When most people refer to the mean, they are referring to the arithmetic mean. The arithmetic mean is the sum of a list of numbers divided by the number of numbers in the list. It is often simply called the average.
The median is the middle value in a sorted, ascending or descending, list of numbers and can be more descriptive of that data set than the arithmetic mean. The geometric mean is the product of the elements of a set, taken in a specific order, and is often used when the elements are non-negative real numbers and the order of the elements matters. The cumulative mean is the sum of the values of a function up to a given point.
Here is a table that summarizes the key differences between the mean, median, and geometric mean:
| Measure | Definition | Formula |
|—|—|—|
| Mean | The sum of a list of numbers divided by the number of numbers in the list. | $\frac{\sum_{i=1}^n x_i}{n}$ |
| Median | The middle value in a sorted, ascending or descending, list of numbers. | $x_{(n+1)/2}$ |
| Geometric mean | The product of the elements of a set, taken in a specific order. | $\sqrt[n]{x_1 x_2 \cdots x_n}$ |
I hope this helps! Let me know if you have any other questions.