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# Module 3

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### Module 3

1. 1. UNIT –03 DIAGRAMMATIC AND GRAPHIC REPRESENTATION Tabulation is the device of presenting the statistical data in concise, systematic and intelligible from, thus high lighting the salient features. But another important convincing, appealing and easily understood method of presenting the statistical data is the use of diagrams and graphs. DIAGRAMMATIC REPRESENTATION OF DATA: Diagrams occupy on important place in statistical methods and have universal utility. Diagrams are found in newspapers, advertisements exhibitions propaganda posters etc. diagrammatic presentation of data enables one to grasp the entire information at a glance. For a common man numbers are not so interesting, he is more attracted by charts, pictures or diagrams that the figures. Diagrams are comparable and appealing both to the eye and the intellect. Diagrams are visual aids which comprise of presenting statistical materials in pictures, geometric figures and curves. They present complex and unwieldy data in a simple and attractive manner. Objects of diagrams The main objects of diagrammatic representation are:- 1. To make a quick, lasting and accurate impression of the significant facts, because diagrams are very attractive impressive and interesting. 2. To make data simple and intelligible because diagrams do not strain the mind of observes. 3. To save time and labour in grasping the facts of the data and in drawing the conclusions. 4. As tools of analysis a diagram is visual guide in the planning mathematical computaltion and general procedure of research study. 5. To make comparison possible. USES OR UTILITY OR ADVANTAGES: Diagrams have universal utility and are widely used in economic, business administration, social and other fields. Diagrams are extremely useful because of the following reasons. 1. Bird’s eye – view: Diagrams give a birds eye – view of the entire data 2. Direct appeal: A diagram provides a clear picture with a more direct appeal. 3. Attractive and impressive: Diagrams facilitate comparison of statistical data relating to different periods of time of different regions. 4. Comparison: Diagrams facilitate comparison of statistical data relations to different periods of time or different regions. 5. Make data simple and intelligible: As a tool of presentation of data diagrams render complex data simple and easily understandable. 6. More information: Diagrams give more information than the data presented in the tabular form. 7. Specific knowledge:- No knowledge of mathematics is required to draw and understand them. Limitations:- Statistical diagrams have the following limitations 1. Need of same characteristics for comparison The diagrams constructed with same characteristics are only comparable. 2. To present a precise difference is not possible: It is not possible to present a precise difference between two sets of data in a diagram 3. Limited to two or three aspects A diagram can vividly show only a limited amount of information and it is limited to the portrayal of two or three aspects of a set of data. 27
2. 2. 4. Not of much use to a statistician:- Diagrams are meant mostly to explain and impress quantitative facts to the general public. For statistician, these diagrams are simply a tool and not the substitute for statistical analysis. 5. Easily Misinterpreted. The diagrams can be easily misinterpreted. Because diagrams cannot be accepted without a close inspection of the bonafides. 6. Illusory – Diagrams drawn on false – base line are illusory 7. Need of double presentation: Use of an inappropriate diagram may distort the facts and mislead the reader by giving a wrong impression, if he does not have a knowledge of the tabulated data. To avoid this confusion on has to adopt a practice of double presentation – the tables for detailed reference and the diagrams for rapid understanding. General Rules for drawing diagrams The following are some of the important points to be kept in mind which constructing a diagram. 1. Selection of proper diagram:- Extreme care should be taken in the selection of a proper diagram after careful study of the data. 2. Suitable title:- There should be a brief title of the diagram. 3. Attractive:- The diagrams should be attractive and self. Explanatory. 4. Proper scale:- A proper scale should be used and must be placed at the side of diagram so the some idea of the magnitude of each item is obtained. 5. Index:- The index of the colours or symbols used in the diagram should be given on the top corner of the diagrams. 6. Size: A diagram should neither be too big nor too small 7. Accuracy: Diagrams should be made with the help of geometrical tools. 8. Use of colour:- The drawing of the diagrams should be neat and clean with different colours. 9. Comparison: Every care should be taken to make the diagrams of a particular type so that comparisons can be made easily. Types of diagrams: There are various forms of presenting the data diagrammatically. The following are the common methods of diagrams. I. On the bases of dimension: 1. One – dimensional diagrams [bars] 2. Two – dimensional diagrams [squares & Rectangles] 3. Three – dimensional diagrams [cubes, cylinders etc] II.On the bases of view a. Pictograms b. Cartograms. [Note:- As per syllabus, we have only one – dimensional & two dimensional diagrams] ONE DIMENSIONAL DIAGRAMS: In one dimensional diagrams we consider only one dimension. These are the simplest and one of the most common diagrams, such diagrams are in the form of 1. Bar diagrams 2. Line diagrams [not included in syllabus] BAR DIAGRAMS:- A bar is a merely a thick line in one way In bar diagrams only the length or height is taken into consideration. The data is represented by the thick bars of uniform width leaving the uniform gaps in between the two bars. The length or height of the bars are taken proportional to the values they represent. The bars originate from common base line and may be drawn vertically or horizontally. The Bar diagrams can be classified into four main types. 1. Simple Bar diagram 2. Multiple Bar diagram 3. Sub – divided Bar diagram 4. Sub – divided Bar diagram based on percentage. Simple bar diagram:- 28
3. 3. Simple bar diagrams are used to represent individual observations, time series and spatial series when the comparison of magnitude of different items is done, the simple bar diagrams are widely used. The values of variable are taken either in ascending order or in descending order. We can use different colours or shades for each bar to identify the data and to make the diagram attractive. Problem No: 01 Draw a Vertical simple bar diagram from the following data relating to the number of small scale industrial units in various states during the year 2004. States: Karnataka T. Nadu Kerala Andhra Maharashtra MP UP No. of small scale units 70 80 65 50 90 80 95 (000) Solution: Scale for oy axix: 1 cm = 10000 units, 1mm = 1000 Diagrams showing number of small scale units in different states. Y 100 90 80 M A 70 H A K 60 R A K A R E 50 T. S N R N A H A A 40 A N T T L U D D R M A A HR P U- A P 30 K 6 A 9 8 9 8 A7 5 5 5 0 0 0 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 10 0 0 0 0 X MULTIPLE BAR DIAGRAMS: These diagrams are also known as compound bar diagrams. When two or more adjacent bars are drawn, such a diagram is called multiple bar diagram. In each set, we may have two or more bars joined together depending upon the attributes similar attributes in each period are presented for the purpose of comparison. For the identification, of different attributes, an index is prepared. ILLUSTATION: 02 Draw the Multiple bar diagram, showing the working population of me, women, and children during the year 2004. Population Karnataka Andhra Maharashtra Men 39000 45000 50000 Women 25000 30000 38000 Children 18000 20000 22000 29
4. 4. Solution: Diagrams showing working population in three states during the year 2004 Index Men Women Children Y 50 45 40 35 5 30 4 3 0 3 5 25 9 3 0 8 0 2 0 2 0 0 0 20 1 0 2 2 0 3 0 0 0 0 15 8 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 05 0 0 X KARNATAKA ANDHRA MAHARASHTRA SUB – DIVIDED BAR DIAGRAM: These diagrams are also known as component bar diagram. Each bar is sub – divided according to the components consisting in it. In each bar the different portions are made from the bottom of the bar to distinguish different components. The complete bar represents the total values of variable along with the various values of components. Each component can be distinguished from the other by different colour. ILLUSTRATION: 03 Following is the information relating to the number of students registered with the Bangalore university during the three years prepare the sub – divided bar diagram. Year Arts Science Commerce Total 2001 20000 30000 40000 90000 2002 30000 30000 45000 105000 2003 35000 30000 45000 11000 30
5. 5. Solution:- Diagrams showing the no. of students registered with Bangalore University Index Arts Science Commerce Y Scale for oy axis 1cm = 10000, 1mm=1000 110 100 45000 90 45000 80 70 40000 60 45000 50 45000 40 30 30000 20 30000 35000 10 20000 0 X 2001 2002 2003 ILLUSTRATION :04 The following table shows the results of MBA students of a university for the last three years. Represent the data in a sub – divided bar diagram Year First Class Second Class Third Class Failed Total 2002 60 160 260 120 600 2003 180 210 310 100 800 2004 200 250 300 150 900 Solution: Sub – divided bar diagram, showing the result of MBA in three years. 31
6. 6. Scale for oy axiz 1cm =100, 1mm = 10 I Class II Class III Class Fail 900 800 700 600 500 400 300 200 100 0 2002 2003 2004 SUB – DIVIDED PERCENTAGE BAR DIAGRAM In these bars, absolute variations in the values of variable are not depicted as regards different components. To have the relative changes in the components, we are converting the values of variable into percentages. Now all the bars look equal in heights representing the value 100 as a percentage. The components parts for each division are also depicted in percentages in each bar these diagrams are just simlar to that of sub – divided simple bars, but based on percentages. ILLUSTRATION = 05 Represent the following by sub divided bars drawn on percentage basis. Cost, proceeds, profit or loss per chair during 2001, 2002 and 2003. 2001 2002 2003 Particulars Rs. Rs. Rs. Cost per chair Wages 45 75 105 Other cost 30 51 70 Polishing 15 24 35 Total Cost 90 150 210 Proceeds per Chair 100 150 200 Profit (+) Loss (-) +10 Nil - 10 32
7. 7. SOLUTION = 06 Assume selling price of a chair as 100% and calculate percentage. 2001 2002 2003 Item Rs. % age Rs. %age Rs. % age Wage 45 45 75 50 105 52.5 Other Cost 30 30 51 34 70 35.5 Polishing 15 15 24 16 35 17.0 Total Cost 90 90 150 100 210 105 Profit/ Loss +10 10 -- -- -10 -5.0 Selling Price 100 100 150 100 200 100 Draw the required percentage sub – divided bar diagrams Scale for oy axis 1 cm = 10% 1 mm= 1% Polishing Other cost Wages Profit 100% 90 80 70 60 50 40 30 20 10% 0 2001 2002 2003 ILLUSTRATION = 07 Represent the following data by sub- divided bar diagram drawn on percentage basis. Monthly Expenditure in three families Items FAMILY A FAMILY B FAMILY C Food 360 300 200 Clothing 162 140 120 House Rent 126 120 100 Fuel & light 72 60 40 Miscellaneous 180 130 140 Total 900 750 600 Solution: To draw a sub – divided bar diagram based on percentage all components are changed into percentages on the basis of their total expenditure taken as 100% 33
8. 8. FAMILY – A FAMILY – B FAMILY - C C=cumulative Items Rs % C.% Rs. % C.% Rs. % C% Sub – Food 360 40 40 300 40 40 200 33 33 divided Clothing 162 18 58 140 19 59 120 20 53 diagrams House Rent 126 14 72 120 16 75 100 17 70 based on Fuel & Light 72 8 80 60 8 83 40 7 77 percentage Mise. 180 20 100 130 17 100 140 23 100 showing Total 900 100 750 100 600 100 monthly expenditures of three families Y 100 Food 90 80 Clothing 70 60 House Rent 50 40 Fuel& Light 30 20 Misce 10 0 A B C TWO DIMENSIONAL DIAGRAMS When area characterizes the data i.e. when length & breath both have to be taken into account, the diagrams are called two dimensional diagrams or area or surface diagrams. The most common forms of area diagrams are. 1. Squares Not included in syllabus 2. Circles 3. Rectangles RECTANGLES A Rectangle is a four sided figure with four right angles with adjacent sides unequal. The Rectangle represent the relative magnitude of two or more values. They are placed side by side like bars and are modified form of bar diagrams. Like bar diagrams, rectangles are also sub – divided and shown in percentages as regards the components. The area of a rectangle is equal to the product of height and width. There are two types of rectangles diagrams. 1. Simple sub – divided Rectangles 2. Percentage sub – divided Rectangles Simple Sub – divided Rectangles Under this method, the breadth and height of the bars vary according to the values proportionately. The figures are represented as they are given. Such diagrams are generally used to show three related phenomena – per unit cost, quantity of sales and value of sales. 34
9. 9. ILLUSTRATION = 08 Present the following data by means of sub – divided Rectangular diagram. Particulars Product – A Product – B Quantity sold 200 units 240 Units Selling price per Unit Rs. 500 Rs.600 Cost or Raw Material per Unit Rs.200 Rs.300 Wages per Unit Rs.150 Rs.120 Other cost per unit Rs.100 Rs.90 Profit per unit Rs.50 Rs.90 Solution Statement showing the total cost and the total sales of A and B products. Product A 200 Units Product B 240 units Particulars of Cost Cost per unit Total Cost per unit Total Rs Rs. Rs. Rs. Raw Material 200 40000 300 72000 Labour 150 30000 120 28800 Other costs 100 20000 90 21600 Total Cost 450 90000 510 122400 Profit 50 10000 90 21600 SALES 500 100000 600 144000 Rectangles showing the total cost and sales of products A & B Scale for oy axis 1cm = 20000, 1mm=2000 A B Length 100000 144000 Width 200 : 240 5 : 6 2.5 : 3 160 Profit 120 Other cost 100 Profit Labour Other cost 80 60 Labour 40 Raw Materials 20 Raw Materials 0 2.5c.m 3 cm PERCENTAGE SUB- DIVIDED RECTANGLES Under this method, the widths of the rectangles are kept proportionately according the total values in the ratio. As regards the length of rectangles, are equal to 100% and all the values are converted into percentages. The data can also be shown by simple sub – divided rectangles. 35
10. 10. ILLUSTRATION : 09 Present the following data by a percentage sub – divided Rectangular diagram. Family A Family – B Item Income Rs. 5000 PM Income –Rs.8000PM Food 1200 2000 Clothing 800 1600 Education 600 1200 Medicine 500 800 Rent 600 600 Fuel 800 600 Others 600 400 Total Expenses 5100 7200 Deficit/Surplus --100 +800 Monthly Income 5000 8000 Solution Values are converted into percentages as under. Monthly income = 100% Items of expenses Family – A Rs.=5000 Family – B Rs. =8000 Rs. Percentage % age Rs. Percentage % age Food 1200 1200/5000 x 100 24 2000 2000/8000 x 100 25 Clothing 800 800/5000 x 100 16 1600 1600/8000 x 100 20 Education 600 600/5000 x 100 12 1200 1200/8000 x 100 15 Medicine 500 500/5000 x 100 10 800 800/8000 x 100 10 Rent 600 600/5000 x 100 12 600 600/8000 x 100 7.5 Fuel 800 800/5000 x 100 16 600 600/8000 x 100 7.5 Others 600 600/5000 x 100 12 400 400/8000 x 100 5.0 Total expenses 5100 102 7200 90 Deficit /surplus -100 -2 +800 10 Monthly income 5000 100 8000 100 Length of each Rectangle equal to 100% Width of each Rectangle = 5000 : 8000 5 : 8 Food 2.5 : 4cm 100 Clothing 90 Education 80 70 Medicine 60 Rent 50 Fuel 40 Other 30 20 10 Surplus 0 2.5 cm 4 c.m Deficit 36
11. 11. ILLUSTRATION – 10 Particulars 2002 2003 2004 Material cost 320 320 450 Labour cost 240 280 400 Polishing 160 200 250 Total Cost 720 800 1100 Sale proceeds 800 800 1000 Profit/ Loss +80 ----- -100 Present the above data by means of percentage Rectangular form. Solution: Given values are converted into percentage as under 2002 2003 2004 Particulars Rs. %age Rs. %age Rs. %age Materials 320 40 320 40 450 45 Labour 240 30 280 35 400 40 Polishing 160 20 200 25 250 25 Total Cost 720 90 800 100 1100 110 Profit /Loss +80 10 Nil --- -100 -10 Sales Proceeds 800 100 800 100 1000 100 Length of each Rectangles equal to 100% Width of each Rectangle = 800: 800: 1000 4 : 4:5 2 : 2 : 2.5 Scale for oy axis = 1cm = 10%, 1mm = 1% Rectangles showing the total cost and sales of a product in three periods. Y 100 90 Material 80 70 Labour 60 Polishing 50 40 30 20 10 Profit 0 Loss 2 cm 2cm 2.52.5cm cm MODEL QUESTIONS OR TERMINAL QUESTIONS (5, 10 & 15 MARKS) 1. Explain the importance of diagrammatic presentation of statistical data 2. What are the points to be taken into consideration while presenting a statistical data diagrammatically? 3. What are the merits and limitations of a diagrammatic representation of statistical data? 4. Briefly explain various types of diagrams used to represent statistical data 37
12. 12. 5. Briefly explain the following diagrams a. Percentage sub – divided diagram b. Percentage sub – divided rectangular diagram c. Multiple bar diagram. 6. Represent the following data by sub – divided bars drawn as percentage basis. The total cost, sale proceeds and profit or loss per chair during 2001, 2002, 2003 are as follows. Particulars 2001 2002 2003 Wages 64 80 88 Polishing 32 32 40 Other cost 64 80 128 Total cost 160 192 256 Sale proceeds per Chair 192 192 240 Profit or Loss per chair +32 --- -16 7. Draw a multiple bar diagram to represent the following data Year Sales (in 000 of Rs.) Gross Profit (in 000 of Rs.) Net profit (in 000Rs.) 2000 100 30 10 2001 120 40 15 2002 130 45 25 2003 150 50 30 8. Draw a Rectangular diagram to represent the following information Product A Product B Particulars Rs. Rs. Number of units sold 500 800 Selling price per unit 30 25 Cost of Raw Materials 5000 9600 Labour and other cost 6000 6400 Profit 4000 4000 9. Present the following data by a percentage sub – divided Rectangle diagram. Particulars Product – A Product – B Product – C Material cost 3100 4000 4500 Labour Cost 2300 2800 4000 Direct Expenses 1600 2000 2500 Factory Expenses 1500 1200 600 Other Expenses 1000 1000 600 Total Cost 9500 11000 12500 Sales proceeds 10000 11000 12000 Profit / Loss + 500 Nil -500 10. From the below given details of the monthly expenditure of two families, prepare Rectangle diagram on percentage basis. Items of Expenditure Family A income Rs.10000 Family B income Rs.8000 Food 2800 2400 Clothing 1600 1600 House Rent 2000 1200 Education 600 800 Fuel 800 400 Miscellaneous 800 800 Savings 1400 800 10000 8000 GRAPHIC REPRESENTATION OF DATA: 38
13. 13. A Graph is a vivid or intense or bright form of presentation of data. It is a simplest and commonest aid to the numerical reading which gives a picture of numbers in such a way that the relations between the two series can be easily compared. Graphic method of representation of data is becoming more effective and powerful than the diagrammatic representation. Graphs bring to light the facts that are hidden. They are becoming more and more powerful in all the fields of study. For clear and effective exposition and appreciation of quantitative data, graphical presentations play an important role by facilitating comparison of values trends and relations. Merits of graphic presentation Following are the merits 1. A graph is more attractive and impressive than a table of figures 2. With the help of graphs, comparison between two or more phenomena can be made very easy. 3. Since the data become visible at a glance in a graph, one can understand it easily and can study the tendency and fluctuations of data. 4. The impressions created by the figures presented in a tabular than those created by the figures presented in a tabular form. 5. Graphs are also helpful in interpretation, extra population and forecasting. 6. Correlation between two series can be studied easily with the help of graphs. 7. Certain statistical measures like, median, mode, quartiles etc can be determined by drawing the graphs of frequency distribution. 8. It needs no special knowledge of mathematics to understand a graph. 9. Apart from simplicity, it saves the time and energy of the statistician as well as observer. LIMITATIONS:- It suffers from the following limitations 1. Graphs may be misused by taking false scale 2. Since a curve shows tendency and fluctuations, actual values are not knows 3. Accuracy is rather not possible in a graph 4. Graphs cannot be quoted in support of a statement 5. Only one or two characteristics can be depicted in a graph. If more features are shown, the graph become difficult to follow. General rules of constructing graph: To represent statistical data by a graph the following points should be born in mind. 1. Every graph must have a title, indicating the facts presented by the graph. 2. It is necessary to plot the independent variables on the horizontal axis and dependent variable on the vertical axis 3. The principle of drawing graph is that the vertical scale must start from zero. 4. Problem arises regarding the choice of a suitable scale the choice must accommodate the whole data. 5. For showing proportional relative changes in the magnitude, the ratio or log arithmetic scale should be used. 6. The graph must not be over crowed with curves. 7. If more than one variable is plotted on the same graph it is necessary to distinguish them by different line like dotted line, broken lines etc. 8. Index should be given to show the scales and the meaning of different curves 9. If should be remembered that for every value of independent variable, there is a corresponding value of the dependent variable 10. Source of information should be mentioned as foot note. 39
14. 14. Difference between diagrams and graphs The diagrams and graphs are statistical techniques of representing the data. However, we can make the following distinctions between the two DIAGRAMS GRAPHS 1. Both plain paper and graph sheet can be used. 1. Graph sheets are used. 2. A diagram does not represent any mathematical 2. A graph represents mathematical relationship relationship between two variable. between two variables. 3. Diagrams are constructed for categorically 3. Graphs are more appropriate for representing data, including time series and spatial series. time series and frequency distribution. 4. Diagrams are more attractive to the eye and as 4. Graphs are less attractive to the eye. such are better suited for publicity & 5. Graphs are very much used in statistical propaganda. analysis. 5. Diagrams are not helpful in statistical analysis. 6. The value of median and mode can be 6. Median and mode cannot be estimated. estimated. 7. It is used for comparison only. 7. It represents a mathematical relationship 8. Data’s are represented by bars, Rectangles. between two variables. 8. Data are presented by points or dots line. Types of graphs A large number of graphs are used in practices. They can be broadly classified into two heads. 1. Graphs of time series [not included in syllabus] 2. Graphs of frequency distribution Graph of frequency Distribution Graphical representation can be advantageously employed to bring out clearly the statistical nature of frequency distribution may be discrete or continuous. The most commonly used graphs 1. Histogram 2. Frequency polygon 3. Frequency curve 4. Ogive HISTORGRAM: One of the most important and useful methods of presenting frequency distribution of continuous series is known as Histogram. In this, the magnitude of the class interval is plotted along the horizontal axis and the frequency on the vertical axis. Each class gives two equal vertical lines representing the frequency. Upper ends of the lines are joined together. This process will give us rectangles as there are classes and the heights of rectangles are proportional to their frequencies. ILLUSTRATION = 01 From the following data draw a histogram estimate mode graphically. Variable 35 –40 40 – 45 45 – 50 50 – 55 55 – 65 Frequency 12 30 22 30 28 Solution: The first four class intervals are equal, where as the last class interval is not equal. We have to reduce the last class interval as under = 28 x 5 =14 10 Histogram showing The frequency distribution 40
15. 15. Y 30 30 30 25 22 20 15 12 10 14 x 2 05 0 35 40 45 50 55 60 65 X ILLUSTRATION : - 02 Draw a Histogram and determine the mode graphically from the following data and verify the result, Weekly wages in Rs 10 – 15 15 – 20 20 – 25 25 – 30 30 –40 40-60 60-80 No. of workers 14 38 54 30 24 24 16 Solution The last three class intervals are not equal, the frequency of each class is to be adjusted. Histogram show distribution weekly wages & model wages. Y 30 – 40 = 24 x 5/10 = 12 40 – 60 = 24 x 5/20 = 6 60 60 – 80 = 16 x 5/20 = 4 50 Scale for oy axis Z = L1 + f1 – f0 (L2 – L1) 40 1 cm = 20 2f1 – f0 – f2 = 20 + 54 –38 (25) 30 2 x 54 – 38 -30 = 22 20 10 12 x 2 6x 4 0 4x4 15 20 25 30 35 40 45 50 55 60 65 70 75 80 FREQUENCY POLYGON It is a device of graphic representation of a frequency distribution. It is a simple method of drawing the graph with the help of histogram. Then plot the mid point of the top each rectangle. To make a frequency polygon we have to connect the mid- points of the top of all the rectangles by straight lines. This is done under the assumption that the frequencies in each class interval are evenly distributed. The area of the frequency polygon is equal to the area of the histogram, as the area left outside is geometrically equal to the area included in it. FREQUENCY CURVE With the help of the histogram and frequency polygon, we can also draw a smoothed curve to iron out or eliminate the accidental irregularities in the data. A smoothed frequency curve represents a generalized characterization of the data collected from the population or mass. In smoothing a curve it is important to note that the total area under the curve be equal to the are under the histogram or polygon. When the curve is accurately drawn, we can use it for interpolation of the figure also. 41
16. 16. ILLUSTRATION = 03 Draw a histogram, Frequency polygon and Frequency curve for the following data. Estimate mode graphically and verify the result by direct calculation. Values: 0–5 5 – 10 10 – 15 15 – 20 20 – 25 25 – 35 35 –50 Frequency 25 80 120 160 130 96 60 Solution Note: The last two class intervals are not equal, so frequency of each class is to adjusted. 25 – 35= 96 x 5/10 = 48 35 – 50 = 60 x5/15 = 20 Graph showing Histogram Frequency polygon and Frequency smoothed curve Y 160 Scale for oy axis Based on highest frequency 140 =160 1cm = 20 120 1mm = 2 Z = l1 + f1 – f0 (l2 – l1) 100 Frequency polygon 2f1+ f0 – f2 80 Frequency curve = 15+ 160-120 .(20+5) 2x160-120-130 60 Z = 17.8 40 20 48 x 2 20 x 3 0 5 10 15 20 25 30 35 40 45 50 55 60 65 ILLUSTRATION = 04 Draw a Histogram and Frequency polygon from the following data. Also estimate mode graphically and verify the result by direct calculation. Values – X 10 –19 20 – 29 30 – 39 40 – 49 50 – 59 60 – 69 70 - 79 Frequency 20 45 60 100 80 60 40 Solution:- Given Series is an inclusive series first, we should convert into exclusive series, as under. Values X Frequency L1 – ½ of 1 9.5 – 19.5 20 L2 + ½ of 1 19.5 – 29.5 45 29.5 – 39.5 60 ‘1’ is difference between the proceeding 39.5 – 49.5 100 upper limit and the succeeding lower limit 49.5 – 59.5 80 59.5 – 69.5 60 69.5 – 79.5 40 Graph showing Histogram & Frequency Polygon 42
17. 17. Y Scale for oy axis 100 Highest frequency = 100 90 1cm = 10 1 mm = 1 80 Z = l 1 + f1 – f0 (l2 – l1) 70 2f1-f0-f2 = 39.5 + 100-60 (49.5 – 39.5) 60 2x100-60-80 = 39.5 + . 40 .x 10 50 200 – 140 = 39.5 + 400/60 40 = 39.5 + 6.6 = 46.1 30 20 10 0 9.5 19.5 29.5 39.5 49.5 59.5 69.5 79.5 X Mode = 46.1 ILLUSTRATION = 05 Draw a Histogram from the following. Also estimate the value of mode graphically. Mid Values 05 15 25 35 45 55 Frequency 40 50 80 60 30 20 Solution Note: First of all given Mid values are to be converted into continuous series. Before representing these data. Values 0 – 10 10 – 20 20 –30 30 –40 40 – 50 50 – 60 Frequency 40 50 80 60 30 20 Graph showing Histogram & reading of mode graphically Y Scale for oy axis 80 Highest frequency = 80 1cm = 10 70 1mm = 1 60 Z= l1 + f1 – f0 (l2-l1) 2f1-f0-f2 50 = 20 + 80-50 (30-20) 2x80 – 50 – 60 40 =20 + 30 x 10 160-110 20 = 20+300/50 = 26 10 0 10 20 30 40 50 60 Mode = 26 43
18. 18. OGIVE CURVES It is a graphic presentation of cumulative frequency distribution of a continuous series. This method of drawing the curves is the best among other types as if serves many purposes. Ogive curves or Ogives may be used for the purpose of comparing group of statistics in which time is not a factor. They are primarily drawn for determining the partitioned values life median quartiles deciles etc. Since there are two types of cumulative frequencies, we have accordingly two types Ogives a. Less than Frequency curve (Ogive) b. More than Frequency curve (Ogive) Less than ogive:- It consists in plotting the less than frequencies against the upper limit of the class interval. The points so obtained are joined by a smoothed curve. It is an increasing curve sloping upward from left to right of the graph and it is in the shape of an elongated (s) More than ogive It consists in plotting that more than frequencies against the lower limit of the class intervals. The points so are joined by a smoothed curve. It is a decreasing curve sloping down ward from left to right of the graph and it is in the shape of an elongated upside down (s) Galton’s method of Locating the median:- Francis Galton hag given a graphic method by which median can be located. The vertical line (oy) is divided into equal parts corresponding to the unit of measurements. From the half of the oy – axis, a horizontal line is drawn from the left to the right. This line cuts the less than frequency curve in a particular point. From this point draw a perpendicular line on ox – axis. The intersecting point on ox – axis will be the median value. In a similar way we can also find all the other partitioned values such as quartiles, Deciles etc. Characteristics of ogive Following are the characteristics of less than and more than Ogives. 1. When both the Ogives are plotted on the same graph, they intersected particular point, from this intersecting point, if we draw a perpendicular line on ox – axis, it gives the value of median. 2. Less than Ogive is useful in computation of median, quartiles etc. 3. Ogives give a clear picture about the data by which, we can have the comparative study. Thus, the two Ogives are playing an important role in presenting the data relating to cumulative frequency. ILLUSTRATION = 06 Draw an Ogive curve and from it read the median and quartiles. Marks 10 – 20 20 –30 30 – 40 40 –50 50 – 60 60 – 70 70 – 80 No. of Students 21 19 60 42 24 18 17 Solution To find the partitioned values we have to convert the data into a less than frequency distribution. Marks less than 20 30 40 50 60 70 80 No. of Students 21 40 100 142 166 184 201 44
19. 19. Graph showing median & Quarts Marks of students Ogive = Scale for oy axis 200 Total frequency =201 1cm =20 180 Where m = N/2 = 201/2 = 100.5 160 less than Ogive Where q1=N/4= 201/4 = 50.25 Where q3 = 2N/4 = 3x201/4 = 150.75 140 Median = 40.12 120 Q1 = 31.8 M Q3 = 53.7 100 80 q3 = 15..75 60 50 m>100.550.25 q1= 40 q1 = 50.25 20 10 20 30 40 50 60 70 80 90 q1= 31.8 M = 40.12 Q3 = 53.7 ILLUSTRATION = 07 Draw the two Ogive curves from the following data and Locate the median value. Marks X 20 –40 40 – 60 60 –80 80 –100 100 –120 120 – 140 140 – 160 Total No. of Students 14 18 20 16 22 7 03 100 Solution For the Ogive curves e have to convert the data into less than & more than Marks Cf Marks No.og students Less than 40 14 More than 20 100 ------“----- 60 32 ------“----- 40 86 ------“----- 80 52 ------“----- 60 68 ------“----- 100 68 ------“----- 80 48 ------“----- 120 90 ------“----- 100 32 ------“----- 140 97 ------“----- 120 10 ------“----- 160 100 ------“----- 140 03 45
20. 20. Graph showing less than & more than Ogive curve & median 100 Based on Frequency total = 100 1 cm = 10 90 Less than Ogive 1 mm= 1 80 M = l1 +l2 – l1 (m-c) Where m = N/2 f 70 = 60 + 80 –60 (50-32) 60 20 = 60 + 20/20 x 16 50 = 78 40 30 20 More than Ogive 10 0 20 40 60 80 100 120 140 160 Median = 78 ILLUSTRATION = 08 Draw the two Ogives from the following data and locate median Variable 100-200 200 – 300 300 – 400 400 – 500 500 – 600 600 –700 Frequency 20 40 60 80 100 120 Solution Let us convert the data into less than and more than frequency Table Variables X F Less than Table More than Cum. Fr. Table 100 – 200 20 Below or lee than 200 20 More than 100 420 200 – 300 40 Below or lee than 300 60 -----“------ 200 400 300 – 400 60 ----------“--------- 400 120 -----“------ 300 360 400 – 500 80 ----------“--------- 500 200 -----“------ 400 300 500 – 600 100 ----------“--------- 600 300 -----“------ 500 220 600 - 700 120 ----------“--------- 700 420 -----“------ 600 120 46
21. 21. Graph showing two Ogives & Median 420 Scale for oy axis Frequency = 420 380 1 cm = 40 1 mm = 4 340 Median = 510 300 Less than curve 280 240 200 160 120 More than curve 80 40 0 100 200 300 400 500 600 700 800 ILLUSTRATION = 09 Draw a less than Ogive and find the values of Median & quartiles Wages in less than 20 40 60 80 100 120 140 No. of workers 5 8 20 40 55 60 70 Solution: Gives series is a Less than cumulative frequency Table Y Graph Showing Median & Quartiles 70 Scale for oy 1 cm = 10 60 less than Ogive 1 mm = 1 50 Median = 75 40 Q1 = 55.8 Q3 = 96.6 30 20 10 Q3 0 20 40 60 80 100 120 140 Q1 M=75 MODEL QUESTION OF TERMINAL QUESTIONS (5, 10 & 15 Marks) 1. Define and distinguish between diagrammatic and graphic representation 2. Explain in detail the different modes of graphical representation of frequency distribution. 3. What do you understand by a histogram? 4. What is Ogive curve? How it is constructed? 47
22. 22. 5. Explain Frequency polygon and the Frequency curve 6. Draw a Histogram and Frequency polygon from the following data. Estimate more graphically and verify the result by direct calculations. Values 0 – 10 10 –20 20 – 30 30 – 40 40 –50 50 –70 70 –100 100 – 140 Frequency 20 45 60 95 80 78 60 40 7. Construct a Histogram and locate mode there by graphically from the following data and verify the result Wages ins 10 – 20 20 –30 30 –40 40 –50 50 – 60 60 – 80 80 – 110 110 -150 No. of workers 87 121 154 133 95 82 72 32 8. Construct a Histogram and Frequency polygon from the following data. Also estimate mode graphically. Wages in Rs. 20 – 24 25 – 29 30 – 34 35 – 39 40 –44 45 – 49 50 –54 55 -59 No. of workers 8 10 40 80 30 25 20 15 9. Draw a Histogram from the following data. Also estimate mode graphically and verify the result Marks Obtained 35 45 55 65 75 85 95 No. of students 10 18 20 120 40 30 20 10. Draw two Ogive curves for the following data and determine the value of Median, verify the result by direct calculation. Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100 No. of 12 28 35 55 30 20 20 17 13 10 students 11. Draw an Ogive curve for the following distribution and the values of Median & quartiles. Verify the result. Size 0-4 4-8 8-12 12-16 16-20 20-24 24-28 28-32 32-36 Frequency 4 8 10 15 25 20 14 12 10 12. By using the following table draw an Ogive curve and determine the value of median verify the result by direct calculation. Wages in Rs. 70-79 80-89 90-99 100-109 110-119 120-129 130-139 140-149 150-159 No of worker 12 18 35 42 50 45 20 8 10 ----------------------------------------------------------------------------------------------------------------------------- 48