5.2 Example: Mean as a model

• Mathematical equation
• Summaries data (reduces data/predicts data with fewer parameters)
• Simple model: Mean of the distribution of a variable

\(\bar{y} = \frac{y_{1}+y_{2}+\cdots +y_{n}}{n}=\frac{\sum_{i}^{n} trust2006_{i}}{n} = \frac{40668}{6633} = 6.13\)

\(y_{i} = \color{blue}{\overline{y}} \pm \color{red}{\varepsilon}_{i}\)

\[ \begin{aligned} trust2006_{Anna} = 3 = \color{blue}{\overline{y}} \pm \color{red}{\varepsilon}_{Anna} = \color{blue}{6.13} - \color{red}{3.13} \end{aligned} \]

• Mean (= model) predicts Anna’s value with a certain error
• Q: How well does the model (mean = 6.33) predict person’s that have values of 0, of 3 or of 4?