Cumulative relative frequency

The accumulated relative frequency is the result of adding the relative frequencies of the observations or values of a population or sample. This is represented by the acronym Hi.

To calculate the cumulative relative frequency, you must first calculate the absolute frequency (fi) and the relative frequency (hi) of the population or sample values.

To do this, the data is ordered from smallest to largest and placed in a table. Once this is done, the accumulated relative frequency is obtained by adding the relative frequencies of a class or group in the sample with the previous one (first group + second group, first group + second group + third group and so on until accumulating from the first group to the last).

Cumulative frequency

Example of cumulative relative frequency (Hi) for a discrete variable

Suppose that the grades of 20 students in the first economics course are as follows:

1,2,8,5,8,3,8,5,6,10,5,7,9,4,10,2,7,6,5,10.

Therefore we have:

Xi = Statistical random variable (mark of the first-year economics exam).

N = 20

fi = Absolute frequency (number of times the event is repeated, in this case the exam grade).

hi = Relative frequency (proportion that represents the i-th value in the sample).

Hi = Cumulative relative frequency (Sum of the proportion that represents the i-th value in the sample).

Xi	fi	hi	Hi
1	1	5%	5%
2	2	10%	15%(5+10)
3	1	5%	20%(15+5)
4	1	5%	25%(20+5)
5	4	20%	45%(25+20)
6	2	10%	55%(45+10)
7	2	10%	65%(55+10)
8	3	15%	80%(65+15)
9	1	5%	85%(80+5)
10	3	15%	100%(85+15)
∑	20	100%

The calculation in parentheses in the third column is the result of the corresponding Hi. For example, for the second row our first Hi is 5% and our next hi is 10%. So, for the third row, our Hi is 15% (the result of having accumulated hi = 5% and hi = 10%) and our next hi is 5%. Carrying out this procedure successively, we reach 100%. This is the result of accumulating all the relative frequencies and has to coincide with the total number of observations.

Frequency probability

Example of accumulated relative frequency (Hi) for a continuous variable

Let us suppose that the height of 15 people presenting themselves for the positions of the national police force are the following:

1,82, 1,97, 1,86, 2,01, 2,05, 1,75, 1,84, 1,78, 1,91, 2,03, 1,81, 1,75, 1,77, 1,95, 1,73.

To prepare the frequency table, the values are ordered from lowest to highest, but in this case, given that the variable is continuous and could take any value from an infinitesimal continuous space, the variables must be grouped by intervals.

Therefore we have:

Xi = Statistical random variable (height of applicants to the national police force).

N = 15

fi = Number of times the event is repeated (in this case, the heights that are within a certain interval).

hi = Proportion representing the i-th value in the sample.

Hi = Sum of the proportion that represents the i-th value in the sample.