Frequency and proportion

We encounter the term ‘frequency’ in everyday situations. For instance, think about your school or college days and situations when the professor gave you your exam score. Students often ask themselves: ”How many of my colleagues got the same score?”. In other words, they ask themselves what is the frequency of the score.


Frequency and relative frequency

Let $S = \{s_{1}, \cdots, s_{N}\}$ be the finite population and $X: S \rightarrow O$ a variable, where $O$ is the set of properties. Let $X(S) =\{x_{1}, x_{2}, \cdots, x_{k}\}$ be the set of properties of population (values of a variable).

The absolute frequency (or simply frequency) is the number of times a specific value for a variable has occured. Furthermore,  a relative frequency is the number of times a specific value for a variable has occured in relation to the total number of values for that variable.

Precisely, the definition is:

Definition: Number of elements of a set $X^{-1}(x_{i}), i = 1, \cdots, k$, that is, the number of all elements of population which have the same property $x_{i}$, is called the frequency of property $x_{i}$ and we denote it by  $f_{i}$.

We say that number $p_{i}= \frac{f_{i}}{N}$ is a relative frequency or proportion, where $N$ is a number of elements of population.


$$N = \sum_{i = 1}^{k}f_{i}$$ and $$\sum_{i = 1}^{k}p_{i} = 1, p_{i} = \frac{f_{i}}{\sum_{i = 1}^{k}f_{i}}.$$

We can also express relative frequency as percentage. Thus, a relative frequency of $0.30$ is equivalent to a percentage of $30\%$.

Definition: Function which joins the corresponding (relative) frequency $f_{i} (p_{i}), i = 1, \cdots, k$ to every property $x_{i}$  is called (relative) frequency function, and set of points $$\{(x_{i}, f_{i}), i = 1, \cdots, k\} (\{(x_{i}, p_{i}), i = 1, \cdots, k\})$$ is a graph of that function.

In addition, we can calculate a relative frequency of number of properties by forming a relative frequency table. The first column represents category names, second column the counts and third column relative frequencies.



Example 1:   Observing an eye color of $11$ people

frequency-and-proportion 2022

Example 2:   Let’s say that people $s_{1}, s_{2},\cdots, s_{10}$ make a population and have $20, 18, 18, 30, 25, 20, 20, 18, 25, 20$ years, respectively. Then  properties (years) $18, 20, 25$ and $30$ have frequencies $3, 4, 2, 1$, respectively.

Example 3:   $40$ people were asked what kind of chocolate they like most:

$10$ people prefer dark chocolate

$21$ people prefer milk chocolate

$9$ people prefer white chocolate

Therefore, the relative frequencies are:

dark chocolate: $p_{1}= \frac{10}{40} = 0.25 = 25 \%$

milk chocolate: $p_{2} = \frac{21}{40} = 0.525 = 52.5 \%$

white chocolate: $p_{3} = \frac{9}{40} = 0.225 = 22.5 \%$

Obviously, $0.25 + 0.525 + 0.225 = 1$.

Finally, a relative frequency table is:

frequency-and-proportion 2022


Cumulative sequence

We often represent a numeric variable with finite sequence of its values $y_{1}, \cdots, y_{N}, y_{i} = X (s_{i}), i = 1, \cdots, N$, which we call statistical sequence.

If we sort the properties, i.e. the elements of a statistical sequence by some criterion, we”ll get a grouped statistical sequence. Furthermore, if $f_{1}, \cdots, f_{k}$ are frequencies, we define cumulative sequence:

$$F(x_{1}) = f_{1}$$

$$F(x_{2}) = f_{1} + f_{2}$$


$$F(x_{i}) = f_{1} + \cdots + f_{i}$$


$$F(x_{k}) = f_{1} + \cdots + f_{k} = N.$$

Here is an example of a cumulative frequency.

Example 4:  Suppose you sort your social network friends in several groups by their age: under $18$, $18 – 25$, $25 – 35$ and so on. Furthermore, frequencies for each group are given. Let’s say you want to know how many of your friends are under the age of $25$. In other words, you simply need to sum the frequencies of the first two groups.