrich Gauss independently developed the theory of Normal distribution nearly 100 years later.

• It is appropriate for data that could, in principle, take any value between \(-\infty\) and \(+\infty\).

• The Normal distribution is characterized by two parameters:

  • the mean, \(\mu\)

  • the variance, \(\sigma^2\)

• We can write: \[ \large X \sim N(\large\mu, \sigma^2) \]