The distribution or table of frequencies is a table of the statistical data with its corresponding frequencies.
- Absolute frequency: number of times that a value appears. It is represented as $$f_i$$ where the subscript represents each of the values. The sum of the absolute frequencies is equal to the total number of data, represented bas $$N$$.
$$$f_1+f_2+f_3+\ldots+f_n=N$$$ equivalent to: $$$\sum_{i=1}^n f_i=N$$$
- Relative frequency: the result of dividing the absolute frequency of a certain value by the total number of data. It is represented as $$n_i$$. The sum of the relative frequencies is equal to $$1$$. We can prove this easily by factorizing $$N$$.
$$$n_i=\displaystyle \frac{f_i}{N}$$$
- Cumulative frequency: the sum of absolute frequencies of all the values equal to or less than the considered value. This is represented as $$F_i$$.
- Relative cumulative frequency: the result of dividing the cumulative frequency by the total number of information, which is represented by $$N_i$$ (when we are dealing with cumulative frequencies, the letters to represent them are in capital letters).
$$15$$ students answer the question of how many brothers or sisters they have. The answers are:
$$$1, 1, 2, 0, 3, 2, 1, 4, 2, 3, 1, 0, 0, 1, 2$$$
Then, we can construct a table of frequencies
| Brothers | Absolute frequency $$f_i$$ | Relative frequency $$n_i$$ | Cumulative frequency $$F_i$$ | Relative cumulative frequency $$N_i$$ |
|---|---|---|---|---|
| $$0$$ | $$3$$ | $$\displaystyle \frac{3}{15}$$ | $$3$$ | $$\displaystyle \frac{3}{15}$$ |
| $$1$$ | $$5$$ | $$\displaystyle \frac{5}{15}$$ | $$3+5=8$$ | $$\displaystyle\frac{3}{15}+\frac{5}{15} =\frac{8}{15}$$ |
| $$2$$ | $$4$$ | $$\displaystyle \frac{4}{15}$$ | $$3+5+4=12$$ | $$\displaystyle \frac{12}{15}$$ |
| $$3$$ | $$2$$ | $$\displaystyle \frac{2}{15}$$ | $$3+5+4+2=14$$ | $$\displaystyle \frac{14}{15}$$ |
| $$4$$ | $$1$$ | $$\displaystyle \frac{1}{15}$$ | $$3+5+4+2+1=15$$ | $$\displaystyle\frac{15}{15}$$ |
| $$\sum$$ | $$15$$ | $$1$$ |
Notice that the difference between the cumulative frequency and the relative frequency is only that in the case of the relative we must divide by the total number of data. This can help us avoid unnnecessary calculations.