1. Format for summary is the same as format for prediction
Assign a probability to each possible outcome.
For categorical variables: a listing of the levels and a probability assigned to each.
- For quantitative variables:
- Detailed: the distribution function
- Concise: the 95% summary interval
2. Means and standard deviations
In joke form: > A statistician can have his head in the oven and his feet in the freezer, and he will say that on average he feels fine.
- Vague description of mean w.r.t. data: the best single representative value
- Do we want to encourage students to use just a single value?
- Representative of what?
- Historical connection to the “ideal”
- Standard deviation
- Weird name
- Too small to cover much of data
- No name for mean ± 2 standard deviations
- Refers to center without any need
3. An alternative to the standard deviation
Average square distance (ASD) between all pairs of values
- Corresponds to \(\sqrt{2} \sigma\)
- Easily converted to a simple, robust form: use absolute value instead of square.
Normal PDF:
\[ \frac{1}{\sqrt{2 \pi \sigma^2}} \ e^{-\frac{(x - \mu)^2}{2 \sigma^2}}\]
Why the 2?
Normal PDF written with the ASD:
\[ \frac{1}{\sqrt{\pi S^2}} \ e^{-\frac{(x - \mu)^2}{S^2}}\]