Frequency table is a table which records the lists of items and uses the tally marks and shows us the number of times they occur. Here, each item contains its own particular frequency and each frequency are distributed with an interval between them. In simple words, a tabular display of frequency distribution is called as frequency table.
There are two types of frequency table:

Univariate Frequency Table:

In univariate frequency table, the frequency distributions are displayed as the lists ordered by quantity showing the number of times each value occurs.

Example: If we have a class of 10 students and asked them their favorite subject, then the univariate frequency table is displayed as follows:

Subject  Number of students 
 Maths  2
 Science  3
 Social  1
 English  4

Joint Frequency Table:

In Joint frequency table, the frequency distributions are displayed in two-way table. The total row and the total displays the marginal frequencies, while the body of the table displays the joint frequencies.
Example:

  Dance   TV   
Sports  Total 
Men   4  8  10 22 
 Women  5  12  8  25
 Total  9  20  18  47

In statistics, for arranging the large amount of data, the data is grouped into different classes, so that, we get a clear idea of the distribution. And, the range of such class of data id is called as the class intervals.

Class intervals are equal in width and are mutually exclusive. The ends of the class interval are called as class limits, the middle of the interval is a class mark. And, these are generally used to draw histogram.

Example:

Height(in cm)   Number of students
 146 - 150  2
 150 - 155  3
 155 - 160  5
 160 - 165  7
 165 - 170  6
 170 - 175  2

In the table above, height of 25 students of a class are divided into classes with the size of each class interval being 5.
Frequency is the number of times an item occurs. A frequency table is a table that lists each item with its frequency. Now, lets learn how to make a Frequency Table.

There are few steps to keep in mind while constructing a frequency table. They are:
  1. Collect the specific set of items into a table.
  2. Divide the items into different groups called classes with their particular frequencies.
  3. Draw 1st column showing the items and another column showing the frequencies.
  4. Now, let us tally the item into classes. Each item value falls into one class. 
  5. Count the tally and record the frequencies.

Solved Example

Question: The marks scored by the students of class 8 are as follows:

50, 35, 75, 50, 50, 40, 35, 45, 75, 75
45, 35, 40, 60, 40, 60, 40, 75, 45, 65

Present into information on a frequency table.
Solution:
Step 1: Let us first divide the items into different class and mark as tally symbols.

 Score  Tally Marks 
 30 - 35  III 
 35 - 40  IIII
 41 - 46  III
 46 - 50  III
 51 - 65  III
 66 - 75  IIII

Step 2: Count the tally marks and record the frequencies.

 Score   Tally Marks  Frequencies
 30 - 35  III  3
 35 - 40  IIII  4
 41 - 46  III  3
 46 - 50  III  3
 51 - 65  III  3
 66 - 75  IIII  4


The total frequency through the classes of frequency distribution is called as cumulative frequency table. Cumulative frequency table are very useful during the construction of histogram. They are very easy to calculate from the Frequency table.

Solved Example

Question: Construct a cumulative frequency table for the following data:

Marks of the student  Frequency(f)
 101 - 110  3
 111 - 120  5
 121 - 130  7
 131 - 140  10
 141 - 150  4

Solution:
Marks of the students  Frequency(f)  Cumulative frequency(cf) 
 101 - 110  3  3
 111 - 120  5  3 + 5 = 8
 121 - 130  7  8 + 7 = 15
 131 - 140  10  15 + 10 = 25
 141 - 150  4  25 + 4 = 29


The grouped frequency table is constructed to organize and simplify a larger set of items into smaller groups. The main purpose of the grouped frequency table is to find out how often each value appears within each group of the whole data. Grouped frequency tables are used if variables take a large number of values or the variable is continuous.

The first column displays all the possible number of classes and the second column displays the frequencies of each classes.

 Class(Marks) Frequency 
 10 - 14  1
 15 - 19  5
 20 - 24  3
 25 - 29  3
 30 - 34  5
 35 - 39  2
 40 - 44  2
 45 - 50  6
Total  27

Here, by the frequency, we get to know how many times the data occurs. For example, in the table above, we see that only 2 students have scored marks between 35 - 39.
When a table displays relative frequencies for different categories of a categorical variable, it is called as relative frequency table.

For example, the below 2 tables displays the favorite TV channels of 10 viewers and both are relative frequency table. The table(A) displays the relative frequency as a proportion, and the table(B) displays the relative frequencies as a percentage.

 Favorite Zee  Sony  Star 
 Proportion  0.5  0.3  0.2
               Table(A)

 Favorite Zee  Sony   Star
 Percentage  50  30  20
              Table(B)
Two way table is constructed to deal with bivariate or joint frequency table. It helps us to compare and analyze the data that shows two different categories in much simpler and effective way. Two way frequency tables are often involved with the number of rows and columns in the table.

Solved Example

Question:
Students   Maths  Science Social  Total 
 A  81  34  90  145
 B  64  15  29  108
 C  90  25  15  130
 D  97  43  22  162

From the table, analyze which student has scored the highest total marks.
Solution:
The above table displays 4 rows and 4 columns. And, it is a joint frequency table. By the table, we can clearly conclude that student D has scored the highest total marks of 162.

Given below are few problems based on frequency table:

Solved Example

Question: Prepare a frequency table for the data of marks scored by 30 students in Science test.
30, 35, 90, 95, 50, 40, 35, 35, 45, 70, 90, 30, 75, 90, 45, 80, 29, 80, 45, 65, 75, 95, 62, 62, 45, 60, 80, 29, 80, 75.
Also, find the range of data.
Solution:
First, let us find the range. This can be calculated by finding the difference between the highest value and the smallest value.

Range = Highest value - Smallest value
= 95 - 29
= 66.
$\therefore$ Range = 66.

Now, let us construct a frequency table by dividing the marks into different classes and mark the tally symbol with frequency.

 Score Tally Marks   Frequency
 25 - 30  IIII  4
 31 - 36  III  3
 37 - 41  I  1
 42 - 47  IIII  4
 47 - 55  I  1
 56 - 61  III  3
 62 - 67  I  1
 68 - 76  IIII  4
 77 - 85  IIII  4
 86 - 91  III  3
 92 - 100  II  2

Total Frequency is 30.