Frequency is the number of times an event is repeated in a given period of time. It accumulated, on the other hand, it is the sum, the grouping or the reunion of diverse elements.
As for the idea of cumulative frequency, the concept appears in the field of statistics, where frequency is understood as the number of times that a certain event is repeated in a sample or in an experiment.
This number of repetitions is known as Absolute frecuency. If the absolute frequency is divided by the size of the sample, we get the relative frequency.
From these data, we can calculate two types of cumulative frequency: the cumulative absolute frequency and the cumulative relative frequency.
The cumulative absolute frequency (sometimes simply called cumulative frequency) signals the number of absolute frequencies for all the events that, in an ordered list, are minor or identical that a certain value.
Take the case of the goals scored by a footballer over five years. These data constitute the statistical sample:
14, 12, 14, 11, 15
The absolute frequency of 14, for example, is 2, since the 14 appear 2 times in the sample. This means that the athlete scored 14 goals in 2 seasons different in the last five years. The calculation of the accumulated absolute frequency for that value (14) it is 4: there is 4 values in the sample that are equal to or less than 14.
Another cumulative frequency that we can calculate is the cumulative relative frequency. In this case, we have to divide the accumulated absolute frequency by the total sample. Returning to the previous example, as the accumulated absolute frequency of 14 it is 4 and the total of numbers of the statistical sample is 5, the cumulative relative frequency is 0.8.
In order to formulate the distribution of this concept in a mathematical equation without falling into the use of a table, it is possible to adapt it to what is called cumulative probability distribution. In the fields of statistics and probability, there is talk of probability distribution to refer to a function that is applied to a variable and gives the events that are defined on it different probabilities that they will take place.
Thanks to the use of a equation math rather than a table it is also easier to interpolate and extrapolate values. It is understood by interpolationWithin the numerical analysis, the process that results in a series of points from a discrete set of points. With respect to extrapolationOn the other hand, it is the estimate that exceeds the limits of the observation, that is, it is more uncertain than the interpolation and carries a greater risk of failure.
It is important to note that when extrapolating a distribution accumulated frequency errors can arise such as not following the probability distribution once the observed range is exceeded. In this framework we have different methods to carry out the same process, such as the distribution normal, the exponential, the by Gumbel and the Pareto.
Another option consists of introducing discontinuity between the data, something that can be very beneficial if the extreme values and the tail of the distribution are far from the median mass. One of the applications of this method is found in the analysis of precipitation when the climate changes behavior according to the influence of currents.
Having said all this, it is clear that conducting a prediction based on the cumulative frequency distribution carries a certain margin of error that is not always acceptable. To minimize useless results, it is recommended to avoid those cases in which the conditions of the data ranges to be compared are very different.