### 20.1.13 Cumulative frequencies

Given a list of numbers L, the cumulated frequency at x is the fraction of numbers in the list less than x. The cumulated_frequencies command plots the cumulated frequency of the numbers in a list or given by a matrix.

For numbers in a list:

- cumulated_frequencies takes L, a list of numbers.
- cumulated_frequencies(L) draws the cumulated frequency of the numbers in L, where if L is a matrix, each number in the first column is repeated the number of times given in the second column.

For numbers in a matrix:

- cumulated_frequencies takes M, a matrix.
- cumulated_frequencies(M) (for a matrix with two columns, whose first column consists of numbers and whose second column consists of positive integers) draws the cumulated frequency of the numbers in the first column, where number in the first column is repeated the number of times given in the second column.
- cumulated_frequencies(M) (for a matrix with more than two columns, whose first column consists of numbers and whose remaining columns consist of positive integers) draws the cumulated frequencies for the first column paired with each remaining column.
- cumulated_frequencies(M) (for a matrix with two columns, whose first column consists of ranges a..b and whose second column consists of positive numbers), will normalize the second column so the elements add up to 1 and draw the cumulated frequencies where the second column gives the frequency for the intervals in the first column.

#### Examples

cumulated_frequencies([1,2,1,1,2,1,2,4,3,3])

or:

cumulated_frequencies([[1,4],[2,3],[3,2],[4,1]])

cumulated_frequencies([[1..2,30],[2..4,40],[4..5,30]])

or:

cumulated_frequencies([[1..2,0.3],[2..4,0.4],[4..5,0.3]])

cumulated_frequencies([[1,4,1],[2,3,4],[3,2,1],[4,1,2]])

Here, both the distributions given by [[1,4],[2,3],[3,2],[4,1]] and [[1,1],[2,4],[3,1],[4,2]] are drawn on the same axes.