Lecture 6 (Measure of Dispersion, Rang, M.D and Coeffts).pdf

It is very good to understand probability

1. Introduction To Statistics,
Statistics And Probability
1
Dr.Shabbir Ahmad
Assistant Professor,
Department of Mathematics,
COMSATS University
Islamabad, Wah Campus
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:51 AM
Lecture No. 6

2. Measures of Dispersion
Different Tools (Range, M.D, Coefficient
of Range and MD.
Merits and Demerits of
2
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:51 AM

3. In this lecture
•What is Measure of Dispersion
•Why measures of dispersion?
•Characteristics of a good measure of dispersion
•Types of Dispersion Measures
•Range, Coefficient of Range, Merits and Demerits
•Mean Deviation (M.D) , Merits and Demerits of M.D and Caff. of M.D
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:51 AM
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:51 AM

4. WHAT AND WHY MEASURE OF DISPERSION
•Aquantitythatmeasuresthevariabilityamongthe
data,orhowthedataaredispersedaboutthe
average,knownasMeasuresofdispersion,scatter,
orvariations.
•Instatistics,dispersion(alsocalledvariability,
scatter,orspread)istheextenttowhicha
distributionisstretchedorsqueezed.
•Ameasureofstatisticaldispersionisanonnegative
realnumberthatiszeroifallthedataarethesame
andincreasesasthedatabecomemorediverse.
4
Ameasuresofdispersion:
•Serveasabasisforthe
controlofthevariability.
•Tocomparethevariability
oftwoormoreseries.
•Determinethereliability
ofanaverage.
•Facilitatetheuseofother
statisticalmeasures.
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:51 AM

5. CHARACTERISTICS OF A GOOD MEASURE OF DISPERSION
Anidealmeasureofdispersionisexpectedtopossessthefollowingproperties
1.Itshouldberigidlydefined.
2.Itshouldbebasedonalltheitemsobservationsofthedata.
3.Itshouldnotbeundulyaffectedbyextremeitems.
4.Mustbeeasilysubjectedtofurthermathematicaloperations.
5.Itshouldbesimpletounderstandandeasytocalculate.
5
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:51 AM

6. TYPES OF DISPERSION MEASURES
Measures of dispersion
Absolute
Range
Mean
absolute
deviation
Quartile
Deviation
Variance and
standard
deviation
Relative
Coefficient of
Range
Coefficient of
Quartile
deviation
Coefficient of
mean
deviation
Coefficient of
variation
6
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:52 AM

7. RANGE
•Thisisthesimplestpossiblemeasureofdispersionandisdefinedasthedifference
betweenthelargestandsmallestvaluesofthevariable.Insymbols,
�??????���=??????
�????????????–??????
�??????�.
Where,
??????
�????????????=�??????������??????���and??????
�??????�=��??????������??????���
•Inindividualobservationsanddiscreteseries,??????
�????????????and??????
�??????�areeasilyidentified.
•Incontinuousseries,thefollowingtwomethodsarefollowed.
Method1:
??????
�????????????=UpperC.Bofthehighestclassand??????
�??????�=LowerC.Bofthelowest
class.
Method2:
??????
�????????????=Midvalueofthehighestclass.and??????
�??????�=Midvalueofthelowest
class.
7
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:52 AM

8. Example2:
Calculaterangefromthefollowingdistribution.
Size:60−63,63−66,66−69,69−72,72−75
f: 5, 18, 42, 27, 8
Solution:
X
max=UpperC.Bofthehighestclass=75,X
min=LowerC.Bofthelowestclass=60
Range=X
max–X
min
Range=75–60=15
8
Example1:
Theyields(kgperplot)ofacottonvarietyfromfiveplotsare8,9,8,10and11.
Findtherange.
Solution:
As,X
max=11andX
min=8
Range=X
max–X
min=11−8=3
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:52 AM

9. Therelativemeasureofdispersionbasedonrangeisthecoefficientofrangealsoknownasrange
coefficientofdispersion.
Formula:
�����??????�??????������??????���=
??????
�????????????–??????
�??????�
??????
�????????????+??????
�??????�
Example:
Theyields(kgperplot)ofacottonvarietyfromfiveplotsare8,9,8,10and11.Findthecoefficientof
range.
Solution:
As,??????
�????????????=11and??????
�??????�=8
�����??????�??????������??????���=
??????
�????????????–??????
�??????�
??????
�????????????+??????
�??????�
=
11−8
11+8
=
3
19
=0.1578
Coefficient of Range
9
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:52 AM

10. MERITS AND DEMERITS OF RANGE
Merits
1.Itissimpletounderstand.
2.Itiseasytocalculate.
3.Incertaintypesofproblemslikequalitycontrol,weatherforecasts,
sharepriceanalysis,etc.,rangeismostwidelyused.
10
Demerits
1.Itisverymuchaffectedbytheextremeitems.
2.Itisbasedononlytwoextremeobservations.
3.Itcannotbecalculatedfromopen-endclassintervals.
4.Itisnotsuitableformathematicaltreatment.
5.Itisaveryrarelyusedmeasure.
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:52 AM

11. MEAN DEVIATION (M.D)
•Adeviationisthedifferencebetweenindividualobservationwithaverage(Mean,
Median,Mode)ofthedata,i.e.??????
??????−??????���??????��.
•Theaverageoftheabsolutevaluesofdeviationfromthemean(medianormode)is
calledmeandeviationormeanabsolutedeviation.
Formula: �.�=
σ??????
??????−????????????�????????????��
�
11
M.Dfrommeanforsampledata: �.�=
σ??????
??????−ത??????
�
M.Dfrommeanforpopulationdata:�.�=
σ??????
??????−µ
�
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:52 AM

12. FORMULAS OF M.D
Ungrouped data
M.D from Mean:
�.�=
σ??????
??????−ത??????
�
M.D from Median:
�.�=
σ??????
??????−���????????????�
�
M.D from Mode:
�.�=
σ??????
??????−����
�
Grouped data
M.D from Mean:
�.�=
σ�??????
??????−ത??????
σ�
M.D from Median:
�.�=
σ�??????
??????−���????????????�
σ�
M.D from Mode:
�.�=
σ�??????
??????−����
σ�
12
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:52 AM

13. Example 1: Let the ages of 5 boys in a class is 12, 14, 14, 15, 18. Calculate the M.D from mean.
Solution: first calculating the mean age:
ഥX=
12+14+14+15+18
5
=14.6
Now using formula of M.D: ??????.??????=
σ????????????−ഥ??????
�
M.D=
12−14.6+14−14.6+14−14.6+15−14.6+18−14.6
5
M.D=
−2.6+−0.6+−0.6+0.4+3.4
5
M.D=
2.6+0.6+0.6+0.4+3.4
5
M.D=
7.6
5
=1.52
13
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:52 AM

14. Example 2: Calculate the M.D from mean for the following frequency distribution:
14
??????????????????????????????�???????????? ?????? ????????????
??????
??????−ഥ??????
??????
??????-
156.1667
ȁ??????
??????
�??????
??????−���.���
110−109114.5 1 114.5−41.666741.66667 41.66667
120−129124.5 4 498−31.666731.66667 126.6667
130−139134.5 17 2286.5−21.666721.66667 368.3333
140−149144.5 28 4046−11.666711.66667 326.6667
150−159154.5 25 3862.5−1.666671.666667 41.66667
160−169164.5 18 29618.3333338.333333 150
170−179174.5 13 2268.518.3333318.33333 238.3333
180−189184.5 6 110728.3333328.33333 170
190−199194.5 5 972.538.3333338.33333 191.6667
200−209204.5 2 40948.3333348.33333 96.66667
210−219214.5 1 214.558.3333358.33333 58.33333
??????�??????????????????������������
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:52 AM
Solution:
ത??????=
σ�??????
σ�
=
18740
120
=
156.1667
Now using formula of M.D:
�.�=
σ�??????
??????−ത??????
σ�
Putting values from table:
�.�=
1810
120
=15.0833

15. MERITS AND DEMERITS OF M.D
Merits
1.Simplifies calculations.
2.Can be calculated by mean, median and mode.
3.Is not affected by extreme measures.
4.Used to make healthy comparisons.
Demerits
1.Mathematically illogical to assume all negatives as positives.
2.Not reliable.
3.Not suitable for comparing series.
15
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:52 AM

16. COEFFICIENT OF MEAN DEVIATION
TherelativemeasureofdispersionwegetbydividingMeanDeviationbyMeanor
Median,iscalledCoefficientofMeanDeviation.
Formula: �����??????�??????������.�=
�.??????
��??????��??????���????????????�
16
Example:Lettheagesof5boysinaclassis12,14,14,15,18.Calculatethecoefficientof
M.Dfrommean.
Solution:
firstcalculatingthemeanage:ഥX=14.6
NowusingformulaofM.D:M.D=1.52
CoefficientofM.D=
M.D
ഥX
=
1.52
14.6
=0.1041
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:52 AM

17. MERITS & DEMERITS OF COEFFICIENT OF M.D
Merits
1.Better Result than Range & Quartile Coefficient.
2.Least sampling fluctuation.
3.Rigidly defined.
Demerits:
1.Fractional Average.
2.Cannot be used for sociological studies
3.Less reliable than Coefficient of Variation
17
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:52 AM

18. Question1:Thefollowingsampledataofthenumberofcommunicationsaretakenfromlogsof
communicationwithDistanceEducationstudents:5,9,5,23,27,55,34,7,30,15,22,60,14,52,297,8,
51,15,51,35,15,39,137,43,38,14,93,7.Calculatemeandeviationfrommean.
Question2:Thefollowingistheagedistributionof1000personsworkinginanorganization
Calculatemeandeviationfrommeanandmeandeviationfrommedian.
Age Group Number of Persons
20-25 30
25-30 160
30-35 210
35-40 180
40-45 145
45-50 105
50-55 70
55-60 60
60-65 40
18
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:52 AM

19. ANY QUESTION
19
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 30-Sep-21 10:40:52 AM