2.3.1 Normal distribution model

The normal distribution model always describes a symmetric, unimodal, bell-shaped curve. However, these curves can look different depending on the details of the model. Specifically, the normal distribution model can be adjusted using two parameters: mean and standard deviation. As you can probably guess, changing the mean shifts the bell curve to the left or right, while changing the standard deviation stretches or constricts the curve. Figure LABEL:twoSampleNormals shows the normal distribution with mean $0$ and standard deviation $1$ in the left panel and the normal distributions with mean $19$ and standard deviation $4$ in the right panel. Figure LABEL:twoSampleNormalsStacked shows these distributions on the same axis.

If a normal distribution has mean $\mu$ and standard deviation $\sigma$, we may write the distribution as $N(\mu ,{\sigma }^2)$. The two distributions in Figure LABEL:twoSampleNormalsStacked can be written as

Because the mean and standard deviation describe a normal distribution exactly, they are called the distribution’s parameters.