Overview
Presentation of Data 25
3
Presentation of Data
3.1 Introduction
3.2 Classification
3.3 Tabulation
3.4 Frequency Distribution
3.5 Cumulative Frequency
3.6 Diagram and Graphs
(a) Bar diagram
(b) Pie Diagram
(c) Histogram
(d) Frequency Curve or Frequency Polygon
(e) Ogives
Review Question
In previous we learnt about primary data, secondary data, tools of collecting primary data and various sources of secondary data. Once the data has been collected the logical next step is to present it in a form that looks attractive and helps the researcher analyze it with ease. The commonly used forms of presentation of the data are:
1. Classification,
2. Tabulation,
3. Frequency Distribution,
4. Cumulative Frequency Distribution,
5. Diagrams and graphs.
This form of presentation is used when a large body of numerical data is given in textual form or data arrays. The data is divided into smaller groups or classes on the basis of size or degree or some common characteristic of the variant.
Classification is invariably accompanied by tabulation and involves a process called frequency distribution.The number of times a certain value of the variant occurs in the raw data is called its frequency.
For example, in the raw data 8, 7, 5, 8, 4, 8, 3, 10, 8, the value 8 occurs 4 times. So frequency of 8 is 4.
The process of placing classified data into tabular form is known as tabulation. A table is a symmetric arrangement of statistical data in rows and columns. Rows are horizontal arrangements whereas columns are vertical arrangements. It may be simple, double or complex depending upon the type of classification.
Types of Tabulation:
The types of tabulation are explained as follows:
(1) Simple Tabulation or Oneway Tabulation: When the data are tabulated to one characteristic, it is said to be a simple tabulation or oneway tabulation. For example: Tabulation of data on the population of the world classified by one characteristic like religion is an example of a simple tabulation.
(2) Double Tabulation or Twoway Tabulation: When the data are tabulated according to two characteristics at a time, it is said to be a double tabulation or twoway tabulation. For example: Tabulation of data on the population of the world classified by two characteristics like religion and sex is an example of a double tabulation.
(3) Complex Tabulation: When the data are tabulated according to many characteristics, it is said to be a complex tabulation. For example: Tabulation of data on the population of the world classified by three or more characteristics like religion, sex and literacy, etc. is an example of a complex tabulation.
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.
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. Frequency distribution helps us:
To analyze the data.
To estimate the frequencies of the population on the basis of the ample.
To facilitate the computation of various statistical measures.
Frequency distribution is divided into two: Grouped and Ungrouped
(1) Grouped: Grouped frequency is frequency where class interval are formed. The values of class interval is called Class Limits. The minimum value is lower class limit and maximum value is upper class limit. Grouped frequency is again divided into two:
(a) Inclusive Method: In this, the class intervals are not in continuous formed. It is known as Discrete Classes.
(b) Exclusive Method: In this, the class intervals are in continuous formed. It is known as Continuous Classes.
(2) Ungrouped: Ungrouped Frequency is frequency where no class interval is formed. Thus, there is no lower and upper limits.
Solved Examples
Ex. (1) In a ‘tree plantation’ project of a certain school there are 45 students of 'Harit Sena.' The record of trees planted by each student is given below:
3, 5, 7, 6, 4, 3, 5, 4, 3, 5, 4, 7, 5 , 3, 6, 6, 5, 3, 4, 5, 7, 3, 5, 6, 4, 4, 3, 5, 6, 6, 4, 3, 5, 7, 3, 4, 5, 7, 6, 4, 3, 5, 4, 4, 7.
Prepare an ungrouped frequency distribution table of the data.
Ex. (2) The value of up to 50 decimal places is given below:
3.14159265358979323846264338327950288419716939937510
From this information prepare an ungrouped frequency distribution table of digits appearing after the decimal point.
Ex. (3) In a college, 46 students of 12thstandard, were told to measure the lengths of the pen in their compassboxes in centimetres. The data collected was as follows.
16, 15, 7, 4.5, 8.5, 5.5, 5, 6.5, 6, 10, 12, 13, 4.5, 4.9, 16, 11, 9.2, 7.3, 11.4, 12.7, 13.9, 16, 5.5, 9.9, 8.4, 11.4, 13.1, 15, 4.8, 10, 7.5, 8.5, 6.5, 7.2, 4.5, 5.7, 16, 5.7, 6.9, 8.9, 9.2, 10.2, 12.3, 13.7, 14.5, 10
By taking inclusive classes 05, 510, 1015.... Prepare a grouped frequency distribution table.
Ex. (4) In the tables given below, classmark and frequencies is given. Construct the frequency tables taking inclusive and exclusive classes:
(1)
Class Width (x)  Frequency (f) 
5  9 
15  9 
25  15 
35  13 
Solution:
Class Width = The difference between any two consecutive class.
= 15 – 5
= 10
Class Width = 10
Divide =
Now, Subtract 5 to each class and add 5 to each class to get class interval.
Frequency Distribution Table,
By Exclusive Method:
Class Interval  Class Marks (x)  Frequency (f) 
0 – 10  5  9 
10 – 20  15  9 
20 – 30  25  15 
30 – 40  35  13 
By Inclusive Method:
Class Interval  Class Marks (x)  Frequency (f) 
0.5 – 9.5  5  9 
10.5 – 19.5  15  9 
20.5 – 29.5  25  15 
30.5 – 39.5  35  13 
(2)
Class Width (x)  Frequency (f) 
22  6 
24  7 
26  13 
28  4 
Solution:
Class Width = The difference between any two consecutive class.
= 24 – 22
= 2
Class Width = 2
Divide =
Now, Subtract 1 to each class and add 1 to each class to get class interval.
Frequency Distribution Table,
By Exclusive Method:
Class Interval  Class Marks (x)  Frequency (f) 
21 – 23  22  6 
23 – 25  24  7 
25 – 27  26  13 
27  29  28  4 
By Inclusive Method:
Class Interval  Class Marks (x)  Frequency (f) 
21.5 – 22.5  22  6 
23.5 – 24.5  24  7 
25.5 – 26.5  26  13 
27.5 – 28.5  28  4 
Cumulative frequency is the running total of the frequencies. Cumulative Frequencies are number of observations above or below of a certain value.
There are two types of Cumulative Frequency.
(1) Cumulative Frequency – Less than type: Number of observation less than the upper boundary of a class is called “Less than type” cumulative frequency of that class.
(2) Cumulative Frequency – More than type: Number of observation more than or equal to the lower boundary of a class is called “More than type” cumulative frequency of that class.
Solved Examples:
Ex. (1) Calculate less than type cumulative frequency:
Class Interval  Frequency 
0 – 10  3 
10 – 20  10 
20 – 30  7 
30 – 40  8 
40 – 50  2 
Solution:
Class Interval  Frequency  Cumulative Frequency Less than type 
0 – 10  3  3 
10 – 20  10  3 + 10 = 13 
20 – 30  7  13 + 7 = 20 
30 – 40  8  20 + 8 = 28 
40 – 50  2  28 + 2 = 30 
 N = 30 

Ex. (2) Calculate more than type cumulative frequency:
Class Interval  Frequency 
0 – 10  3 
10 – 20  10 
20 – 30  7 
30 – 40  8 
40 – 50  2 
Solution:
Class Interval  Frequency  Cumulative Frequency More than type 
0 – 10  3  27 + 3 = 30 
10 – 20  10  17 + 10 = 27 
20 – 30  7  10 + 7 = 17 
30 – 40  8  2 + 8 = 10 
40 – 50  2  2 
 N = 30 

Ex. (3) Calculate Less than and More than type cumulative frequency:
Class Interval  Frequency 
150 – 153  5 
153 – 156  7 
156 – 159  15 
159 – 162  10 
162 – 165  5 
165 – 168  3 
Solution:
Class Interval  Frequency  Cumulative Frequency Less than type  Cumulative Frequency More than type 
150 – 153  5  5  40 + 5 = 45 
153 – 156  7  5 + 7 = 12  33 + 7 = 40 
156 – 159  15  12 + 15 = 27  18 + 15 = 33 
159 – 162  10  27 + 10 = 37  8 + 10 = 18 
162 – 165  5  37 + 5 = 42  3 + 5 = 8 
165 – 168  3  42 + 3 = 45  3 
 N = 45 


Ex. (4) Calculate Less than and More than type cumulative frequency:
Class Interval  Frequency 
1000 – 5000  45 
5000 – 10000  19 
10000 – 15000  16 
15000 – 20000  2 
20000 – 25000  5 
Solution:
Class Interval  Frequency  Cumulative Frequency Less than type  Cumulative Frequency More than type 
1000 – 5000  45  45  42 + 45 = 87 
5000 – 10000  19  45 + 19 = 64  23 + 19 = 42 
10000 – 15000  16  64 + 16 = 80  7 + 16 = 23 
15000 – 20000  2  80 + 2 = 82  5 + 2 = 7 
20000 – 25000  5  82 + 5 = 87  5 
 N = 87 


Diagrams and graphs are geometrical representations of a data. They do not involve any complex calculations and so are easy to draw. They are pleasing to the eye, give a bird’s – eye view of the whole data and highlight all its salient features. They are easy to interpret and facilitate comparisons between different sets of data. Some of the commonly used diagrams and graphs are:
(a) Bar Diagrams
(b) Pie Diagram (or Pie chart)
(c) Histogram
(d) Frequency Curve or Frequency Polygon
(e) Ogives (cumulative frequency curve)
A bar graph (also known as a bar chart or bar diagram) is a visual tool that uses bars to compare data among categories. A bar graph may run horizontally or vertically. The important thing to know is that the longer the bar, the greater its value.
Bar graphs consist of two axes. On a vertical bar graph, as shown above, the horizontal axis (or xaxis) shows the data categories. In this example, they are years. The vertical axis (or yaxis) is the scale. The coloured bars are the data series.Bar diagrams are of many types such as:
(1) Simple bar diagram: A simple bar chart is used to represent data involving only one variable classified on a spatial, quantitative or temporal basis. In a simple bar chart, we make bars of equal width but variable length, i.e. the magnitude of a quantity is represented by the height or length of the bars.
(2) Multiple bar diagram: In a multiple bars diagram two or more sets of interrelated data are represented (multiple bar diagram facilitates comparison between more than one phenomena). The technique of making a simple bar chart is used to draw this diagram but the difference is that we use different shades, colours, or dots to distinguish between different phenomena.
(3) Subdivided bar diagram: A subdivided or component bar chart is used to represent data in which the total magnitude is divided into different or components. In this diagram, first we make simple bars for each class taking the total magnitude in that class and then divide these simple bars into parts in the ratio of various components.
(4) Percentage bar diagram: In drawing a percentage bar chart, bars of length equal to 100 for each class are drawn in the first step and subdivided into the proportion of the percentage of their component in the second step. The diagram so obtained is called a percentage component bar chart or percentage stacked bar chart. This type of chart is useful to make comparisons in components holding the difference of total constants.
Solved Examples:
Ex. (1) With the help of given statistical data prepare a simple bar graph:
Years  1971  1981  1991  2001  2011 
Population (in crores)  55  68  85  103  121 
Solution:
Ex. (2) Draw a simple bar graph by to represent the following data:
Years  2000 – 2001  2001  2002  2002 – 2003  2003 – 2004  2004 – 2005 
Per Capita Income (in Rs.)  18000  19000  20000  24000  27000 
Solution:
1
Ex. (3) Following data gives imports and exports of India for a few years. Construct a suitable diagram to represent the given data:
(BMS, April, 2017)
Year  Rupees in crores  
Imports  Exports  
1994 – 1995  17,170  11,850 
1995 – 1996  19,760  11,010 
1996 – 1997  20,080  12,550 
1997 – 1998  22,330  15,710 
Solution: