91-appendixA.Rmd

# (APPENDIX) Appendix {-}

# Statistical Background {#appendixA}

## Basic statistical terms

### Mean

The mean is the most commonly reported measure of center.  It is commonly called the "average" though this term can be a little ambiguous.  The mean is the sum of all of the data elements divided by how many elements there are. If we have $n$ data points, the mean is given by: $$Mean = \frac{x_1 + x_2 + \cdots + x_n}{n}$$

### Median

The median is calculated by first sorting a variable's data from smallest to largest.  After sorting the data, the middle element in the list is the **median**.  If the middle falls between two values, then the median is the mean of those two values.

### Standard deviation

We will next discuss the **standard deviation** of a sample dataset pertaining to one variable.  The formula can be a little intimidating at first but it is important to remember that it is essentially a measure of how far to expect a given data value is from its mean:

$$Standard \, deviation = \sqrt{\frac{(x_1 - Mean)^2 + (x_2 - Mean)^2 + \cdots + (x_n - Mean)^2}{n - 1}}$$

### Five-number summary

The **five-number summary** consists of five values:  minimum, first quantile (25^th^ percentile), median (50^th^ percentile), third quantile (75^th^) quantile, and maximum.  The quantiles are calculated as

- first quantile ($Q_1$): the median of the first half of the sorted data
- third quantile ($Q_3$): the median of the second half of the sorted data

The _interquartile range_ is defined as $Q_3 - Q_1$ and is a measure of how spread out the middle 50% of values is. The five-number summary is not influenced by the presence of outliers in the ways that the mean and standard deviation are.  It is, thus, recommended for skewed datasets.

### Distribution

The **distribution** of a variable/dataset corresponds to generalizing patterns in the dataset.  It often shows how frequently elements in the dataset appear.  It shows how the data varies and gives some information about where a typical element in the data might fall.  Distributions are most easily seen through data visualization.

### Outliers

**Outliers** correspond to values in the dataset that fall far outside the range of "ordinary" values.  In regards to a boxplot (by default), they correspond to values below $Q_1 - (1.5 * IQR)$ or above $Q_3 + (1.5 * IQR)$.

Note that these terms (aside from **Distribution**) only apply to quantitative variables.