• Descriptive statistics provide numbers that capture important features of frequency distributions
  • Location
  • Spread
  • Shape
  • Proportion

Means

Arithmetic mean

• Given a data set $X=\{x_1,x_2,...,x_n\}$, the arithmetic mean (or simply mean) is given by

$$ A=\frac{1}{n}\sum^n_{i=1}x_i=\frac{x_1+x_2+...+x_n}{n} $$

• Denoted as $\mu$ for population mean and $\bar x$ for sample mean of $X$
• Measure of central tendency
  • $(x_1-\bar x)+...+(x_n-\bar x)=0$
  • Important implications for residuals

<aside> ⁉️ Challenge 1 What is the sum of adding 1, 2, …, to 100?

• Answer: </aside>

Geometric mean

$$ G=(\prod^n_{i=1}x_i)^{\frac{1}{n}}=(x_1x_2...x_n)^{\frac{1}{n}} $$

• More descriptive of the central tendency of a data set $X$, when $x_1,x_2,...,x_n$ have a multiplicative or exponential relationship (e.g. a geometric series)
• Useful in real world data that is derived from proportions and rates
• A particular feature is that it can average across values on completely different scales

Harmonic mean

$$ H=(\sum^n_{i=1}\frac{1}{x_i})^{-1}=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}} $$

• Equivalent to the reciprocal of the arithmetic mean of the reciprocals of the dataset
• Reciprocals ****are particularly useful for multiplicative or fraction relationships
• Similar to geometric mean in this sense