# Cumulative Relative Frequency Distribution

The cumulative relative frequency distribution of a quantitative variable is a summary of frequency proportion below a given level.

The relationship between cumulative frequency and relative cumulative frequency is:

#### Example

In the data set faithful, the cumulative relative frequency distribution of the eruptions variable shows the frequency proportion of eruptions whose durations are less than or equal to a set of chosen levels.

#### Problem

Find the cumulative relative frequency distribution of the eruption durations in faithful.

#### Solution

We first find the frequency distribution of the eruption durations as follows. Further details can be found in the Frequency Distribution tutorial.

> duration = faithful\$eruptions
> breaks = seq(1.5, 5.5, by=0.5)
> duration.cut = cut(duration, breaks, right=FALSE)
> duration.freq = table(duration.cut)

We then apply the cumsum function to compute the cumulative frequency distribution.

> duration.cumfreq = cumsum(duration.freq)

Then we find the sample size of faithful with the nrow function, and divide the cumulative frequency distribution with it. As a result, the cumulative relative frequency distribution is:

> duration.cumrelfreq = duration.cumfreq / nrow(faithful)

The cumulative relative frequency distribution of the eruption variable is:

> duration.cumrelfreq
[1.5,2) [2,2.5) [2.5,3) [3,3.5) [3.5,4) [4,4.5) [4.5,5)
0.18750 0.33824 0.35662 0.38235 0.49265 0.76103 0.98529
[5,5.5)
1.00000

#### Enhanced Solution

We can print with fewer digits and make it more readable by setting the digits option.

> old = options(digits=2)
> duration.cumrelfreq
[1.5,2) [2,2.5) [2.5,3) [3,3.5) [3.5,4) [4,4.5) [4.5,5)
0.19    0.34    0.36    0.38    0.49    0.76    0.99
[5,5.5)
1.00
> options(old)    # restore the old option

We then apply the cbind function to print both the cumulative frequency distribution and relative cumulative frequency distribution in parallel columns.

> old = options(digits=2)
> cbind(duration.cumfreq, duration.cumrelfreq)
duration.cumfreq duration.cumrelfreq
[1.5,2)               51                0.19
[2,2.5)               92                0.34
[2.5,3)               97                0.36
[3,3.5)              104                0.38
[3.5,4)              134                0.49
[4,4.5)              207                0.76
[4.5,5)              268                0.99
[5,5.5)              272                1.00
> options(old)

#### Exercise

Find the cumulative frequency distribution of the eruption waiting periods in faithful.