Measures of center
The mean, median, and mode are different ways to summarize the middle of data, and they can be far apart. “Average” usually means the mean. The geometric mean averages growth factors and ratios, and the expected value is the mean of a probability distribution rather than of data.
Definitions
Missing definition: arithmetic_mean
Think of a list of numbers, like the heights of the students in a class. An average of the list is a single number that represents a typical value of it. Usually, “average” means the arithmetic mean.
Missing definition: median
Missing definition: mode
A weighted average is an average in which some values count more than others.
The weighted average of values $x_i$ with nonnegative weights $w_i$ is
\[\frac{\sum_i w_i x_i}{\sum_i w_i}.\]Values with larger weights count more.
Properties
- With equal weights, the weighted average is the ordinary mean.
- A weighted average lies between the smallest and largest of the values $x_i$.
Missing definition: geometric_mean
Missing definition: expected_value
Examples
Each cell is the ___ of the data or distribution in that row.
The data $1, 1, 2, 3, 100$
| arithmetic_mean | $\frac{1 + 1 + 2 + 3 + 100}{5} = 21.4$ |
|---|---|
| median | $\operatorname{median}(1, 1, 2, 3, 100) = 2$ |
| mode | $\operatorname{mode}(1, 1, 2, 3, 100) = 1$ |
| geometric_mean | $\sqrt[5]{1 \cdot 1 \cdot 2 \cdot 3 \cdot 100} = \sqrt[5]{600} \approx 3.59$ |
The roll of a fair die
| median | $\frac{3 + 4}{2} = 3.5$ |
|---|---|
| mode | Every outcome, $1$ through $6$ |
| expected_value | $\mathbb{E}[X] = 3.5$ |
Missing example: credit_weighted_grades
Missing example: investment_growth