## Introduction

In lesson Frequency and proportion we defined frequency as the number of times a particular value occurs in a data. For instance, if $5$ people in a chosen group have blue eyes, then the blue color of eyes is said to have frequency of $5$.

Sometimes we would like to collect and organize given data in a way that it shows frequencies of data values. In that cases we use **frequency tables**. They can show **qualitative** or **quantitative (numeric) variables**.

The first step in drawing a frequency table is to determine how many **categories** or **classes **we need. Categories are shown in the first column. In addition, collected data values should be shown in **ascending order**. Furthermore, each data value belongs to exactly one class.

In making frequency table we often use **tally marks**, which are written in the second column.

Finally, the third column are **frequencies** of given values.

Similarly, **relative frequency table **has three columns; first column contains classes, second column contains frequencies of given values and third column contains relative frequencies. We defined relative frequencies in lesson Frequency and proportion.

## Frequency table – examples

* Example 1: *The number of absent students in math class during one semester is given with the following data:

$$2, 5, 1, 1, 3, 4, 4, 4, 2, 3, 3, 4, 0, 0, 4, 4, 3, 6, 1, 4, 2, 2, 4, 3, 2, 1, 3, 2, 2, 5, 4, 0, 3, 2, 1, 2, 4, 1, 3, 3.$$

Construct a frequency table.

**Solution: **

To make the frequency table, we first need to write the categories (number of absent students) in one column.

The next step is to tally the numbers in each category. For instance, the number $0$ appears $3$ times so we put $3$ tally marks: $\vert \vert \vert$. When we reach to fifth tally mark, we draw another tally line through the previous $4$ marks. In that way we can easier read our results.

Finally, we simply sum the tally marks and write the frequencies in the last column.

* Example 2: *The key of the letters listed below is: C = cinema, T = theatre, M = museum. Furthermore, each letter represents a student who prefers that type of amusement.

$$C, \ T, \ T, \ M, \ C, \ C, \ T, \ T, \ M, \ C, \ M, \ T, \ T, \ C, \ C, \ T, \ C, \ C .$$

Draw a frequency table. Which type of amusement from the above has the minimum frequency and which the maximum frequency? Which is the most popular?

**Solution:**

Museum has the minimum frequency and cinema the maximum frequency.

In conclusion, the most popular type of amusement is cinema.

## Frequency table – example with classes

When we have a data set whose values are spread out, it is too complicated to insert every data value in a frequency table. Therefore, we should organize given data values by sorting them into classes.

The frequency of a class is the number of data values that belong to that class.

* Example 3: * The number of calls from customers per day for customer service was recorded for one month:

$$ 200, \ 215, \ 29, \ 47, \ 123, \ 128, \ 36, \ 40, \ 55, \ 178,$$

$$\ 174, \ 184, \ 113, \ 202, \ 217, \ 66, \ 74, \ 98, \ 92, \ 180,$$

$$\ 41, \ 37, \ 76, \ 80, \ 99, \ 105, \ 110, \ 207, \ 31, \ 30 .$$

Construct a frequency table.

**Solution:**

As mentioned in the introduction, we first need to figure out **how many categories (classes)** we need. The formula is:

$$\frac{log (number \ of \ observations)}{log(2)}.$$

We need to **round up** the result to the next integer.

Therefore,

$$\frac{log(30)}{log(2)} \approx 4.9 = 5.$$

After that, we need to calculate the **difference between the maximum and minimum value** of given data.

As we can see, the lowest data value is number $29$ and the highest is number $217$. Therefore, their difference is $217 – 29 = 188$.

Furthermore, we divide that difference by the calculated number of classes and round that number up to the following integer. In this way we will get the width of the class. In other words,

$$\frac{188}{5} \approx 37.6 = 38.$$

Since our minimum value is number $29$ and the width of the each class is $38$, the first class is: $29 – 66$. The frequency table is: