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.