Unit-7 biostatistics Standard_Deviation.pdf

Unit-7 biostatistics Standard_Deviation.pdf

1. B. Sc (P) Life Science III year Semester VI
DSE-1: Analytical Techniques in Plant Sciences
Dr Madhu Rani
Department of Botany, DBC, DU
Unit 7:Biostatistics
Measures of Dispersion-
Standard Deviation
Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655

2. •Whenthedeviatescoresaresquaredinvariance,
theirunitofmeasureisalsosquared.
•E.g.-IfweightsofindividualsaremeasuredinKg,then
thevarianceoftheweightswouldbeexpressedinKg
2
(or
squaredKg)
•Sincedealingwithsquaredunitsofmeasureareoften
problematic,thesquarerootofvarianceisoftenused.
•Thestandarddeviationisthesquarerootof
variance
Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655

3. •Whenthedeviatescoresaresquaredinvariance,theirunit
ofmeasureisalsosquared.
•E.g.-IfweightsofindividualsaremeasuredinKg,thenthe
varianceoftheweightswouldbeexpressedinKg
2
(orsquaredKg)
•Sincedealingwithsquaredunitsofmeasureareoften
problematic,thesquarerootofvarianceisoftenused.
•Thestandarddeviationisthesquarerootofvariance
Standard deviation = variance
Variance = standard deviation
2
Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655

4. DefinitionofStandarddeviation(SD)
•It is also called root mean square deviation.
•It is represented by Greek symbol σand in short form by SD.
•It represents the extent to which individual values differ from the
average or Mean.
•Itisbasedonallobservations.
Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655
Standarddeviationofaseriesis‘thepositivesquarerootof
thearithmeticmeanofthesquaresofdeviationsofthevarious
itemsfromthearithmeticmeanoftheseries’.

5. Calculation of Standard deviation
Formula
Where

2
= population variance,
X = a variable,
ഥXor μ= population mean, and
N = t0tal number of variables.
σ² =
σ??????−ത??????
2
??????
or σ² =
σ??????−μ
2
??????
Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655

6. Merits of SD
•It summarizes the deviation of a large distribution from Mean in one figure.
•It is most reliable and dependable Measure of Dispersion.
•It helps in calculating the Standard Error.
•It helps in finding the suitable size of sample for valid conclusions.
•It is less affected by fluctuations in sampling.
•SD is rigidly defined, and its values are always definite.
Demerits of SD
•It gives more weightage to extreme value and left to the values that are closer to
mean.
•The process of acquiring deviations and then taking square root involves lengthy
calculations. Hence its calculation is not easy.
Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655

7. Significance of SD
•It isthemostwidelyused measure of dispersion.
•It is based on all the observations.
•Thesquaringofthedeviationsremovethedrawbacksofignoringthesignsofdeviations
incomputingtheMeanDeviation.
•SDisbestamongallthemeasuresofdispersion,becauseitisleastaffectedby
fluctuationsofsampling.
•AlargevalueofSDshowsthatthemeasurementoffrequencydistributionarewidely
spreadoutfromtheMean,whilesmallvaluesofshowsthatobservationsareclosely
spreadinthevicinityofMean.
•SDindicateswhetherthevariationofdifferenceofanyindividualobservationfromthe
Meanisnaturalorrealduetosomespecificreasons.
•IthelpsinfindingtheStandardErrorwhichdetermineswhetherthedifferencebetween
theMeansoftwosimilarsamplesisbychanceforreal.
Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655

8. Step 1-arithmetic mean is calculated using the formula...
Calculation of SD
Step 2-Deviationor difference of each observation from the Mean is
calculated using the formula....
Step 3-This difference between observation and Mean is squared...
step 4-All the squared values are added to calculate the sum of squared
deviations, i.e., or
Step 5-Calculate the variance (σ
2
) by using the formula…
where N -1is the number of observations minus one.
Step 6-Find the square root of the variance to get Standard Deviation.
i.e.,
µ or
Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655

9. Calculation of SD from ungrouped data
1. Indirect method.
1.Indirect method-SDisobtained from Mean using the following formula:
2. Direct method.
When N ≥ 30 When N < 30
Wheredxor x= deviation obtained from actual Mean, i.e.,
N=totalnumberofobservations
2.Directmethod-SDisobtainedfromAssumedMean(A)insteadof
ActualMean(ഥX).SDiscalculatedbyusingtheabove-mentionedformulae,
replacingഥXwithA.
Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655

10. Calculation of SD from grouped data
For grouped data, SD iscalculated either using Arithmetic Mean (Long
method) or Assumed Mean (Short method) using the following formula:
Wherex= ??????−ത??????
f = mid-point of each class
??????=Variables
????????????=
σ????????????
2
σ??????
????????????=
??????????????????−ത??????
2
????????????
or
Long method
Short method
????????????orσ=??????.
σ??????.??????′
2
.??????
2
σ??????
Where??????′= DeviationcalculatedfromAssumedMean
c = correction
??????=Class interval
Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655

11. Coefficient of Standard Deviation
CoefficientofSDisobtainedforcomparativestudy.Thefollowing
formulaisused.
CoefficientofSD=
StandardDeviation
ArithmeticMean
=
SD
ഥX
Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655

12. Abiologistwasinterestedintheaverageheightandstandarddeviationofaplant
species.Thefollowingdataaretheheightsforasampleofn=20plants.
Theaverageheightis10.53cm,roundedtotwoplaces.Findthesamplestandard
deviation,roundedtotwodecimalplaces.
Practiceproblem
Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655
10 119.510 111011.51110.511.5
9 1010.511.5119.510.511 1110.5

13. Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655
Thevariancemaybecalculatedbyusingatable.Thenthestandarddeviationis
obtainedbytakingthesquarerootofthevariance.
Solution

14. The sample variance, σ
2
, is equal to the sum of the last column (9.7375)
divided by the total number of data values minus one, which is 19.
??????
2
=
9.7375
19
= 0.5125
The sample standard deviation ??????is equal to the square root of the sample
variance:
??????= 0.5125= 0.715891
Rounded to two decimal places, ??????= 0.72.
Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655

15. Example-2.CalculatetheStandardDeviationfromthefollowingobservations
relatedtoyieldofseeds(gm)perplantinaspecies.
240.12 240.13 240.15 240.12 240.17
240.15 240.17 240.16 240.22 240.21
Example -3. Calculate the Mean and Standard Deviation from the following data.
Leaf length (cm)90-9980-8970-7960-6950-5940-4930-39
Frequency 2 12 22 20 14 2 1
Submit the
answers in
Google classroom
till 24
th
April
2020
Dr Madhu Rani (BOTANY) e-mail [email protected]
Contact/Whatsapp-9412209655