A frequency distribution is an orderly arrangement of data classified according to the magnitude of the observations. When the data are grouped into classes of appropriate size indicating the number of observations in each class we get a frequency distribution. By forming frequency distribution, we can summarize the data effectively. It is a method of presenting the data in a summarized form. Frequency distribution is also known as Frequency table.
Uses: Frequency distribution helps us
1. To analyze the data.
2. To estimate the frequencies of the population on the basis of the ample.
3. To facilitate the computation of various statistical measures.
Frequency distribution table (also known as frequency table) consists of various components.
Classes: A large number of observations varying in a wide range are usually classified in several groups according to the size of their values. Each of these groups is defined by an interval called class interval. The class interval between 10 and 20 is defined as 1020.
Class limits: The smallest and largest possible values in each class of a frequency distribution table are known as class limits. For the class 1020, the class limits are 10 and 20. 10 is called the lower class limit and 20 is called the upper class limit.
Class limit: Class limit is the midmost value of the class interval. It is also known as the mid value.
$\text{Mid value of each class}$ =
$\frac{(\text{lower limit + Upper limit})}{2}$.
If the class is 010, lower limit is 0 and upper limit is 10. So the mid value is
$\frac{(0 + 10)}{2}$ = $\frac{10}{2} $ = $5$.
Magnitude of a class interval: The difference between the upper and lower limit of a class is called the magnitude of a class interval.
Class frequency: The number of observation falling within a class interval is called class frequency of that class interval.
Construct a Frequency Distribution
A frequency distribution table is one way to organize data so that it makes more sense. The data so distributed is
called frequency distribution and the tabular form is called frequency
distribution table. Let us see with the help of example how to construct distribution table.
The frequency distribution table lists all the marks and also show how many times (frequency) they occurred.
The number which tells us how many times a particular data appears is
called the frequency. For example, 2 marks have been scored by five
students which means marks 2 occurs five times. Therefore, the frequency
of score 2 is five. Similarly, the frequency of marks 5 is three
because three students scored five marks.
A relative frequency distribution is a distribution in which relative frequencies are recorded against each class interval. Relative frequency of a class is the frequency obtained by dividing frequency by the total frequency. Relative frequency is the proportion of the total frequency that is in any given class interval in the frequency distribution.
Relative Frequency Distribution Table
If the frequency of the frequency distribution table is changed into
relative frequency then frequency distribution table is called as
relative frequency distribution table. For a data set consisting of n values. If f is the frequency of a particular value then the ratio '$\frac{f}{n}$' is called its relative frequency.
Solved Example
Question: Find the relative frequency from the data given below:
Class interval 
Frequency 
2025 
10 
2530 
12 
3035 
8 
3540 
20 
4045 
11 
4550 
4 
5055 
5 
Solution:
Relative frequency distribution table for the given data.
Here n = 70
Class interval 
Frequency (f) 
Relative Cumulative Frequency ($\frac{f}{n}$)

2025 
10 
10 / 70 = 0.143 
2530 
12 
12 / 70 = 0.171 
3035 
8 
8 / 70 = 0.114 
3540 
20 
20 / 70 = 0.286 
4045 
11 
11 / 70 = 0.157 
4550 
4 
4 / 7 0 = 0.057

5055 
5 
5 / 70 = 0.071 
Total 
n = 70 

One of the important type of frequency distribution is Cumulative frequency distribution. In cumulative frequency distribution, the frequencies are shown in the cumulative manner. The cumulative frequency for each class interval is the frequency for
that class interval added to the preceding cumulative total. Cumulative
frequency can also defined as the sum of all previous frequencies up to
the current point.
Cumulative Relative Frequency Distribution
Cumulative relative frequency distribution is one type of frequency distribution. The relative cumulative frequency is the cumulative frequency divided by the total frequency.
Simple frequency distribution is used to organized orderly the larger
data sets. When the number of cases being studied is large, it is
inconvenient to list them separately because the list would be too long. A simple frequency distribution shows the number of times each score occurs in a set of data. To find the frequency for score count how many times the score occurs.
A grouped frequency distribution is an ordered listed of a variable $X$, into groups in one column with a listing in a second column, the frequency column. A grouped frequency distribution is an arrangement class intervals and corresponding frequencies in a table.
There are certain rules to be remembered while constructing a grouped frequency distribution
1. The number of classes should be between $5$ and $20$.
2. If possible, the magnitude of the classes must be $5$ or multiple of $5$.
3. Lower limit of first class must be multiple of $5$
4. Classes are shown in the first column and frequencies in the second column.Grouped Frequency Distribution Table
Inclusive type of frequency distribution can be converted into exclusive type as in Table (b)
A frequency distribution with an interval width of 1 is reffered to an ungrouped frequency distribution. Ungroped frequency distribution is an arrangement of the observed values in ascending order. The ungrouped frequency distribution are those data, which are not arranged in groups. They are known as individual series. When the ungrouped data are grouped, we get the grouped frequency distribution.
For Example: A teacher gave a test to a class of 26 students. The maximum mark is 5. The marks obtained by the pupils are:
3

2

3

3

4

3

1

2

5

1 
5 
4 
2 
1 
1 
3 
3 
4 
1 
2 
1 
4 
5 
4 
2 
2 

Such data as above is called ungrouped (or raw) data.
We may arrange the marks in ascending or descending order. The data so represented is called an array.
1 
1

1

1

1

1

1

2

2

2

2

2



3 
3 
3 
3 
3 
3 
4 
4 
4 
4 
4 
4 
5

5

The difference between the greatest and the smallest
number is called range of the data. Thus for the above data, the range
is 5  1 which equals 4 marks.
A histogram is sometimes known as frequency distribution chart. It is a set of vertical bars whose areas are proportional to the frequencies. While constructing histogram, the variable is always taken on the horizontal axis and frequencies on the vertical axis. The width of the bars in the frequency distribution chart will be proportional to the class interval. The bars are drawn without leaving spaces between them. A frequency distribution chart or histogram in general represents a continuous curve.
The graphs of frequency distribution are frequency graphs that are used to reveal the characteristics of discrete and continuous data. Such graphs are more appealing to eye than the tabulated data. This helps us to facilitate comparative study of two or more frequency distributions. We can compare the shape and pattern of the two frequency distributions.
The most commonly used graphs of frequency distribution are
1. Histogram
2. Frequency polygon
3. Frequency Curve
4. Ogives (Cumulative frequency curves)
Mean of frequency distribution can be find by multiplying each midpoint by its frequency, and then divide by the total number of values in the frequency distribution, we have an estimate of the mean.
Mean =
$\frac{\sum\ f\ \times\ x}{n}$where, $f$ = frequency in each class
$n$ = sum of the frequencies.
Solved Example
Question: Find the mean for this set of data.
Solution:
Frequency distribution table for the set of data:
x

f  f * x

2 
5  10 
3 
2  6 
5 
6  30 
7

7  49 
Sum  n = $\sum f$ = 20
 $\sum fx$ = 95

Mean =
$\frac{\sum fx}{n}$ =
$\frac{95}{20}$=
$\frac{95}{20}$
=
$\frac{19}{4}$.
Solved Examples
Question 1: Given the following frequency distribution, Find the standard deviation of the data.
Solution:
x

f
 fx  x^{2}
 x^{2} f

6 
2
 12  36
 72 
7 
3  21  49  147 
8 
3  24  64  192 
9 
2  18  81  162 
Sum  $\sum$f = 10  $\sum$fx = 75   $\sum$ x^{2} f = 573 
Step 1:
($\sum$fx)^{2} = (75)^{2}
= 5625
Step 2:
The variance formula is:
s^{2} = $\frac{\sum x^{2}f  \frac{(\sum fx)^2}{n}}{n}$
=> s^{2 }= $\frac{573  \frac{5625}{10}}{10}$
= $\frac{573  562.5}{10}$
= $\frac{10.5}{10}$
= 1.05
and the standard deviation is
s = $\sqrt{1.05}$
= 1.03
Hence the standard deviation is 1.03
Question 2: The set of data below shows the marks of 30 students. Draw a cumulative frequency table for the data.
Solution:
The cumulative frequency table for the data.
Class (marks) 
Frequency (f) 
Cumulative Frequency (cf)

1015 
2

2 
1520 
2 
2 + 2 = 4

2025 
4 
4 + 4 = 8

2530 
3 
8 + 3 = 11

3035 
6 
11 + 6 = 17

3540 
6 
17 + 6 = 23

4045 
4 
23 + 4 = 27

4550 
3 
27 + 3 = 30

Question 3: Draw the frequency distribution table for the given ungrouped data.
Consider the following ungrouped marks (out of 50) given to 30 students:
24
 30
 36
 35
 42
 40
 26
 23

36  36  12  45  29  21  34  40 
16  47  28  32  33  44  19  34 
30  36  35  49  20  14 


Solution:
Frequency distribution table for the given data:
The
range for the above ungrouped data is 49  12 = 37. Normally it is
desirable to divide the range into 6 to 10 classes. Consider the class
11  15. If a student scores 11 marks or 15 marks, he will be put in
this class. For this class, 11 is the lower limit and 15 is the upper
limit and both are included in the class.
Question 4: Consider this example for exclusive type of distribution. The
following is a survey of the pocket money of 40 students in a school
(pocket money in rupees per week):
Solution:
The range for the above ungrouped data is obtained. The range is 78  27, which equals 51. We make intervals of 20  30, 30  40, 40  50,…
For class 20  30, we read it to mean 20 and above but less than 30. For class 30  40, we read it to mean 30 and above but less than 40 and so on. Now frequency distribution table is obtained.
Here, overlapping classes are selected. Therefore, lower limits and actual lower limits as also the upper limits and actual upper limits are the same.