Frequency data occur when we classify statistical data in respect of either a variable or an attribute. A frequency distribution is a tabular representation of statistical data, usually in an ascending order relating to a measurable characteristic according to the individual value or a group of values of the characteristic under study.

Frequency distribution is a statistical table that distributes the total frequency to a number of classes. 
In case, the characteristic under consideration is an attribute, then the tabulation is made by allocating numerical figures to the different classes the attribute belongs to. When tabulation is done in respect of a discrete random variable, it is known as Discrete or ungrouped frequency distribution and in case the characteristic under consideration is a continuous variable, the classification is termed as Grouped frequency distribution.

Given below are the important terms associated with a frequency distribution table.

Class Limit (CL): Corresponding to a class interval, the class limit is defined as the minimum value and the maximum value, the class interval may contain. The minimum value is known as the lower class limit (LCL) and the maximum value is known as the upper class limit (UCL).

Class Boundary (CB): Class boundaries are found by adding 0.5 to the upper class limit and subtracting 0.5 from the lower class limit.

Mid-point or class mark: Corresponding to a class interval, midpoint is the total of the two class limits or class boundaries to be divided by two.
Mid-point = $\frac{LCL + UCL}{2}$ 

Width or size of a class interval: Width of a class interval is the difference between the upper class boundary and the lowest class boundary of the class interval. 
For the construction of a frequency distribution we need to go through the following steps.
  1. Find the largest and the smallest observations and obtain the difference between them, known as range, in case of a continuous variable.
  2. Form a number of classes depending on the number of isolated values assumed by a discrete variable. In case of a continuous variable, find the number of class intervals using the relation, number of class interval 'x', class length approximate range.
  3. Present the class interval in a table known as frequency distribution table.
  4. Apply tally mark, i.e. a stroke against the occurrence of a particular value in a class interval.
  5. Count the tally marks and present these numbers in the next column, known as frequency column and finally check whether the total of all these frequencies tally with the total number of observations.
Relative frequency of a class is the frequency of the class divided by the total number of frequencies of the class. Relative frequency is expressed as a percentage. A relative frequency distribution table is constructed in a similar manner to that of the frequency distribution table.

Solved Example

Question: The weight of 30 persons were given in the study considered. Find the relative frequency.
 Weight (in Kgs) 0 - 4
 5 - 9  10 - 14  15 - 19
 Number of persons $f_{i}$  3  5  10  12

Solution:
Weight (in Kgs)
Number of persons
Relative frequency
 0 - 4  3 $\frac{3}{30}$   or 0.1    or 10%

 5 - 9  5  $\frac{5}{30}$  or 0.17  or 17%

 10 - 14  10  $\frac{10}{30}$  or 0.33 or 33%

 15 - 19  12  $\frac{12}{30}$  or 0.4  or 40%
 Total $\sum f_{i}$ = 30
 


The total frequency of all classes less than the upper class boundary of a given class is known as the cumulative frequency of that class. A table showing cumulative frequencies is known as cumulative frequency distribution. Cumulative frequency distribution are of two types.

Less than cumulative frequency distribution:
This is obtained by adding successively the frequencies of all the previous classes including the class against which it is written. The cumulative is started from the lowest to the highest size.

More than cumulative frequency distribution:
This is obtained by finding the cumulative total of frequencies starting from the highest to the lowest class.
A cumulative relative frequency distribution is the aggregate proportion of observations less than or equal to the category. It is used to know the percentage of observations falling below a certain specified value. Relative frequency is the proportion or fraction of the total number of items and it is found using a frequency table. Frequency table sorts the data into groups counting the frequency of each group within the data set.

Given below is an example of cumulative relative frequency distribution.

Solved Example

Question: 15 students were asked how many hours they worked per day. Their responses in hours are listed below:
2, 2, 4, 7, 6, 6, 4, 4, 7, 3, 5, 5, 4, 3, 2.
Solution:
The given data of 15 students should be arranged in ascending order and a cumulative frequency table should be constructed.

Data
Frequency
Less than cumulative frequency
Cumulative relative frequency
 2  3  3  $\frac{3}{15}$ =  0.2 or 20%

 3  2  5   $\frac{5}{15}$ = 0.33 or 33%

 4  4  9   $\frac{9}{15}$ = 0.6 or 60%

 5  2  11   $\frac{11}{15}$ = 0.73 or 73%

 6  2  13   $\frac{13}{15}$ =  0.87 or 87%

 7  2  15   $\frac{15}{15}$ = 1 or 100%
 Total     $\sum f_{i}$ = 15    


In a grouped frequency distribution, the data will be sorted and separated into groups called classes. Given below are some points to be remembered, while constructing a grouped frequency distribution table.
  • The classes must be non-overlapping and continuous.
  • The classes must be of equal width. Else, the frequency distribution will have a distorted view of the data. It is good if a class width is an odd number, as the mid points will be integers instead of decimals.
  • All the data values must be included. So, the classes must be exhaustive.
  • In grouped frequency distribution, number of classes can be between 5 and 20.
Number of newspapers sold at a local shop over the past 20 days are as follows 
16, 17, 16, 18, 19, 25, 25, 35, 30, 29, 42, 18, 44, 32, 27.

Count of each number and their associated frequencies are shown below:
Papers sold
15  
16  
 17   18  
 19   20  
21  
 22   23  
  24  
25    26   27  
28  
 29   30  
31  
32  
 33   34  
 35  36  
  37  38  
39  
40  
41  
42  
43  
44
Frequency 0  2  1  2  0  0  1  0  0    0
 2  0 1
 0  1  1  0  1   0  0 1 0  0 0 0 0 0 10
 1

Given below is the table of grouped values in 5's.

Papers Sold  
Frequency
 15 - 19  5
 20 - 24  1
 25 - 29  4
 30 -  34  2
 35 - 39  1
 40 - 44  2
Ungrouped frequency distribution is used for data that is discrete or for continuous data that can be treated as discrete data. It is convenient to make a listing, that pairs each data value with the number of times that a data value occurs. It is used for data that can be enumerated and when the range of values in the data set is small and the sample size is large.

Follow these steps to form an ungrouped frequency distribution table.
  • List all the possible unique data elements.
  • For each data value in the data set, find the frequency for the corresponding data element.
  • List all the results in a tabular form.

Solved Example

Question: A greenhouse worker inspects 25 plants and counts the number of flower bud on each plant. The results are as shown below:
5, 2, 1, 8, 4, 3, 5, 1, 2, 3, 5, 2, 3, 3, 4, 7, 7, 8, 4, 7, 1, 2, 6, 6, 7.
Solution:
Data  Frequency
 1  3
 2  4
 3  4
 4  3
 5  3
 6  2
 7  4
 8  2


A frequency distribution is the organization of raw data in a table form using classes and frequencies. Frequency distribution graphs are used to organize data in a meaningful and intelligent way.

There are four most common graphs used in an frequency distribution.
  1. Histogram
  2. Ogives
  3. Frequency Polygon
  4. Frequency curves.
A very convenient way to represent a frequency distribution is a histogram. Histogram helps us to get an idea of the frequency curve of the variable under study. Some statistical measures can be obtained using a histogram. A comparison among the frequencies for different class intervals is possible in this mode of diagrammatic representation. 

In order to draw a histogram, the class limits are first converted to the corresponding class boundaries and a series of adjacent rectangles, one against each class interval, with the class interval as the base and the frequency (when the Class interval are not uniform) as length is erected.

Solved Example

Question: Plot the histogram for the marks of students in Mathematics paper. 
Class Interval
Frequency
 0 - 5  4
 5 - 10  10
 10 - 15  18
 15  - 20  8
 20 - 25   6

Solution:
Histogram plot for marks of students in mathematics is as follows:
Frequency Distribution Histogram

In a frequency distribution, mean is found by adding all the values together and dividing by the number of values in the set. Assuming that the values in each group are spread evenly throughout the group, mean will be approximately equal to the midpoint for each class. If we multiply each midpoint by its frequency and then divide by the total number of values in the frequency distribution, we get an estimate of mean.

$\bar{x}_{g}$ = $\frac{(\sum f\times m)}{n}$
where, f: frequency in each class
m: mid point of each class
n: Total of the frequencies
$\bar{x}_{g}$: Estimate of mean from a frequency distribution.
Given below are some of the examples on frequency distribution.

Solved Examples

Question 1: Convert the following distribution into more than frequency distribution.
Weekly wages (less than '00 Rs)
20  
40  
 60   80  
 100  
Number of workers  41   92  156  194   201

Solution:
We are given less than cumulative frequency distribution. To obtain the more than cumulative frequency distribution, we shall first convert it into continuous frequency distribution as shown below:

More than cumulative frequency distribution:
Weekly wages
(in '00 RS)
Number of workers
            (f)
More than cumulative frequency distribution  
 0 - 20  41  160 + 41 = 201
 20 - 40  92 - 41 = 51  109 + 51 = 160
 40 - 60  156 - 92 = 64  45 + 64 = 109
 60 - 80  94 - 156 = 38  7 + 38 = 45
 80 - 100  01 - 194 = 7                                 7
   N = 201
 

Given below is the more than cumulative frequency distribution table:

Weekly wages
(in '00 RS)
More than cumulative frequency distribution
 0  201
 20  160
 40  109
 60  45
 80  7
 100  0


Question 2: Following figure relates to the weekly wages of workers in a factory.
                           Wages (in '00 Rs)
100
100
101
 102 106
86
82
 87 109
104
 75  89  99  96  94  93  92  90  86  78
 79  84  83  87  88  89  75  76  76  79
 80  81  89  99  104  100  103  104  107  110
 110  106  102  107  103  101  101 101  86  94
 93  96  97  99  100  102  103  107  107  108
 109  94  93  97  98  99  100  97  88  86
 84  83  82  80  84  86  88  91  93  95
 95  95  97  98  100  105  106  103  85  84
 77  78  80  93  96  97  98  98  98  87
Prepare a frequency table by taking a class interval of 5.
Solution:
In the above distribution, the minimum value of the variable 'X', wages (in '00 Rs) is 75 and the maximum value is 110.
Given class interval = 5.
Since wages is a continuous variable, the frequency distribution with exclusive method would be appropriate.
As minimum value is 75, it is convenient to take as the lower limit of first class. The class interval are taken as 75 - 80, 80 - 85, .........110 -115.
The upper limit of each class is included in the next class.

The frequency distribution is given below.

 Weekly wages
    (in '00 Rs)
Number of workers
            (f)
 75 - 80  9 
 80 - 85  12
 85 - 90  15
 90 - 95  11
 95 -100   20
 100 - 105  20
 105 - 110  11
 110 - 115 2
   N = 100