Week 4 - Measure of Variability (variance, SD, IQR, etc)

Biostatistics and Epidemiology

1. DESCRIPTIVE STATISTICS-MEASURES
OF VARIABILITY
Lecture Week 4

2. LEARNING OUTCOMES:
1.Explain the uses of the different descriptive
statistical tools.
2.Interpret results of the different statistical
tools

3. MEASURES OF VARIABILITY
Describes how far apart data points lie
from each other and from the center of a
distribution
Aka spread, scatter or dispersion

4. IMPORTANCE
➢Variabilitydetermineshowwellyoucan
generalizeresultsfromthesampletoyour
population.
➢Lowvariabilitymeansyouhavemore
consistentvalues/clusteredvalues
➢Highvariabilitymeansyouhaveless
consistentvalues

6. RANGE
Tells you the spread of your data from the
lowest to the highest value in the distribution.
Difference between the lowest and the highest
observations.
R= (H-L)

7. Example: Data on the birth (in ounces) of the newly
born child at the certain hospital are as follows 112,
111, 107, 119, 92, 80, 81, 84, 118, 106, 103, 94.
Compute the range.

8. INTERQUARTILE RANGE
The range of observations of specified part of the
total group usually the middle 50 percent of the
cases lying between Q1 and Q3.
Gives you the spread of the middle of your
distribution
I��= �3 −�1

9. EXAMPLE:
ComputetheIQRofthetimes(inminutes)12
randomlyselectedBSMT/MLSstudenttookto
commutefromhometoUSL.Usethefollowing14,
25,23,45,20,32,42,19,26,30,40,24.

10. 1.Arrange the times in order as follows 14, 19, 20,
23, 24, 25, 26, 30, 32, 40, 42, and 45.
2.Solve for quartiles:
➢Q1= 3
RD
+4
TH
/2
➢Q3= 9
TH
+ 10
TH
/2
3.IQR=

11. QUARTILE DEVIATION (QD)
➢Theaveragedistancefromthemediantothefirsttwoquartiles
➢Usedwhenthereareextremelylowandhighobservations
especiallywhentherearebiggapsbetweenobservations.

12. EXAMPLE:
ComputetheIQRofthetimes(inminutes)12
randomlyselectedBSMT/MLSstudenttookto
commutefromhometoUSL.
Usethefollowing14,25,23,45,20,32,42,19,
26,30,40,24.

13. AVERAGE DEVIATION
➢Measureofabsolutevariabilitythatisaffectedby
everyindividualobservation.
➢Itisthemeanoftheabsolutedeviationsofthe
individualobservationsfromthemean.

14. WHERE:
AD = average deviation
∑ = symbol for "summation"
Xi = ithindividual observations
??????̅= sample mean
n = total number of observations

15. EXAMPLE:
➢Theweight(inpounds)lostbysevennewly
recruitedfacultymembersattheSHASinUSLatthe
endoffirstthreemonthsmembershipasfollows:15,
23,18,10,20,17,and10.
➢ComputefortheAD.

17. Stepsindeterminingtheaveragedeviationrawdataareas
follows:
1.Computethemeanfromrawobservations.
2.Subtractthemeanfromtheindividualobservationstoget
thedeviation.
3.Getthesumofthedeviationregardlessofsigns.
4.Dividestep3by(??????-1).Thequotientistheaverage
deviation.

18. STANDARD DEVIATION
➢Measureofdispersionthatinvolvesall
observationsinthedistributionratherthan
throughextremeobservations
➢Themostaccuratemeasureofdispersion

19. WHERE:
SD = average deviation
∑ = symbol for "summation“
Xi = ithindividual observations
??????̅= sample mean
n = total number of observations

20. EXAMPLE:
ThescoresofsixSHASstudentsin
Biostatisticsare22,33,45,25,37,and
48.
ComputeaSD.

22. Thestepsindeterminingthestandarddeviationfromraw
dataareasfollows:
1.Findthemean
2.Subtractthemeanfromtheindividualobservationstoget
thedeviation
3.Squarethedeviationsandthen,getthesumofthesquared
deviation
4.Dividethesumofthesquareddeviationsby(n-1)
5.Extractthesquarerootoftheanswerinstep4.Theanswer
isthestandarddeviation.

23. COEFFICIENT OF VARIATION
➢Ratio of the standard deviation to its mean
➢usually expressed in percentage; unitless
➢A measure of relative variability

24. ➢Usedwhenonewishestocomparethescatterofone
distributionwithanotherdistribution.
➢Whentheunitsofmeasurementofthevariablesbeing
comparedaredifferent,e.g.,heightincentimetersandweight
inkilograms,orwhenthemeansdiffermarkedly,e.g,mean
ageof5yearsoldandmeanageof15yearsold
➢Wecancomparewhichvariableishighertheircoefficientof
variation.

25. Example:Themeanheightandweight(withSD)of
medicalstudentsofSaintLouisUniversity..Compute
andinterprettheCV.

27. INTERPRETATION:
EventhoughiftheSDoftheweightislower
thantheheight,itismorevariablethanthe
latterasprovenbyitshighercoefficientof
variation.

28. Z-SCORE (STANDARD SCORE)
➢How many SDs an observation is above or below
the mean?
Where:
▪X: individual observation
▪??????̅= sample mean
▪SD: Standard deviation

29. Z-scoreInterpretation
Positiveimplies that the given observation is
above(or higher than, or greater than) the
mean
Negativeit means the observation is below (or lower
than, or less than) the mean.
Zero it means the given observation is the same
as the mean

30. In Biostatistics first examination, the mean
grade is 80 and the standard ' deviation is 8.
Find the standard score for a student whose
grade is 70 and 90.

32. ➢Az-scoreof-1.25meansthereis1.25standarddeviation
between70and80.Thenegativesignindicatesthatthegiven
grade,70,isbelowthemeangrade.
➢Also,Z-scoreof1.25meansthereis1.25standard
deviationbetween90and80.Thepositivesignindicatesthat
thegivengrade,90,isabovethemeangrade.

34. MEASURES OF SKEWNESS
Indicates the degree of asymmetry &
direction of distribution.
Positive skewness
(skewed to the right)
Values are concentrated on the
left
Longer tail is directed to the right
Negative skewness
(skewed to the right)
Values are concentrated on the
right
Longer tail is directed to the left