Summary Statistics

When we investigate a variable in a dataset, two things are great at summarizing the dataset: the center and the spread.

Measuring Center

There are two ways to measure center: the mean and the median.

Mean

The mean is the average/expected value of a variable. The sample mean is denoted as ${\bar {X}}$, whereas the population mean is denoted as $\mu _{X}$.

${\bar {X}}={\frac {1}{n}}\sum x_{i}$

where $n$ is the sample size.

Median/Percentiles/Quartiles

The median tell us the literal center of the dataset: 50% of statistics are on the left, 50% on the right. It is denoted with ${\widetilde {X}}$

The quartiles is the same, except at 25% for the first quartile, 50% for the second (also the median), and 75% for the third. They are denoted with $Q_{1},Q_{2},Q_{3}$

The percentiles is also the same, except at a particular percentage. For example, the 80th percentile has 80% of data before it.

To calculate the P-th percentile (and thereby calculating all the other something-tiles), we have

$\left({\frac {P}{100}}\right)(n+1)$

where $n$ is the sample size.

Mode

The mode is the most frequently occurring value. It's pretty neglected lol.