Standard Deviation (Meaning, Characteristics and Calculation)

Standard Deviation basics (Introduction to statistics)-It’s a quantity that indicates by how much the members of a group differ from the mean value for the group.

1. Standard
Deviation
Dr Rajesh Verma
Asst. Prof (Psychology)
Govt. College Adampur,
Hisar, HaryanaPC Mahalanobis, Indian Legend

2. Meaning-Cum-Definition
It’saquantitythatindicatesbyhowmuchthe
membersofagroupdifferfromthemeanvalueforthe
group.
Standarddeviationisandindexofdegreeof
dispersionandanestimateofthevariabilityinthe
populationfromwhichthesampleisdrawn(Guilford
&Fruchter,
1976).

3. Introduction
SDisalsoknownasCoefficientofVariation(CV).It’s
akindofspecialaverageofallthedeviationfromthegroup
mean.SpecialaveragemeansthatSDisbeyondthesimple
arithmeticmean.Alongwiththetotalrange,semi-
interquartilerange,averagedeviationitisthemost
commonlyusedsinglenumbertoindicate
variabilityamongadataset.Variabilityis
thescatterorspreadoftheseparatescores
aroundtheircentraltendency(Garret,2014).
SDisthemoststableindexofvariability
whichiscomputedusingsquareddeviations
thataretakenfrommeanonlyi.e.neither
mediannormode.

4. Characteristics
(i)HighvalueofSDmeansthatmostofthescoresare
awayfromgroupmeanandviceversa.
(ii)Itistheindexofspreadofscoresaboutthemeanthat
involveseveryobservation.
(iii)GreaterSDthanmeanindicates
thatthedistributionhavemajorityof
lowvalues.
(iv)LowerSDthanmeanindicates
thatthedistributionhavemajority
ofhighvalues.
(v)Itcanalsoindicatethatwhether
thedistributionhasdispersionof
extremevaluesorhas
centraltendency.

5. (vi)Itisexpressedinthesameunitsasthedata.
(vii)Itispossibletocalculatethecombinedstandard
deviationoftwoormoregroups.
(viii)TheSDisconsideredmostappropriatestatisticaltool
forcomparingthevariabilityoftwoormoregroups.
(ix)Itisusedasunitofmeasurementinnormal
probabilitydistribution.
(x)SDformsthebasefor
severalstatisticaltechniques
suchskewness,kurtosis,
ANOVAetc.

6. How much SD is Good or Bad
NoamountofSDisgoodorbadbecauseit’sjustan
indicatorofspreadofscores.Ifsomeoneisinterestedtosee
theclosenessofscoresaroundthemeanthanSD≤1isgood,
ontheotherhandifsomeoneisinterestedinlargerspread
thanSD≥1isgood.Acrossbooksandmanualsithasbeen
acceptedthatSD≥1isthe
indicatorofhighvariation
amongthescoresandSD≤1
indicateslowvariation.It’sa
kindofruleofthumb.
Infactitdependupon
thedatasetandobjectivesof
theresearcher.

7. Computation of SD
For Ungrouped Data
Letustakeanhypotheticaldata.
Ex–12,54,32,51,24,58,21,43,31,48
StepI–Calculatethemean.
Mean( ??????)=
∑??????
??????
Where,x=Score
N=totalno.ofscore
Therefore,12+54+32+51+24
+58+21+43+31+48=374
??????=
���
��
=34

8. StepII–Calculatedeviation(d)ofeachscorefrom
themean.
(i)Itisachievedby
subtractinggroupmean
fromeachscore,
(ii)Squaretheeach
Deviation,
(iii)Sumupsquared
Deviations,
StepIII–CalculateSDby
usingthefollowingformula
σ=
∑??????
�
??????
=>
����
��
=>���.���=14.545

9. SD of Grouped Data
LetustakeHypotheticalData
asshowninadjacenttable
StepI-Calculatethemean
Forcalculationofmeanwemust
followthefollowingfewsteps

10. Stepsofcalculatingmean
(i)FindthemidpointsofCI(x)
(ii)Multiplymidpoints(x)withtheirrespective
frequency(f)
(iii)Addfx
(iv)Divideitbythesumof
frequenciesi.e.50
mean=
∑????????????
??????
=
����
��
=33

11. StepII-Subtractmeanfromthemidpointsi.e.??????− ??????
StepIII-Squarethesevalueswhicharedenotedby(x`)
StepIV-Multiplyx`withtheirrespectivefrequencies(fx`)
StepV-Addfx`
Finallysubstitutethevaluesintheformula
σ=
∑????????????`
??????
=
����
��
=���.��
(??????
�
=110.24)
=10.499

12. References:
1. https://dictionary.apa.org/quartile-deviation.
2. Guilford,J.P.andFruchter,B.(1978).FundamentalStatisticsin
PsychologyandEducation,6thed.Tokyo:McGraw-Hill.
3. https://todayinsci.com/M/Mahalanobis_Prasanta/
MahalanobisPrasanta-Quotations.htm.
4. Garrett,H.E.(2014).StatisticsinPsychologyandEducation.New
Delhi:PragonInternational.
5. Levin,J.&Fox,J.A.(2006).ElementaryStatistics.
NewDelhi:Pearson.

13. [email protected]
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Discussion
Concept of
Normality