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.
Obviously,
$$N = \sum_{i = 1}^{k}f_{i}$$ 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.
Examples
Example 1: Observing an eye color of $11$ people
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:
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:
$$F(x_{1}) = f_{1}$$
$$F(x_{2}) = f_{1} + f_{2}$$
$$\vdots$$
$$F(x_{i}) = f_{1} + \cdots + f_{i}$$
$$\vdots$$
$$F(x_{k}) = f_{1} + \cdots + f_{k} = N.$$
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.
