Diagramatic Representation

1. iarammati Nere Senttm
Hqramneti RafruentteniadaKim o Procoa
t oispl shdstinl Tem obtamad .totoet
olleed and bulated olata, visuly
mentad aapm. to undershnd
abetter uAYt up md onm wthin tho-
bgy admInisttr
ma_ConwTCia Seutr tuye of cliagrams
Vari 0diiaL_ Sds Slatifis ja esentias
t undersand ickly abrnt thaPrvre ou

2. A Fa i PietoriaL reretation o#
e a t i v hansb2tuumi twoqantHis
varie a a

3. Graphs and Curves
Line graph and curve
Suppose we have plotted some points with reference to two
perpendicular lines
as axesS,
32
D 28
D
/ B
E1
15
A
10
12
Curve
Line Graph
Fig. 9.5
Fig. 9.6
we will get the proper location of some points, which may be called as A, B, C, D, . . .
etc. Now, if wejoin AB, BC, CD,... by separate lines then we will get a line graph
(fig. 9.5.) and if we join the points A, B, C, D,.. .
by a suitable arch, so that the
turning points atB,C,D,. .
becomes sinooth, then we will get a 'curve' (fig. 9.6.).
Time series
Any series of data related with time is the time series. As for example, data on
Seasonal Variation of sales and purchase of certain commodities is the time series.
Yearly budget of central or any state government, day-wise attendance of students
in a class in a week, population according to census report are all the examples o
a. time series.
Any line graph related to time series is knowiu as
Historigram.
Historigrams may be of single, two or more variables. If the average monthly
income, expenditure and savings of a person are given for last ten years, then taking
any one of the data into account, We imay draw a single liistorigranm, or a double
historigram with twice of them or with all of tlhem a multi-historigram can be drawn.

4. Diagramniatic itefpresentation
209
E x a n n p
alnple 9.5. The average inonthiy income, exjpenditure and saviugs of a skilled
Luhour in a certain company are given 1or 10 years. Express the data graphically in
a s u i t a b l e m a n n e r .
Years Average monthly Average mouthly Average monthly
income expenditure savings
1989-90
1650 1275 375
1990-
1920 4S0 440
1991-92
2340 1620 r20
1992-93 2760 1975 783
1993-94 280 2250 1030
1994-95 6-40 2800 840
1995 7520 4450 3070
1996-997 12640 8500 1140
1997598 18580 2300 6280
1998-199 23200 14700 8500
Solution:
AVERAGE MONTHLY INCOME, EXPENDITURE AND SAVINGS OF A SKILLED LABOUA
25000
20000 Income
15000
- -D- Expenditure
10000
Savings
5000
9 -
1 9
9 0 - 9
2 . 9
9 9
1994-95
1995-9 -99
199
1998
Histograms of two or more variables
Fig.9.7
Example 9.6. First 8 batsnen of two teams in a cricket match scóred as following:
Batsman 1 2 3 4 5 6 7 S
Teain A 87 43 14 29 43 32 20 S
Team B 48 57 64 40 0 22 28 30
onare teani-wise performance by irawing line graph.
1991-92 1993-94 S6-97
1997-98

5. P'robabilaty aulStatisties
Solution:
RUNS SCORED BY FIRST 8 BATSMEN OF TWO TEAMS
00
90
TEAM A
70
- -TEAM B
0
0
Batting order of two teams
Fig. 9.8
Note: The above are the examples of line graphs but these are not the historigrams
as the data is not related with time.
Histogram
To
represent graphically. the frequency distribution of
corresponding class intervals,
the adjacent rectangular bars are used. The
assembly of such adjacent rectangular
bars are known as
Histogram. The length of the rectaugles are
proportional to
corresponding frequencies of classes. Karl PearsOu in 1895 first used this nane.
Types of histogramn
(1) Histogram for ecqual class intervals.
(2) Histogram for
unequal class
intervals
Methods of construction
(1) When the class intervals are cqual, the hight of the rectangles should be
proportional to the
corresponcding trequencies of cach class.
(2) If there be any discrete type ol class
intervals we arc to convert thenu to
continuous type.
(3) Generally class intervals are to be laken alog horizontal axis and frequ-
encies in äny form, along vertical axis.
(4) In case of histogram for inequal class intervals, the breadths of the rer-
tangles are to be taken
projortional to the class width and the heights of
the rectangles, proportional to
freqieney density ot the
respertive class.

6. 3.45
3.24 HISTOGRAM
A Histogram is a
graplh containing a set of rectangles, each being constructedtorepresent the
size ofthe class interval by its width andthe frequency in each class-interval byits height. The area
of each rectangle is
proportional to the frequencyin the respective class-interval andthe total area
of the histogram is
proportionalto thetotal frequency. A
histogram is used to depicta frequency
distribution.
CUISuuu storam a

7. (mapl of rejuene Hctibt.an
Th ak, o4 equinty distibacitn are
deie t
Prelet he chatscrvistie feat ures D
ofigtribuuhions °aooding {hIY
mor req ny
Shape Ond
Th mat Conmmmly URed hs ae
DHistogra
foquany polm
C)Feqwnty Curwa
Cl) Cum 4tue reawny (ure

8. 23. GRAPHIC REPRESENTATION OF A FREQUENCY DISTRIBUTION
It is often useful to represent a frequency distribution by means of a diagramn
which makes the unwieldy data intelligible and conveys to the eye the general run of
the observations. Diagrammatic representation also facilitates the
comparison of
two
or more frequency distributions. We consider below some important types of graphic
representation.
2.31. Histogram. In drawing the histogram of a given continuous frequency
dtstríbution we first mark off along the
x
-
axis all the class intervals
on a
suitabie
scale. On each class interval erect rectangles with heights proportional to the
frequency of the corresponding class interval so that the area of the rectangleis
proportional to the frequency of the class. It, however, the classes are of uncqual
width then the height of the rectangle will be proportional to the ratio of the
frequencies to the width of the classes. The diagram of continuous rectangles so
obtained is called histogran1.
l aictoTram for an ungrouped freauoncy distrib1ution of a variabic
T

9. Comprenensive Slatistical
Methods
TYPEL HISTOGRAM WITH EQUAL CLASS INIERVALS
The sizes of class intervals are drawn on I-axis with equal distances and their respe
Irequencies on y-axis. Class and its frequency taken together torm a
rectangle. The graph at
rectangles is known as
histogram.
Each class has lower and upper values. This gives us two equal ines
representing th
frequencies. Upper ends of the lines are joined together. This
process will give us
rectangles. Te
heights of the rectangles will be
proportional to their frequencies.
Example 36. The monthly profits in rupees of 100 shops distributed as follows:
Profit per shop 0-100 100-200 200300 300-400 400-500 500 600
No. of shops
pective
apa of
be
20 17
12 18 27
Solution. This is the case of
Histogram with equal frequencies.
30
0 100 200 300 400 500 600
Class-Interval [P rofit per shop
Fig. 3.41. Histogram showing monthly prolits.
Example 37.Draw the histogram for the
following data:
Marks
0-
10 10 20 20 30 30 40 40 50 50 60 60 -70
S0 40
No. of Students
Solution. We
represent the class limits along x-axis and
frequencies along y-axis. Taking class
intervals as bases and the
corresponding frequencies (No. of students) as
heights, we construct the
rectangles to
get the
histogram of the given frequency distribution as shown in Fig. 3.42.
20 30 70
0 40
Scale: Along x-axis 1 cm =
10 marks
Along y-a xis: 1 cm =
10 studenis
A
10 20 30 40 50 60 70
Marks-
Flg.3.42. Histogram showing marks oblained by sludents.
TYPE II.
HISTOGRAM WHEN CLASS INTERVALS ARE GIVEN N
EXCLUSI
FORM, i.e., WHEN CLASS
INTERVALS ARE NOT
CONTINUOUS
In a
histogram, it is
necessary that the
adjacent rectangles be attached to each other. If
w
to
represent the given data (in
cxCusivefomm) as such we shall get a bar
diagram as there w
gaps in between the classes. But in a
histogram the bars are
continuous without any Be

10. The histogram of the given data is given by Fig. 3.49.
Note: The crank mark or kink ( ) in the curve on the hornzontal axis means tha
lack of space this small distance representss to
S.
owing
he anu
Example 45. Draw the histogram of the following frequenc)y distribulion and show th
TYPETV HISTOGRAM WITH UNEQUAL CLASS INTERVALS
On your graph which represents the total number of wage-eamers in he age-group 19-n
35-
14 15 16- 17 18-20 21-24 25 29 30-34
0
00
Age group
150 i10 I10
No. of wage-earners 120 140
Solution. Here the class intervals have been marked by class-limits. As a result, the uperli
of one class does not coincide with the lower limit of the next
class. In order to draw a histogm
the upper limit of one class must coincide with the lower limit of the next class. To draw a histogm
n suchcase where the upper limit of one class does not coincide with the lower limit ofthene
class, class limits of all the classes should be extended to their class-boundaries. This ill elt
drawing ofa histogram from a
frequency distribution in which class intervals are marked byclas
limits.
In this example, the classes are not of equal width. Some have less width and some have
more width. So the histogram should be drawn on the basis of frequency density and not on te
basis of frequency.
Calculation of Histogram
Class-boundary Class-width
Frequency densiy
Frequency
(No. of wage-earners)
Class-interval
Age group)
13.5 15.5
15.5 17.5
17.5 20.5
(Age groupP)
14 15
120 0
16- 17
140 70
18- 20
150 0
21 24 20.5 24.5
10 27.5
25 -29 24.5 29.5
110 2
34 29.5 34.5
100 20
35 3 39 34.5 39.5
0 18
13.5 15.5 17.5 19 20.5 24.5 29.5 32 34.5 39.5
Class-boundary (Age In
years
Flg.3.50. Histogram showing the
number of
wage-eatners.
The total number of
wage earnerS in the age group 19 -32 is shown by the sn
in the histogram.
aded area

11. DiagraInmiatiC 1ieprescntation
211
Example 9.7. Construct a
Histogram tron the followino talla.
Weight (in lbs) No. of Men
95-105 105
105-115
210
115-125
125-135
300
175
135-145 255

12. Example 9.8. Draw a Histogram of the followingdata:
Height in cm No. of Persons
120-129 15
130-139 20
140-154 45
155-159 25
160-179 30

13. C o m p r e n E T o m
3.50
Example
42. The following
table presents
the number ofiterate females in the age pro
(10 34) in a town:
group
25-29 30-34
20-24
Age group
10-14
15-19
580 290
800
No. ofFemales : 300
980
Draw a histogram to represent the above data.
discontinuous distribution. In order to draw a histogram
autian. Thedifferenco

14. Diagrammatic and Graphic Presentation of Data
3.15
Example 14: The following figures relate to the costof construction of a huuse in Delhi:
Ttem :Cement Steel Bricks Timber Labour Miscellaneous
Expenditure 20% 18% 10% 15% 25% 12%
Represent the data by a suitablediagram.

15. 3.14
he
Example 13: Draw a pie diagram to represeni ihefollowing data ofinvestmment patterm in the
Five Year Plan: 14%
Agricuiture andCommunity Development
16%
Irrigaiion and Power
Small and Organised Industries and Minerals
Transport and Communication
Social senvices
29%
17%
16%
8 9
Inventories

16. Example 15: Draw a suitable diagramforthe following:
Expenditure on item Percentage of Total expenditure
Food 65
Clothing 10
Housing 12
Fuel andlighting 5
Miscellaneous 8