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Frequency Distribution

Statistics is a very useful subject that is originated from mathematics. Its study is started in middle school and becomes a separate subject in higher-level mathematics. Statistics is a systematic study of collection, organization and analysis of data for a research or a survey. It is quite helpful in the forecast and prediction of various results from the well-organized data.

A statistical data may consists of a list of numbers related to a research. Among those numbers, few may be repeated twice and even more than twice. The repetition of number is a data set is termed as frequency of that particular number or the variable in which that number is assigned. The frequencies of variables in a data are to be listed in a table. This table is known as frequency distribution table and the list is referred as frequency distribution.

This page includes the study of frequency distribution with various examples. Our study on frequency distribution includes a study on cumulative frequency, mean frequency table. Frequency distribution can be defined as the tabulation of the values with one or more variables. A frequency distribution is defined as an orderly arrangement of data classified according to the magnitude of the observations.

There are many types of frequency distributions

  1. Grouped frequency distribution
  2. Ungrouped frequency distribution
  3. Cumulative frequency distribution
  4. Relative frequency distribution
  5. Relative cumulative frequency distribution

Related Calculators
Frequency Distribution Calculator Frequency Calculator
Calculate Relative Frequency Frequency and Wavelength Calculator
 

What is Frequency Distribution?

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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.

Table

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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 10-20.

Class limits: The smallest and largest possible values in each class of a frequency distribution table are known as class limits. For the class 10-20, 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 0-10, 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.

Frequency Distribution

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.

Relative Frequency Distribution

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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
20-25 10
25-30 12
30-35 8
35-40 20
40-45 11
45-50 4
50-55 5


Solution:
 
Relative frequency distribution table for the given data.

Here n = 70

Class interval Frequency (f) Relative Cumulative Frequency ($\frac{f}{n}$)
20-25 10 10 / 70 = 0.143
25-30 12 12 / 70 = 0.171
30-35 8 8 / 70 = 0.114
35-40 20 20 / 70 = 0.286
40-45 11 11 / 70 = 0.157
45-50 4 4 / 7 0 = 0.057
50-55 5 5 / 70 = 0.071
Total n = 70

 

Cumulative Frequency Distribution

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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

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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.

Grouped Frequency Distribution

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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

Grouped Frequency Distribution

Inclusive type of frequency distribution can be converted into exclusive type as in Table (b)

Ungrouped Frequency Distribution

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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.

Chart and Graph

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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

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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.

     x       
        f      
     2       5
     3       2
     5       6
     7
      7


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 = $\frac{\sum fx}{n}$ = $\frac{95}{20}$

= $\frac{95}{20}$

= $\frac{19}{4}$.
 

Example

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Solved Examples

Question 1: Given the following frequency distribution, Find the standard deviation of the data.

x
f
 6  2
 7  3
 8  3
 9  2

Solution:
 

x
f
 fxx2
x2 f
 6  2
 1236
 72
 7  3 21 49 147
 8  3 24 64 192
 9  2 18 81 162
 Sum $\sum$f = 10 $\sum$fx = 75  $\sum$ x2 f = 573

 
Step 1:

($\sum$fx)2 = (75)2

= 5625

Step 2:
The variance formula is:

s2 = $\frac{\sum x^{2}f - \frac{(\sum fx)^2}{n}}{n}$

=> s= $\frac{573 - \frac{5625}{10}}{10}$

= $\frac{573 - 562.5}{10}$

= $\frac{10.5}{10}$

= 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.

Frequency Distribution Example


Solution:
 
The cumulative frequency table for the data.

 Class (marks)  Frequency (f) Cumulative Frequency (cf)
 10-15 2
 2
 15-20  2  2 + 2 = 4
 20-25  4  4 + 4 = 8
 25-30  3  8 + 3 = 11
 30-35  6  11 + 6 = 17
 35-40  6  17 + 6 = 23
 40-45  4  23 + 4 = 27
 45-50  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:

Frequency Distribution Examples


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):

Example of Frequency Distribution


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.

Examples of Frequency Distribution


Here, overlapping classes are selected. Therefore, lower limits and actual lower limits as also the upper limits and actual upper limits are the same.


 

More topics in Frequency Distribution
Bivariate Frequency Distribution Frequency Distribution Example
Related Topics
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