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:
We can express the distribution of an arbitrary normally distributed random variable \(X\) as \(X \sim N(\mu, \sigma^2)\). Therefore, for each plot, we can write down their respective distributions as follows:
- Plot A: \(X \sim N(0, 1^2)\)
- Plot B: \(X \sim N(5, 1^2)\)
- Plot C: \(X \sim N(0, 0.5^2)\)
- Plot D: \(X \sim N(5, 1.5^2)\)
When writing down the distribution, it is sometimes convenient to express the variance as some number squared, so that we can easily see what the standard deviation is. For example, knowing that Plot D's variance is \(\sigma^2 = 1.5^2\), we immediately know that the standard deviation is \(\sigma = 1.5\), since \(\sigma = \sqrt{\sigma^2} = \sqrt{1.5^2} = 1.5.\)
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.