# Chapter 4 The Normal Distribution

At this point, we introduce perhaps the most famous distribution in statistics: the ** Normal Distribution**. You may well have seen it depicted as a bell curve before, similar to the below picture:

Distributions can be specified by key parameters. For the Normal distribution, the key parameters are the mean (\(\mu\)) and the standard deviation (\(\sigma\)) (or alternatively the variance, \(\sigma^2\)). In the above picture, we have that:

- \(\mu = 0\)
- \(\sigma = 1\). This is what is known as a standard normal distribution. However, \(\mu\) and \(\sigma\) can in fact take any value. Below are a few examples:

Comparing Plots A and B, they both have the same standard deviation of 1, but the mean of the distribution in Plot B is higher (5, compared with 0 for Plot A), which is reflected in the scale on the x-axis. Comparing Plots A and C, they both have the same mean of 0, but the standard deviation in Plot C is lower (0.5, compared with 1 for Plot A). This is reflected in that the curve in Plot A appears more spread out than the one in Plot C. On the other hand, Plots B and D both have the same mean (5), but Plot D has a higher standard deviation than Plot B (1.5 compared with 1). Therefore, the curve in Plot D appears more spread out than the curve in Plot B.

We will consider other properties of the Normal distribution in the following sections.