Mean MedianMode Quartile deviation
Whilelookingattheearliermeasuresofdispersionallofthemsuffer fromoneortheotherdemeriti.e. Rangeitsufferfromaseriousdrawbackconsidersonly2valuesand neglectsalltheothervaluesoftheseries. Quartiledeviationconsidersonly50%oftheitemandignoresthe other50%ofitemsintheseries. Meandeviationnodoubtanimprovedmeasurebutignoresnegative signswithoutanybasis. 3
TheconceptofstandarddeviationwasfirstintroducedbyKarlPearsonin 1893. KarlPearsonafterobservingallthesethingshasgivenusamorescientific formulaforcalculatingormeasuringdispersion.WhilecalculatingSDwe takedeviationsofindividualobservationsfromtheirAMandthen eachsquares.ThesumofthesquaresisdividedbytheTotalnumberof observations.Thesquarerootofthissumisknowsasstandard deviation. Thestandarddeviationisthemostusefulandthemostpopularmeasureof dispersion. Itisalwayscalculatedfromthearithmeticmean,medianandmodeisnot considered. 4
D ef in iti on :
StandardDeviationisthepositivesquarerootoftheaverageofsquared deviationtakenfromarithmeticmean. ThestandarddeviationisrepresentedbytheGreekletter(sigma). Formula. Standarddeviation==Standard deviation == Alternatively =
CA LC U LA TI O N O F ST AN D AR D D EV IA TI O N - IN D IV ID U A L O BS ER VA TI O N Tw o M et ho ds :- By ta ki ng d ev ia tio n of th e ite m s fr om th e ac tu al m ea n. By ta ki ng d ev ia tio n of th e ite m s fr om a n as su m ed m ea n.
CASE-I. When the deviation are taken from the actual mean. DIRECTMETHOD Standarddeviation== or =valueofthevariableofobservation, =arithmeticmean =totalnumberofobservations.
Example:Findthemeanrespirationrateperminuteanditsstandarddeviationwhenin4 casestheratewasfoundtobe:16,13,17and22. Solution: HereMean = 16 13 17 22 Standard deviation ====
-1 -4 0 5
1 16 0 25
Short-Cut Method Standard deviation ==
CASE-II. When the deviation are taken from the Assumed mean.
= = = = = 16.398
Example:Blood serum cholesterol levels of 10 persons are as under: 240, 260, 290, 245, 255, 288, 272, 263, 277, 251. calculation standard deviation with the help of assumed mean. Value A=264 240 260 290 245 255 288 272 263 277 251
576
16 676 361
81 576
64 1 169 169
-24 -4 26 -19 -9 24 8 -1 13 13
Here, Mean== = 9 = 263.9 is a fraction.
CA LC U LA TI O N O F ST AN DA RD D EV IA TI O N -D IS CE RE TE S ER IE S O R G RO U PE D D AT A T hr ee M et ho ds a) A ct ua l M ea n M et ho d or D ir ec t M et h od b) A ss um ed M ea n M et ho d or S ho rt -c u t M et ho d c) St ep D ev ia ti on M et ho d
a) A ct ua l M ea n M et ho d or D ir ec t M et ho d
The S.D. for the discrete series is given by the formula. = Whereis the arithmetic mean, is the size of items, is the corresponding frequency andb) A ss um ed M ea n M et ho d or S ho rt -c ut M et ho d
Standard deviation== Whereis the assumed mean, is the corresponding frequency andEx am pl e:
Periods:10111213141516 No. of patients:2711151041So lu tio n:
Period s:(x)No. of patients() 10 11 12 13 14 15 162 7 11 15 10 4 1 TotalN==50
-3 -2 -1 0 1 2 3
-6 -14 -11 0 10 8 3 =-10
9 4 1 0 1 4 9
18 28 11 0 10 16 9 =92
Mean== =13 =12.8isafraction. = = = = = 1.342
c) S te p D ev ia tio n M et ho d
We divide the deviation by a common class interval and use the following formula Standard deviation==× Wherecommon class interval, is assumed mean f is the respective frequency.=× =× =× =× = 1.235× =4.94 mm Hg.
Ex am pl e:
B.P.(mmHg):102106110114118122126 No. of days:39253517101So lu tio n :
B.P.(mmHg)No. of days () 102 106 110 114 118 122 1263 9 25 35 17 10 1 TotalN=100
-3 -2 -1 0 1 2 3
-9 -18 -25 0 17 20 3 =-12
27 36 25 0 17 40 9 =154
A.M= × mm Hg
S.D. of Continues Series can be calculated by any one of the methods discussed for discrete frequency distribution But Step Deviation Method is mostly used. Standard deviation==× Wherecommon class interval, is assumed mean f is the respective frequency.
Ex am pl e:
I.Q.10-2020-3030-4040-5050-6060-7070-80 No. of students:51215201042So lu tio n:
I.Q.No. of students:()Mid-value 10-20 20-30 30-40 40-50 50-60 60-70 70-805 12 15 20 10 4 2 Total=N=68
-3 -2 -1 0 1 2 3
-15 -24 -15 0 10 8 6 =-30
45 48 15 0 10 16 18 =152
Standard deviation= = = =
15 25 35 45 55 65 75
It is possible to compute combined mean of two or more than two groups. Combined Standard Deviation is denoted by = Wherecombined standard deviation ,
a) Combined S.D. = combined Mean = = == 55
Thefollowingaresomeoftheparticularsofthe distributionofweightofboysandgirlsinaclass: a)Findthestandarddeviationofthecombineddata b)whichofthetwodistributionsismorevariable
BoysGirls Numbers10050 Meanweight60kg45kg Variance()94 = = b) C.V (Boys)= C.V (Girls)=