Descriptive Statistics

Summarizing a single variable

Once data are collected, the first job is to describe them compactly. Descriptive statistics fall into two families: measures of central tendency (where is the center?) and measures of dispersion (how spread out are the values?). Suppose GreenLeaf Gym records the number of visits each of its 11 members made last week: $$3,\ 5,\ 2,\ 8,\ 4,\ 4,\ 6,\ 12,\ 3,\ 5,\ 7.$$

Central tendency

The mean is the arithmetic average, $\bar{x} = \dfrac{\sum x}{n}$. Here $\sum x = 59$ and $n = 11$, so $$\bar{x} = \frac{59}{11} \approx 5.36 \text{ visits}.$$ The median is the middle value when data are ordered: sorting gives $2,3,3,4,4,\mathbf{5},5,6,7,8,12$, so the median is $5$. The mode, the most frequent value, is $3, 4,$ and $5$ each appearing... actually $3$ appears twice,