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. 27
Transcript
UNIT –03DIAGRAMMATIC 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 diagramsThe 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 limitations1. 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 diagram3. 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.
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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 diagramsThe 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 viewa. 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 diagrams2. 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 diagram3. Sub – divided Bar diagram4. Sub – divided Bar diagram based on percentage.
Simple bar diagram:-
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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: 01Draw 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 (000)
70 80 65 50 90 80 95
Solution:Scale for oy axix:
1 cm = 10000 units, 1mm = 1000Diagrams showing number of small scale units in different states.
Y100
90
80
70
60
50
40
30
20
10 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 MaharashtraMen 39000 45000 50000
Women 25000 30000 38000Children 18000 20000 22000
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KARNATAKA7000
T. NADU-80000
KERALA 65000
ANDHRA50000
MAHARASHTRA 90000
MP 80000
UP 95000
Solution:Diagrams showing working population in three states during the year 2004
IndexMen Women Children
Y50
45
40
35
30
25
20
15
10
050 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: 03Following 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 Total200120022003
200003000035000
300003000030000
400004500045000
9000010500011000
30
39000
23000
18000
45000
30000
20000
50000
38000
22000
Solution:- Diagrams showing the no. of students registered with Bangalore University
IndexArts Science Commerce
Y Scale for oy axis 1cm = 10000, 1mm=1000
110 100
90
80
70
60
50
40
30
20
10 0 X
2001 2002 2003ILLUSTRATION :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 Total2002 60 160 260 120 6002003 180 210 310 100 8002004 200 250 300 150 900
Solution:Sub – divided bar diagram, showing the result of MBA in three years.
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4500045000
20000
30000
4000045000
30000 35000
45000
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 DIAGRAMIn 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 = 05Represent the following by sub divided bars drawn on percentage basis.Cost, proceeds, profit or loss per chair during 2001, 2002 and 2003.
Particulars2001Rs.
2002Rs.
2003Rs.
Cost per chairWages
Other costPolishing
453015
755124
1057035
Total Cost 90 150 210Proceeds per Chair 100 150 200Profit (+) Loss (-) +10 Nil - 10
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SOLUTION = 06Assume selling price of a chair as 100% and calculate percentage.
Item2001 2002 2003
Rs. % age Rs. %age Rs. % ageWageOther CostPolishing
Draw the required percentage sub – divided bar diagramsScale 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
2003ILLUSTRATION = 07
Represent the following data by sub- divided bar diagram drawn on percentage basis.
ItemsMonthly Expenditure in three families
FAMILY A FAMILY B FAMILY CFoodClothingHouse RentFuel & lightMiscellaneous
36016212672180
30014012060130
20012010040140
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%
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C=cumulative
Y100 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. SquaresNot included in syllabus
2. Circles3. Rectangles
RECTANGLESA 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 Rectangles2. Percentage sub – divided Rectangles
Simple Sub – divided RectanglesUnder 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.
Present the following data by means of sub – divided Rectangular diagram.Particulars Product – A Product – B
Quantity soldSelling price per Unit
Cost or Raw Material per Unit
Wages per UnitOther cost per unit
Profit per unit
200 unitsRs. 500
Rs.200Rs.150Rs.100Rs.50
240 UnitsRs.600
Rs.300Rs.120Rs.90Rs.90
SolutionStatement showing the total cost and the total sales of A and B products.
Particulars of CostProduct A 200 Units Product B 240 units
Cost per unitRs
TotalRs.
Cost per unitRs.
TotalRs.
Raw MaterialLabourOther costs
200150100
400003000020000
30012090
720002880021600
Total Cost Profit
45050
9000010000
51090
12240021600
SALES 500 100000 600 144000Rectangles showing the total cost and sales of products A & B
Scale for oy axis1cm = 20000, 1mm=2000
A BLength 100000 144000Width 200 : 240
5 : 62.5 : 3
160
120
100
80
60
40
20
0 2.5c.m 3 cm
PERCENTAGE SUB- DIVIDED RECTANGLESUnder 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.
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Profit
Raw Materials
Labour
Other cost
Raw Materials
Labour
Other cost
Profit
ILLUSTRATION : 09Present the following data by a percentage sub – divided Rectangular diagram.
ItemFamily A
Income Rs. 5000 PMFamily – B
Income –Rs.8000PMFoodClothingEducationMedicineRentFuelOthers
1200800600500600800600
200016001200800600600400
Total Expenses 5100 7200Deficit/Surplus --100 +800
Monthly Income 5000 8000Solution
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 % ageFoodClothing EducationMedicineRentFuelOthers
1200800600500600800600
1200/5000 x 100800/5000 x 100600/5000 x 100500/5000 x 100600/5000 x 100800/5000 x 100600/5000 x 100
24161210121612
200016001200800600600400
2000/8000 x 1001600/8000 x 1001200/8000 x 100800/8000 x 100600/8000 x 100600/8000 x 100400/8000 x 100
252015107.57.55.0
Total expenses 5100 102 7200 90Deficit /surplus -100 -2 +800 10Monthly income 5000 100 8000 100Length of each Rectangle equal to 100%Width of each Rectangle = 5000 : 8000
5 : 8 Food2.5 : 4cm
100 Clothing 90
Education 80
70 Medicine
60 Rent
50Fuel
40Other
30
20
10
02.5 cm 4 c.m
Deficit
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Surplus
ILLUSTRATION – 10Particulars 2002 2003 2004
Material costLabour costPolishing
320240160
320280200
450400250
Total Cost Sale proceeds
720800
800800
11001000
Profit/ Loss +80 ----- -100Present the above data by means of percentage Rectangular form.Solution:
Given values are converted into percentage as under
Sales Proceeds 800 100 800 100 1000 100Length 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.Y100
90Material
80
70 Labour
60 Polishing
50
40
30
20
10
0
2 cm 2cm 2.5 cm
MODEL QUESTIONS OR TERMINAL QUESTIONS (5, 10 & 15 MARKS)1. Explain the importance of diagrammatic presentation of statistical data2. 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
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Profit
Loss2.5cm
5. Briefly explain the following diagramsa. 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.
GRAPHIC REPRESENTATION OF DATA: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 presentationFollowing are the merits
1. A graph is more attractive and impressive than a table of figures2. 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 knows3. Accuracy is rather not possible in a graph4. Graphs cannot be quoted in support of a statement5. 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 axis3. 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 curves9. If should be remembered that for every value of independent variable, there is a corresponding value of
the dependent variable10. Source of information should be mentioned as foot note.
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Difference between diagrams and graphsThe diagrams and graphs are statistical techniques of representing the data. However, we can
make the following distinctions between the twoDIAGRAMS GRAPHS
1. Both plain paper and graph sheet can be used.2. A diagram does not represent any mathematical
relationship between two variable.3. Diagrams are constructed for categorically
data, including time series and spatial series.4. Diagrams are more attractive to the eye and as
such are better suited for publicity & propaganda.
5. Diagrams are not helpful in statistical analysis.6. Median and mode cannot be estimated. 7. It is used for comparison only. 8. Data’s are represented by bars, Rectangles.
1. Graph sheets are used.2. A graph represents mathematical relationship
between two variables. 3. Graphs are more appropriate for representing
time series and frequency distribution.4. Graphs are less attractive to the eye.5. Graphs are very much used in statistical
analysis.6. The value of median and mode can be
estimated. 7. It represents a mathematical relationship
between two variables.8. Data are presented by points or dots line.
Types of graphsA 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 DistributionGraphical 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. Histogram2. Frequency polygon3. Frequency curve4. 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 = 01From the following data draw a histogram estimate mode graphically.
SolutionThe last three class intervals are not equal, the frequency of each class is to be adjusted.Histogram show distribution weekly wages & model wages.
Y
60
50
40
30
20
10
015 20 25 30 35 40 45 50 55 60 65 70 75 80
FREQUENCY POLYGONIt 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 CURVEWith 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.
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30 – 40 = 24 x 5/10 = 1240 – 60 = 24 x 5/20 = 660 – 80 = 16 x 5/20 = 4
Scale for oy axis1 cm = 20
14 x 2
12 x 26x 4
4 x 4
Z = L1 + f1 – f0 (L2 – L1) 2f1 – f0 – f2
= 20 + 54 –38 (25) 2 x 54 – 38 -30 = 22
ILLUSTRATION = 03Draw a histogram, Frequency polygon and Frequency curve for the following data. Estimate mode graphically and verify the result by direct calculation.
SolutionNote: The last two class intervals are not equal, so frequency of each class is to adjusted.
25 – 35= 96 x 5/10 = 4835 – 50 = 60 x5/15 = 20
Graph showing Histogram Frequency polygon and Frequency smoothed curveY160
140
120
100Frequency polygon
80 Frequency curve
60
40
20
0 5 10 15 20 25 30 35 40 45 50 55 60 65
ILLUSTRATION = 04Draw 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 - 79Frequency 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 Frequency9.5 – 19.5 20 19.5 – 29.5 4529.5 – 39.5 6039.5 – 49.5 10049.5 – 59.5 8059.5 – 69.5 6069.5 – 79.5 40
Graph showing Histogram & Frequency Polygon
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Scale for oy axisBased on highest frequency
=1601cm = 201mm = 2
Z = l1 + f1 – f0 (l2 – l1)2f1+ f0 – f2
= 15+ 160-120 .(20+5) 2x160-120-130Z = 17.8
48 x 2
20 x 3
L1 – ½ of 1L2 + ½ of 1
‘1’ is difference between the proceeding upper limit and the succeeding lower limit
Y100
90
80
70
60
50
40
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.
Graph showing Histogram & reading of mode graphicallyY
80
70
60
50
40
20
10
010 20 30 40 50 60 Mode = 26
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Z = l1 + f1 – f0 (l2 – l1) 2f1-f0-f2
= 39.5 + 100-60 (49.5 – 39.5) 2x100-60-80
= 39.5 + . 40 .x 10 200 – 140
= 39.5 + 400/60= 39.5 + 6.6= 46.1
Scale for oy axisHighest frequency = 1001cm = 101 mm = 1
Scale for oy axis Highest frequency = 801cm = 101mm = 1
Z= l1 + f1 – f0 (l2-l1)2f1-f0-f2
= 20 + 80-50 (30-20)2x80 – 50 – 60
=20 + 30 x 10160-110
= 20+300/50 = 26
OGIVE CURVESIt 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 = 06Draw 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 – 80No. of Students 21 19 60 42 24 18 17
SolutionTo 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 80No. of Students 21 40 100 142 166 184 201
SolutionLet us convert the data into less than and more than frequency Table
Variables X F Less than Table More than Cum. Fr. Table100 – 200200 – 300300 – 400400 – 500500 – 600600 - 700
20406080100120
Below or lee than 200Below or lee than 300----------“--------- 400----------“--------- 500----------“--------- 600----------“--------- 700
2060120200300420
More than 100-----“------ 200-----“------ 300-----“------ 400-----“------ 500-----“------ 600
420400360300220120
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Based on Frequency total = 1001 cm = 101 mm= 1
M = l1 +l2 – l1 (m-c) Where m = N/2 f
= 60 + 80 –60 (50-32) 20
= 60 + 20/20 x 16 = 78
Graph showing two Ogives & Median420
380
340
300 Less than curve
280
240
200
160
120 More than curve
80
40
0100 200 300 400 500 600 700 800
ILLUSTRATION = 09Draw a less than Ogive and find the values of Median & quartiles
Wages in less than 20 40 60 80 100 120 140No. of workers 5 8 20 40 55 60 70Solution:
Gives series is a Less than cumulative frequency TableY Graph Showing Median & Quartiles 70 Scale for oy
1 cm = 1060 less than Ogive 1 mm = 1
50
40
30
20
10Q3
020 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 representation2. 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?
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Scale for oy axisFrequency = 4201 cm = 401 mm = 4Median = 510
Median = 75Q1 = 55.8Q3 = 96.6
5. Explain Frequency polygon and the Frequency curve6. 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 – 140Frequency 20 45 60 95 80 78 60 407. Construct a Histogram and locate mode there by graphically from the following data and verify the resultWages ins 10 – 20 20 –30 30 –40 40 –50 50 – 60 60 – 80 80 – 110 110 -150No. of workers 87 121 154 133 95 82 72 328. 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 -59No. of workers 8 10 40 80 30 25 20 159. Draw a Histogram from the following data. Also estimate mode graphically and verify the resultMarks Obtained 35 45 55 65 75 85 95No. of students 10 18 20 120 40 30 2010. 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 90-100No. of students
12 28 35 55 30 20 20 17 13 10
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 32-36Frequency 4 8 10 15 25 20 14 12 1012. 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 90-99 100-109 110-119 120-129 130-139 140-149 150-159No of worker 12 18 35 42 50 45 20 8 10-----------------------------------------------------------------------------------------------------------------------------
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