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# Introduction to Quantitative Techniques

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It includes introduction to quantitative techniques; Meaning, Importance applications and Limitations of statistics. Primary vs Secondary Data and their collection methods, Different graphs and their examples. Classification of data, types of data/series etc.

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### Introduction to Quantitative Techniques

1. 1. INTRODUCTION - STATISTICS
12. 12. 12 BirinderSingh,AssistantProfessor,PCTE Ludhiana
13. 13. 13 BirinderSingh,AssistantProfessor,PCTE Ludhiana
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18. 18. COURSE – UNIT 1  Introduction to statistics: Meaning, scope, importance and limitations, applications of inferential statistics in managerial decision-making.  Analysis of data: Source of data, collection, classification, tabulation, depiction of data.  Measures of Central tendency: Arithmetic, weighted, geometric mean, median and mode.  Measures of Dispersion: Range, Quartile deviation, Mean deviation, Standard deviation Coefficient of variation, Skewness and Kurtosis 18 BirinderSingh,AssistantProfessor,PCTE Ludhiana
19. 19. COURSE – UNIT 2  Sampling and Sampling Distribution: Concept and definitions, census and sampling, probability samples and non-probability samples, relationship between sample size and errors, simple numerical only.  Hypothesis Testing: Sampling theory, Formulation of Hypotheses, Application of Z-test, t-test, F-test and Chi-Square test, Techniques of association of attributes & testing. Test of significance for small sample 19 BirinderSingh,AssistantProfessor,PCTE Ludhiana
20. 20. COURSE – UNIT 3  Correlation Analysis: Significance, types, methods of correlation analysis: Scatter diagrams, Graphic method, Karl Pearson’s correlation co-efficient, Rank correlation coefficient, Properties of Correlation.  Regression analysis: meaning, application of regression analysis, difference between correlation & regression analysis, regression equations, standard error and Regression coefficients.  Index Number: Definition, and methods of construction, tests of consistency, base shifting, splicing and deflation, problems in construction and importance of index number. 20 BirinderSingh,AssistantProfessor,PCTE Ludhiana
21. 21. COURSE – UNIT 4  Time Series Analysis: Meaning, Components and various methods of time series analysis, Trend analysis: Least Square method - Linear and Non- Linear equations, Applications in business decision-making.  Theory of Probability: Definition, basic concepts, events and experiments, random variables, expected value, types of probability, classical approach, relative frequency and subjective approach to probability, theorems of probability, addition, Multiplication and Bayes Theorem and its application.  Theoretical Distributions: Difference between frequency and probability distributions, Binomial, Poisson and normal distribution Note: Relevant Case Studies should be discussed in class 21 BirinderSingh,AssistantProfessor,PCTE Ludhiana
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23. 23. STATISTICS  It is set of procedures and rules…for reducing large masses of data to manageable proportions and for allowing us to draw conclusions from those data  It helps businessmen in drawing inferences from the available data and take the decisions accordingly BirinderSingh,AssistantProfessor,PCTE Ludhiana 23
24. 24. MEANING  PLURAL SENSE – It refers to numerical statements of facts relating to any field of enquiry such as data relating to production, income, expenditure, population, prices, etc.  SINGULAR SENSE – It refers to a science in which we deal with the techniques or methods for collecting, classifying, presenting, analyzing and interpreting the data. BirinderSingh,AssistantProfessor,PCTE Ludhiana 24
25. 25. WHAT CAN STATS DO?  Make data more manageable  Group of numbers: 6, 1, 8, 3, 5, 4, 9  Average is: 36/7 = 5.14  Graphs: 0 10 20 30 40 50 60 70 80 90 1st Qtr 2nd Qtr 3rd Qtr 4th Qtr East West North BirinderSingh,AssistantProfessor,PCTE Ludhiana 25
26. 26. WHAT CAN STATS DO?  Allow us to draw conclusions from the data  Sachin’s Scores: 60, 100, 80, 30, 50  Sachin’s Average is 350/5 = 70  Sehwag’s Scores: 10, 150, 25, 5, 110  Sehwag’s Average is 300/5 = 60  Allows us to do this objectively and quantitatively BirinderSingh,AssistantProfessor,PCTE Ludhiana 26
27. 27. APPLICATIONS OF STATISTICS IN DAY TO DAY LIFE & IN BUSINESS  Weather Forecasts  Emergency Preparedness  Political Campaigns  Insurance  Consumer Goods  Family Budgets  Consumer Goods  Stock Market  Quality Testing BirinderSingh,AssistantProfessor,PCTE Ludhiana 27
28. 28. FEATURES OF STATISTICS AS SCIENCE (STAGES OF STATISTICS) Collection of Data Organization of Data Presentation of Data Analysis of Data Interpretation of Data BirinderSingh,AssistantProfessor,PCTE Ludhiana 28
29. 29. SCOPE OF STATISTICS Scope Nature Science Art Subject Matter Descriptive Statistics Inferential Statistics Limitations BirinderSingh,AssistantProfessor,PCTE Ludhiana 29
30. 30. FUNCTIONS OF STATISTICS  To Present Facts in Definite Form  Precision to the Facts  Comparisons  Forecasting  Policy Making  Enlarges Knowledge  To Measure Uncertainty  Establishes relationship between facts  Helps other sciences in testing their laws BirinderSingh,AssistantProfessor,PCTE Ludhiana 30
31. 31. IMPORTANCE OF STATISTICS  Administrative Policies  Industry/Business  Agriculture  Economics / Economic Planning  Politicians  Science & Research  Banking & Insurance  Education BirinderSingh,AssistantProfessor,PCTE Ludhiana 31
32. 32. LIMITATIONS OF STATISTICS  Study of numerical facts only  Study of aggregates only  Homogeneity of Data  Can be used only be experts  Qualitative Aspect Ignored  It does not depict entire story of phenomenon  Misuse of Statistics is possible  Results are true only on average  Statistical results are not always beyond doubt; only means & not a solution BirinderSingh,AssistantProfessor,PCTE Ludhiana 32
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34. 34. TYPES OF DATA Primary Data • Data collected by the investigator for his own purpose, for the first time. It is also called first hand data • Primary data includes information collected from interviews, experiments, surveys, questionnaires, focus groups and measurements Secondary Data • It is widely available and obtained from another party. It is also called second hand data • Secondary data can be found in publications, journals and newspapers. BirinderSingh,AssistantProfessor,PCTE Ludhiana 34
35. 35. PRIMARY VS SECONDARY DATA Basis for Comparison Primary Data Secondary Data Data Real time data Past data Process Very involved Quick and easy Source Surveys, observations, experiments, questionnaire, personal interview, etc. Government publications, websites, books, journal articles, internal records etc. Cost effectiveness Expensive Economical Collection time Long Short Specific Always specific to the researcher's needs. May or may not be specific to the researcher's need. Available in Crude form Refined form Accuracy and Reliability More Relatively less BirinderSingh,AssistantProfessor,PCTE Ludhiana 35
36. 36. METHODS OF COLLECTING PRIMARY DATA Observation Method • Structured & Unstructured Observation • Participant, Non Participant & Disguised Observation • Controlled & Uncontrolled Observation Interview Method • Personal Interview • Telephonic Interview • Video Conferencing Questionnaire Schedules filled through Enumerators BirinderSingh,AssistantProfessor,PCTE Ludhiana 36
37. 37. OBSERVATION METHOD  The observation method is the most commonly used method specially in studies relating to behavioral science.  Observation becomes a scientific tool and the method of data collection for the researcher, when it serves a formulated research purpose, is systematically planned and recorded and is subjected to checks and controls on validity and reliability.  It is also a process of recording the behavior patterns of people, objects, and occurrences without questioning or communicating with them. BirinderSingh,AssistantProfessor,PCTE Ludhiana 37
38. 38. OBSERVATION METHOD  Structured Observation: This means observation of an event personally by the observer when it takes place. This method is flexible and allows the observer to see and record subtle aspects of events and behaviour as they occur. He is also free to shift places, change the focus of the observation. Example: Observer is physically present to monitor  Unstructured Observation: This does not involve the physical presence of the observer, and the recording is done by mechanical, photographic or electronic devices. Example : Recording customer and employee movements by a special motion picture camera mounted in a department of large store. BirinderSingh,AssistantProfessor,PCTE Ludhiana 38
39. 39. OBSERVATION METHOD  Participant Observation: In this observation, the observer is a part of the phenomenon or group which observed and he acts as both an observer and a participant. Example: a study of tribal customs by an anthropologist by taking part in tribal activities like folk dance. The person who are observed should not be aware of the researcher’s purpose. Then only their behaviour will be ‘natural.’ BirinderSingh,AssistantProfessor,PCTE Ludhiana 39
40. 40. OBSERVATION METHOD  Non - Participant Observation: In this method, the observer stands apart and does not participate in the phenomenon observed. Naturally, there is no emotional involvement on the part of the observer. This method calls for skill in recording observations in an unnoticed manner. Example: Use of recording devices to examine the details of how people talk and behave together. BirinderSingh,AssistantProfessor,PCTE Ludhiana 40
41. 41. OBSERVATION METHOD  Disguised Observation: In this method, the observer observes in such a manner that his presence is unknown to the people he is observing. Example: Investigation done in Police Custody BirinderSingh,AssistantProfessor,PCTE Ludhiana 41
42. 42. OBSERVATION METHOD  Controlled Observation: Controlled observation is carried out either in the laboratory or in the field. It is typified by clear and explicit decisions on what, how, and when to observe. It is primarily used for inferring causality, and testing casual hypothesis.  Uncontrolled Observation: This does not involve over extrinsic and intrinsic variables. It is primarily used for descriptive research. Participant observation is a typical uncontrolled one. BirinderSingh,AssistantProfessor,PCTE Ludhiana 42
43. 43. OBSERVATION METHODS  Not Biased  Data is not affected by past behavior  Natural behavior of the group can be recorded  Expensive  Limited Information  Unforeseen factors may interfere with the observational task Merits Demerits BirinderSingh,AssistantProfessor,PCTE Ludhiana 43
44. 44. INTERVIEW METHOD  Personal Interview: It is a face to face two way communication between the interviewer and the respondents. Generally the personal interview is carried out in a planned manner and is referred to as ‘structured interview’. This can be done in many forms e.g. door to door or as a planned formal executive meeting.  Telephonic Interview: the information is collected from the respondent by asking him questions on the phone is called as telephone interview.  Video Conferencing: The combination of video camera and computer is used for conducting this interview. BirinderSingh,AssistantProfessor,PCTE Ludhiana 44
45. 45. INTERVIEW METHOD  Accuracy of data  Reliability of data  Flexibility of questions  Originality of data  Biased  Costly  Not proper for wide areas  Wrong Conclusions Merits Demerits BirinderSingh,AssistantProfessor,PCTE Ludhiana 45
46. 46. QUESTIONNAIRE  In this method, a list of questions relating to the survey is prepared.  It can be sent to the interviewee in the following ways: o Through post o Through E Mail o Through personal presence o Online Surveys BirinderSingh,AssistantProfessor,PCTE Ludhiana 46
47. 47. QUESTIONNAIRE  Economical  Originality of data  Wider Area  Lack of Interest  Lack of flexibility  Limited use  Biased  Less Accuracy Merits Demerits BirinderSingh,AssistantProfessor,PCTE Ludhiana 47
48. 48. QUALITIES OF A GOOD QUESTIONNAIRE  Limited number of questions  Simplicity  Proper Order of the Questions  No Undesirable Questions  Avoid Calculations  Pre testing  Clear Instructions  Cross Verifications of Questions  Request for return BirinderSingh,AssistantProfessor,PCTE Ludhiana 48
49. 49. SECONDARY DATA  Government / Semi Government Publications  Reports of Committees & Commissions  Publications of Trade Associations  Publications of Research Associations  Journals & Papers  Publications of Research Scholars  International Publications  These data are collected by the government & private organizations and is not published. These are used as secondary data too. Published Sources Unpublished Sources BirinderSingh,AssistantProfessor,PCTE Ludhiana 49
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51. 51. CLASSIFICATION OF DATA  It is the process of arranging the data into different classes or groups according to their common characteristics.  According to Spurr & Smith, “Classification is the grouping of related facts into classes” BirinderSingh,AssistantProfessor,PCTE Ludhiana 51
52. 52. OBJECTIVES OF CLASSIFICATION  To make comparisons  To arrange the data in such a way that their similarities and dissimilarities become very clear  To point out the most important features of the data at a glance  To present the data in a brief form  To enable statistical treatment of the collected data  To make data attractive and effective BirinderSingh,AssistantProfessor,PCTE Ludhiana 52
53. 53. METHODS OF CLASSIFICATION Geographical Classification Chronological Classification Qualitative Classification Quantitative Classification BirinderSingh,AssistantProfessor,PCTE Ludhiana 53
54. 54. GEOGRAPHICAL CLASSIFICATION Number of Colleges in 2016 Punjab 70 Haryana 30 Jammu & Kashmir 20 Himachal Pradesh 25 BirinderSingh,AssistantProfessor,PCTE Ludhiana 54
55. 55. CHRONOLOGICAL CLASSIFICATION Population of India (in Cr.) Year 1971 54.8 Year 1981 68.4 Year 1991 84.4 Year 2001 102.8 Year 2011 121.0 BirinderSingh,AssistantProfessor,PCTE Ludhiana 55
56. 56. QUALITATIVE CLASSIFICATION  In this, data are classified on the basis of some attribute or quality such as sex, literacy, religion etc.  It is of two types:  Simple Classification: When only one attribute is studied i.e. Classification of population on the basis of Male & Female  Manifold Classification: When more than one attribute is studied i.e. Classification of population on the basis of Male & Female along with Rural & Urban. BirinderSingh,AssistantProfessor,PCTE Ludhiana 56
57. 57. QUANTITATIVE CLASSIFICATION  When data are classified on the basis of some characteristics which is capable of direct quantitative measurement such as height, weight, income, marks etc.  It is also called numerical or grouped classification Weight (in kgs.) No. of persons 70-80 50 80-90 30 90-100 15 100-110 5 BirinderSingh,AssistantProfessor,PCTE Ludhiana 57
58. 58. QUANTITATIVE CLASSIFICATION Variable • Characteristic capable of direct quantitative measurement • Eg. Height, Weight, Marks, Production, Consumption etc. Frequency • It is the quantity linked with the variable • Eg. No. of persons in weight range 70-80 kgs is 50 Weight (in kgs.) No. of persons 70-80 50 80-90 30 90-100 15 100-110 5 BirinderSingh,AssistantProfessor,PCTE Ludhiana 58
59. 59. FREQUENCY DISTRIBUTION Marks No. of students 10 5 12 6 15 10 18 3 20 1 Discrete Frequency Distribution Grouped Frequency Distribution Marks: Out of 20 Total Students: 25 Marks No. of students 0-20 1 20-40 6 40-60 10 60-80 5 80-100 3 Marks: Out of 100 Total Students: 25 BirinderSingh,AssistantProfessor,PCTE Ludhiana 59
60. 60. TERMINOLOGY ASSOCIATED WITH GROUPED FREQUENCY DISTRIBUTION  Class Interval/Class: Its group of numbers in which items are placed. Eg. 0-20, 20-40, 40-60 etc.  Class Frequency (f): The number of observation falling within a class. Eg. “10” against 40-60  Class Limits: Each class is located between two limits. The lower value of a class is called lower limit & higher value is called upper limit. Eg. In 10-20: 10 is LL (l1) & 20 is UL.(l2) Marks No. of students 0-20 1 20-40 6 40-60 10 60-80 5 80-100 3 BirinderSingh,AssistantProfessor,PCTE Ludhiana 60
61. 61. TERMINOLOGY ASSOCIATED WITH GROUPED FREQUENCY DISTRIBUTION  Class Mark/ Mid Value: It is the average value of l1 & l2. MV = (l1 + l2)/2  Class Size: The width or class size or magnitude of a class is the difference between its lower and upper class limits. It is denoted by i= l2 – l1 Marks No. of students 0-20 1 20-40 6 40-60 10 60-80 5 80-100 3 BirinderSingh,AssistantProfessor,PCTE Ludhiana 61
62. 62. TYPES OF SERIES IN GROUPED FREQUENCY DISTRIBUTION Exclusive Series Inclusive Series Open End Series Mid Value Series Cumulative Frequency Series BirinderSingh,AssistantProfessor,PCTE Ludhiana 62
63. 63. EXCLUSIVE SERIES Marks No. of students 0-20 1 20-40 6 40-60 10 60-80 5 80-100 3 BirinderSingh,AssistantProfessor,PCTE Ludhiana 63
64. 64. INCLUSIVE SERIES Marks Frequency 10-14 4 15-19 5 20-24 8 25-29 5 30-34 4 Total 26 Marks Frequency 9.5-14.5 4 14.5-19.5 5 19.5-24.5 8 24.5-29.5 5 29.5-34.5 4 Total 26 Inclusive Series To Exclusive Series BirinderSingh,AssistantProfessor,PCTE Ludhiana 64
65. 65. OPEN ENDED SERIES Marks Frequency Less than 5 4 5-10 5 10-15 8 15-20 5 20 and above 4 Total 26 Marks Frequency 0-5 4 5-10 5 10-15 8 15-20 5 20-25 4 Total 26 Open Ended Series To Exclusive Series BirinderSingh,AssistantProfessor,PCTE Ludhiana 65
66. 66. FREQUENCY SERIES CONTAINING MID VALUE Mid Value Frequency 5 4 15 5 25 8 35 5 45 4 Total 26 Marks Frequency 0-10 4 10-20 5 20-30 8 30-40 5 40-50 4 Total 26 Mid Value Series To Exclusive Series BirinderSingh,AssistantProfessor,PCTE Ludhiana 66
67. 67. CUMULATIVE FREQUENCY SERIES Marks Frequency Less than 10 4 Less than 20 20 Less than 30 40 Less than 40 48 Less than 50 50 Marks Frequency 0-10 4 10-20 20 – 4 = 16 20-30 40 – 20 = 20 30-40 48 – 40 = 8 40-50 50 – 48 = 2 Total 50 Less Than Type Series To Exclusive Series BirinderSingh,AssistantProfessor,PCTE Ludhiana 67
68. 68. CUMULATIVE FREQUENCY SERIES Marks Frequency More than 0 50 More than 20 48 More than 40 40 More than 60 20 More than 80 6 Marks Frequency 0-20 50 – 48 = 2 20-40 48 – 40 = 8 40-60 40 – 20 = 20 60-80 20 – 6 = 14 80-100 6 Total 50 More Than Type Series To Exclusive Series BirinderSingh,AssistantProfessor,PCTE Ludhiana 68
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70. 70. INTRODUCTION  Whenever verbal problems involving a certain situation is presented visually before the learners, it makes easier for the learner to understand the problem and attempt its solution.  Similarly, when the data are presented pictorially (or graphically) before the learners, it makes the presentation eye-catching and more intelligible.  The learners can easily see the salient features of the data and interpret them. BirinderSingh,AssistantProfessor,PCTE Ludhiana 70
71. 71. INVENTOR OF GRAPHS  William Playfair (1759-1823), Scottish engineer and political economist, is the principal inventor of statistical graphs.  In 1786, he published “Commercial and Political Atlas” that contained 44 charts.  He invented three of the four basic forms of graph: o The statistical line graph o The bar graph o The pie graph BirinderSingh,AssistantProfessor,PCTE Ludhiana 71
72. 72. TYPES OF GRAPHS  Line graphs -Polygraph  Bar graphs  Pie graphs  Flow Charts BirinderSingh,AssistantProfessor,PCTE Ludhiana 72
73. 73. LINE GRAPH  The line graphs are usually drawn to represent the time series data Example: temperature, rainfall, population growth, birth rates and the death rates. 1000 1500 1750 1250 1000 2000 0 500 1000 1500 2000 2500 2011 2012 2013 2014 2015 2016 P r i c e Year Share Price of Co. ABC Ltd. BirinderSingh,AssistantProfessor,PCTE Ludhiana 73
74. 74. 20 18 19 10 18 20 17 15 16 15 0 5 10 15 20 25 Test 1 Test 2 Test 3 Test 4 Test 5 Comparative Score Card Ashima Dhruv BirinderSingh,AssistantProfessor,PCTE Ludhiana 74
75. 75. LINE GRAPH BirinderSingh,AssistantProfessor,PCTE Ludhiana 75
76. 76. 1.2. POLYGRAPH • Polygraph is a line graph in which two or more than two variables are shown on a same diagram by different lines. It helps in comparing the data. Examples which can be shown as polygraph are: – The growth rate of different crops like rice, wheat, pulses in one diagram. – The birth rates, death rates and life expectancy in one diagram. – Sex ratio in different states or countries in one diagram. BirinderSingh,AssistantProfessor,PCTE Ludhiana 76
77. 77. POLYGRAPH 1000 1500 1750 1250 1000 2000 375 625 750 55 450 850 1500 1600 1800 1600 1500 1900 0 500 1000 1500 2000 2500 2011 2012 2013 2014 2015 2016 P r i c e Year Share Price of 3 Companies ABC Ltd. XYZ Ltd. PQR Ltd. BirinderSingh,AssistantProfessor,PCTE Ludhiana 77
78. 78. POLYGRAPH BirinderSingh,AssistantProfessor,PCTE Ludhiana 78
79. 79. BAR GRAPHS It is also called a columnar diagram. The bar diagrams are drawn through columns of equal width. Following rules were observed while constructing a bar diagram: (a) The width of all the bars or columns is similar. (b) All the bars should be placed on equal intervals/distance. (c) Bars are shaded with colors or patterns to make them distinct and attractive. BirinderSingh,AssistantProfessor,PCTE Ludhiana 79
80. 80. TYPES OF BAR GRAPHS  Three types of Bar Graphs are used to represent different data sets: o The simple bar diagram o Compound bar diagram o Polybar diagram BirinderSingh,AssistantProfessor,PCTE Ludhiana 80
81. 81. THE SIMPLE BAR DIAGRAM  A simple bar diagram is constructed for an immediate comparison. It is advisable to arrange the given data set in an ascending or descending order and plot the data variables accordingly. However, time series data are represented according to the sequencing of the time period. BirinderSingh,AssistantProfessor,PCTE Ludhiana 81
82. 82. 8 14 16 25 32 37 34 32 30 26 20 14 0 5 10 15 20 25 30 35 40 Temperature (in C) Temperature (in C) BirinderSingh,AssistantProfessor,PCTE Ludhiana 82
83. 83. THE SIMPLE BAR DIAGRAM BirinderSingh,AssistantProfessor,PCTE Ludhiana 83
84. 84. COMPOUND BAR DIAGRAM  When different components are grouped in one set of variable or different variables of one component are put together, their representation is made by a compound bar diagram. In this method, different variables are shown in a single bar with different rectangles. BirinderSingh,AssistantProfessor,PCTE Ludhiana 84
85. 85. COMPOUND BAR GRAPH 0 50 100 150 200 250 300 2013 2014 2015 2016 80 75 85 100 85 81 88 90 90 82 95 100 Scoresachievedineachsubject Year Subject A Subject B Subject C BirinderSingh,AssistantProfessor,PCTE Ludhiana 85
86. 86. COMPOUND BAR DIAGRAM BirinderSingh,AssistantProfessor,PCTE Ludhiana 86
87. 87. POLYBAR DIAGRAM  The line and bar graphs as drawn separately may also be combined to depict the data related to some of the closely associated characteristics such as the climatic data of mean monthly temperatures and rainfall. BirinderSingh,AssistantProfessor,PCTE Ludhiana 87
88. 88. POLYBAR DIAGRAM BirinderSingh,AssistantProfessor,PCTE Ludhiana 88
89. 89. PIE GRAPHS  Pie diagram is another graphical method of the representation of data. It is drawn to depict the total value of the given attribute using a circle. Dividing the circle into corresponding degrees of angle then represent the sub– sets of the data. Hence, it is also called as Divided Circle Diagram. BirinderSingh,AssistantProfessor,PCTE Ludhiana 89
90. 90. PIE CHART 29% 23% 22% 26% Sales in Cr. of ABC Ltd. in 2015-16 1st Qtr 2nd Qtr 3rd Qtr 4th Qtr BirinderSingh,AssistantProfessor,PCTE Ludhiana 90
91. 91. PIE GRAPHS BirinderSingh,AssistantProfessor,PCTE Ludhiana 91
92. 92. FLOW MAPS/CHART  Flow chart is a combination of graph and map. It is drawn to show the flow of commodities or people between the places of origin and destination. It is also called as Dynamic Map.  Transport map, which shows number of passengers, vehicles, etc., is the best example of a flow chart. BirinderSingh,AssistantProfessor,PCTE Ludhiana 92
93. 93. CONCLUSION  If the information is presented in tabular form or in a descriptive record, it becomes difficult to draw results.  Graphical form makes it possible to easily draw visual impressions of data.  The graphic method of the representation of data enhances our understanding.  It makes the comparisons easy.  Besides, such methods create an imprint on mind for a longer time BirinderSingh,AssistantProfessor,PCTE Ludhiana 93