diagrammatic and graphical representation of data

BSM - Manglore University

2. Oneofthemostconvincingandappealingwaysin
whichstatisticalresultsmaybepresentedisthrough
diagramsandgraphs.
Justonediagramisenoughtorepresentagiven
datamoreeffectivelythanthousandwords.
Iteasytounderstanddiagramsevenforordinary
people.
Introduction

3. Adiagramisavisualformforpresentationofstatistical
data,highlightingtheirbasicfactsandrelationship.
Ifwedrawdiagramsonthebasisofthedatacollected
theywilleasilybeunderstoodandappreciatedbyall.
Itisreadilyintelligibleandsaveaconsiderableamount
oftimeandenergy.
Diagrams

4. Significance of Diagrams and
Graphs
1.They are attractive and impressive.
2.They make data simple and intelligible.
3.They make comparison possible
4.They save time and labour.
5.They have universal utility.
6.They give more information.
7.They have a great memorizing effect.

5. General rules for constructing
diagrams
1.Adiagramshouldbeneatlydrawnandattractive.
2.Themeasurementsofgeometricalfiguresusedin
diagramshouldbeaccurateandproportional.
3.Thesizeofthediagramsshouldmatchthesizeofthe
paper.
4.Everydiagrammusthaveasuitablebutshortheading.
5.Thescaleshouldbementionedinthediagram.
6.Diagramsshouldbeneatlyaswellasaccuratelydrawn
withthehelpofdrawinginstruments.
7.Indexmustbegivenforidentificationsothatthereader
caneasilymakeoutthemeaningofthediagram.
8.Footnotemustbegivenatthebottomofthediagram.
9.Economyincostandenergyshouldbeexercisedin
drawingdiagram.

6. Types of
Diagrams
One-
dimensional
Diagrams
Two-
dimensional
Diagrams
Three-
dimensional
Diagrams
Pictograms
and
cartograms

7. Insuchdiagrams,onlyone-dimensionalmeasurement,i.e.
heightisusedandthewidthisnotconsidered.
Line
diagram
Simple
diagram
Multiple
bar
diagram
Sub-divided
bar
diagram
Percentage
bar
diagram

8. Line Diagram
•Linediagramisusedincasewheretherearemanyitemsto
beshownandthereisnotmuchofdifferenceintheirvalues.
•Suchdiagramispreparedbydrawingaverticallineforeach
itemaccordingtothescale.
•Thedistancebetweenlinesiskeptuniform.
•Linediagrammakescomparisoneasy,butitislessattractive.

9. Simple Bar Diagram
•Simplebardiagramcanbedrawneitheronhorizontalor
verticalbase.
•Barsmustbeuniformwidthandinterveningspacebetween
barsmustbeequal.
•Whileconstructingasimplebardiagram,thescaleis
determinedonthebasisofthehighestvalueintheseries.
•Tomakethediagramattractive,thebarscanbecoloured.
•Bardiagramareusedinbusinessandeconomics.

10. Simple Bar Diagram (Cont)
•However,animportantlimitationofsuchdiagramsisthatthey
canpresentonlyoneclassificationoronecategoryofdata.
•Forexample,whilepresentingthepopulationforthelastfive
decades,onecanonlydepictthetotalpopulationinthe
simplebardiagrams,andnotitssex-wisedistribution.

11. Multiple Bar Diagram
•Multiplebardiagramisusedforcomparingtwoormoresets
ofstatisticaldata.
•Barsareconstructedsidebysidetorepresentthesetof
valuesforcomparison.
•Inordertodistinguishbars,theymaybeeitherdifferently
colouredorthereshouldbedifferenttypesofcrossingsor
dotting,etc.
•Anindexisalsopreparedtoidentifythemeaningofdifferent
coloursordottings.

13. Sub-divided Bar Diagram
•Inasub-dividedbardiagram,thebarissub-dividedinto
variouspartsinproportiontothevaluesgiveninthedataand
thewholebarrepresentthetotal.
•SuchdiagramsarealsocalledComponentBardiagrams.
•Thesubdivisionsaredistinguishedbydifferentcolorsor
crossingsordotting.
•Themaindefectofsuchadiagramisthatallthepartsdonot
haveacommonbasetoenableonetocompareaccurately
thevariouscomponentsofthedata.

14. Example 4 Represent the following data by a sub-
divided bar diagram.

15. Percentage Bar Diagram
•Thecomponentsarenottheactualvaluesbutpercentagesof
thewhole.
•Themaindifferencebetweenthesub-dividedbardiagramand
percentagebardiagramisthatintheformerthebarsareof
differentheightssincetheirtotalsmaybedifferentwhereasin
thelatterthebarsareofequalheightsinceeachbar
represents100percent.
•Inthecaseofdatahavingsub-division,percentagebar
diagramwillbemoreappealingthansub-dividedbardiagram.

18. Intwo-dimensionaldiagramsthearearepresentthedata
andsothelengthandbreadthhavebothtobetakeninto
account.
Suchdiagramsarealsocalledareadiagramsorsurface
diagrams.
Theimportanttypesofareadiagramsare:
1.Rectangles.
2.Squares.
3.Pie-diagrams

19. Rectangles
•Rectanglesareusedtorepresenttherelativemagnitudeof
twoormorevalues.
•Theareaoftherectanglesarekeptinproportiontothe
values.
•Rectanglesareplacedsidebysideforcomparison.
•Wemayrepresentthefiguresastheyaregivenormay
convertthemtopercentagesandthensubdividethelength
intovariouscomponents.

20. •Solution: The items of expenditure will be converted into percentage as
shown below

22. Squares
•Therectangularmethodofdiagrammaticpresentationis
difficulttousewherethevaluesofitemsvarywidely.
•Themethodofdrawingasquarediagramisverysimple.
•Onehastotakethesquarerootofthevaluesofvariousitem
thataretobeshowninthediagramsandthenselecta
suitablescaletodrawthesquares.

24. Pie Diagram or Circular Diagram
•Insuchdiagrams,boththetotalandthecomponentpartsor
sectorscanbeshown.Theareaofacircleisproportionalto
thesquareofitsradius.
•Whilemakingcomparisons,piediagramsshouldbeusedon
apercentagebasisandnotonanabsolutebasis.
•Example8:DrawaPiediagramforthefollowingdataof
productionofsugarinquintalsofvariouscountries.

26. Itconsistsofcubes,cylinders,spheres,etc.
Insuchdiagramsthreethings,namelylength,widthandheight
havetobetakenintoaccount.
Ofallthefigures,makingofcubesiseasy.Sideofacubeis
drawninproportiontothecuberootofthemagnitudeofdata.
Cubesoffigurescanbeascertainedwiththehelpoflogarithms.
Thelogarithmofthefigurescanbedividedby3andtheantilog
ofthatvaluewillbethecube-root.

28. Pictogramsarenotabstractpresentationsuchaslines
orbarsbutreallydepictthekindofdatawearedealing
with.
Picturesareattractiveandeasytocomprehendandas
suchthismethodisparticularlyusefulinpresenting
statisticstothelayman.
WhenPictogramsareused,dataarerepresented
throughapictorialsymbolthatiscarefullyselected.

29. Cartogramsorstatisticalmapsareusedtogive
quantitativeinformationasageographicalbasis.
Theyareusedtorepresentspatialdistributions.
Thequantitiesonthemapcanbeshowninmany
wayssuchasthroughshadesorcolorsordotsor
placingpictogramineachgeographicalunit.

30. Graphs
Agraphisavisualformofpresentationof
statisticaldata.
Agraphismoreattractivethanatableoffigure.
Evenacommonmancanunderstandthemessage
ofdatafromthegraph.
Comparisonscanbemadebetweentwoormore
phenomenaveryeasilywiththehelpofagraph.

31. Graphs
Histogram
Frequency
Polygon
Frequency CurveOgive
Lorenz Curve

32. A. Histogram
Ahistogramisabarchartorgraphshowingthefrequency
ofoccurrenceofeachvalueofthevariablebeing
analyzed.
Inhistogram,dataareplottedasaseriesofrectangles.
Classintervalsareshownonthe‘X-axis’andthe
frequenciesonthe‘Y-axis’.
Theheightofeachrectanglerepresentsthefrequencyof
theclassinterval.Eachrectangleisformedwiththeotherso
astogiveacontinuouspicture.Suchagraphisalsocalled
staircaseorblockdiagram.
wecannotconstructahistogramfordistributionwithopen-
endclasses.Itisalsoquitemisleadingifthedistributionhas
unequalintervalsandsuitableadjustmentsinfrequencies
arenotmade.

34. B. Frequency Polygon
Ifwemarkthemidpointsofthetophorizontalsidesof
therectanglesinahistogramandjointhembya
straightline,thefiguresoformediscalledaFrequency
Polygon.
Thisisdoneundertheassumptionthatthefrequenciesin
aclassintervalareevenlydistributedthroughoutthe
class.
Theareaofthepolygonisequaltotheareaofthe
histogram,becausethearealeftoutsideisjustequalto
theareaincludedinit.

36. C. Frequency Curve
Ifthemiddlepointoftheupperboundariesofthe
rectanglesofahistogramiscorrectedbyasmooth
freehandcurve,thenthatdiagramiscalledfrequency
curve.
Thecurveshouldbeginandendatthebaseline.

38. D. Ogives
Forasetofobservations,weknowhowtoconstructa
frequencydistribution.Insomecaseswemayrequire
thenumberofobservationslessthanagivenvalueor
morethanagivenvalue.
Thisisobtainedbyaaccumulating(adding)the
frequenciesupto(orabove)thegivevalue.This
accumulatedfrequencyiscalledcumulativefrequency.
Thesecumulativefrequenciesarethenlistedinatable
iscalledcumulativefrequencytable.Thecurvetableis
obtainedbyplottingcumulativefrequenciesiscalleda
cumulativefrequencycurveoranogive.

39. Ogive
The ‘ less
than ogive’
method
The ‘more
than ogive’
method.

40. •Inlessthanogivemethodwestartwiththe
upperlimitsoftheclassesandgoaddingthe
frequencies.Whenthesefrequenciesare
plotted,wegetarisingcurve.
•Inmorethanogivemethod,westartwiththe
lowerlimitsoftheclassesandfromthetotal
frequencieswesubtractthefrequencyof
eachclass.Whenthesefrequenciesare
plottedwegetadecliningcurve.

42. Class
limit
Less than ogive Morethan ogive
20 0 110
30 4 106
40 10 100
50 23 87
60 48 62
70 80 30
80 99 11
90 107 3
100 110 0

44. E. Lorenz Curve
Lorenzcurveisagraphicalmethodofstudyingdispersion.It
wasintroducedbyMax.O.Lorenz,agreatEconomistanda
statistician,tostudythedistributionofwealthandincome.
Itisalsousedtostudythevariabilityinthedistributionof
profits,wages,revenue,etc.
Itisspeciallyusedtostudythedegreeofinequalityinthe
distributionofincomeandwealthbetweencountriesor
betweendifferentperiods.
Itisapercentageofcumulativevaluesofonevariablein
combinedwiththepercentageofcumulativevaluesinother
variableandthenLorenzcurveisdrawn.

45. E. Lorenz Curve (Cont)
Thecurvestartsfromtheorigin(0,0)andendsat
(100,100).Ifthewealth,revenue,landetcareequally
distributedamongthepeopleofthecountry,thenthe
Lorenzcurvewillbethediagonalofthesquare.Butthis
ishighlyimpossible.
ThedeviationoftheLorenzcurvefromthediagonal,
showshowthewealth,revenue,landetcarenotequally
distributedamongpeople.