Measures of central tendency: mean, median, mode

In this presentation, I have explained measures of central tendency in simple terms to help you understand them easily. These measures help find the central value of a dataset and include:

📌 Mean – The average of all values.
📌 Median – The middle value when data is arranged in order.
📌 Mode – The most frequently occurring value in the dataset.

⚠️ Caution: While the concepts are explained simply, the answers in the presentation might contain errors. Please check and verify carefully.

1. Mr. Gaurav S. Patil,
M. Pharm (Pharmaceutics)
[email protected]
Measure of Central Tendency:
Mean, Median and Mode-Pharmaceutical Examples
BP 801T: Biostatistics and Research Methodology
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2. Contents
•Measure of central tendency:
•Mean
•Median
•Mode: Pharmaceutical Examples
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3. Measure of central tendency
•Centraltendencyreferstoourintuitionthatthereisacenter
aroundwhichallthesescoresvary.
•Ameasureofcentraltendencyisasinglevaluethatattemptsto
describeasetofdatabyidentifyingthecentralpositionwithin
thatsetofdata.
•Thethreebranchesofcentraltendencyare:
1.Themean,
2.Themedian,and
3.Themode
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4. Measure of central tendency
•Objectivesofmeasureofcentraltendency:
1.Toobtainsinglevaluethatdescribecharacteristicsof
whole/entiregroupofdata.
2.Tocompare:tofacilitatethecomparisonb/wdifferentgroups.
3.Tosummarize:Tocondenselargedataintosinglevalue,making
iteasiertointerpret.
4.Inference:todrawconclusionsaboutpopulationbasedon
characteristicsofsample.
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5. Measure of central tendency
•Types/classificationofmeasureofcentraltendency:
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6. Measure of central tendency (Mean)
1.TheMean:
•TheMeanItistheaverageofthedataorthesumofallvaluesofaset
ofobservationsdividedbythenumberoftheseobservations.
•Themean(oraverage)isthemostpopularandwellknownmeasureof
centraltendencyItcanbeusedwithbothdiscreteandcontinuous
data.
•Calculatedbythisequation:
Mean of sample = =
Meanofpopulation(mu)=
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7. Measure of central tendency
Propertiesofthemean
AdvantagesofMean:
•Uniqueness:Foragivensetofdata,thereisoneandonlyonemean,
providingasinglevaluerepresentation.
•Simpletocompute:Calculatingthemeanisstraightforwardandinvolves
summingallvaluesanddividingbythetotalcount.
DisadvantagesofMean:
•InfluenceofExtremeValues:Extremevalues,whetherveryhighorvery
low,cansignificantlyimpactthemean,potentiallyskewingtheaverage
awayfromthecentraltendencyofthedata.
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8. Measure of central tendency
ExamplesoftheMean
•Problem:Apharmaceuticalcompanywantstodeterminetheaverage
dosageofanewmedicationforaspecificpopulationgroup.The
dosagesadministeredto10patientsareasfollows:25mg,30mg,35
mg,40mg,45mg,50mg,55mg,60mg,65mg,and70mg.
•Solution:
Formula:Mean= (Sumofalldosages)/(Numberofpatients)
•Calculation:
Mean=(25+30+35+40+45+50+55+60+65+70)/10
•Answer:Mean=(485)/10=48.5mg
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9. Measure of central tendency
Problem:Aclinicaltrialisconductedtoassesstheaveragereactiontime
ofpatientstoanewlydevelopeddrug.Thereactiontimes(inseconds)
of15participantsarerecordedasfollows:5.2,5.5,6.1,6.3,6.5,6.7,6.8,
7.0,7.2,7.5,7.8,8.1,8.3,8.5,and8.7.
•Solution:
•Formula:
Mean=(Sumofallreactiontimes)/(Numberofparticipants)
•Calculation:Mean=(5.2+5.5+6.1+6.3+6.5+6.7+6.8+7.0+7.2+
7.5+7.8+8.1+8.3+8.5+8.7)/15
•Answer:Mean=(98.5)/15=6.5667seconds
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10. Measure of central tendency
Theweightedmean:
•Theindividualvaluesinthesetareweightedbytheirrespective
frequencies.
•AweightedmeanisakindofaverageInsteadofeachdatapoint
contributingequallytothefinalmean,somedatapoints
contributemoreweightthanothers. Or
•Where,xi=individualvalue;wi=weightcorrespondingtoeachvalue
•Itcanbeexpressedasthesumofthemeanofeachgroupmultipliedby
itsrespectiveweight(thenineachgroup)dividedbythesumofthe
weight(N).
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11. Measure of central tendency
Exampleofweightedmeanforungroupeddata:
•Apharmaceuticalcompanyproducestabletswithvaryingconcentrationsof
activeingredients.Theconcentrations(inmg/tablet)andthecorresponding
numberoftabletsproducedateachconcentrationlevelareasfollows:
•Findtheweightedmeanconcentrationoftheactiveingredientpertablet.
Concentration
(mg/tablet)
Number of tablets
5 100
10 150
15 200
20 120
25 50
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12. Measure of central tendency
Solution:
Weightedmean:
(5×100)+(10×150)+(15×200)+(20×120)
=
100+150+200+120
(7400)
=
570
= 12.98
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13. Measure of central tendency
Exampleofweightedmeanforgroupeddata:
Problem:Apharmaceuticalcompanyisconductingastudyonthe
distributionofbloodpressurelevelsamongagroupofpatients.The
bloodpressurereadings(inmmHg)aregroupedintointervals,andthe
frequencyofpatientswithineachintervalisrecorded.
BloodPressure
Range (mm/Hg)
Frequencyof patient
120-130 15
130-140 25
140-150 30
150-160 20
160-170 10
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14. Measure of central tendency
Solution:Tofindthemeanbloodpressurelevel,wefirstneedtofind
themidpointofeachinterval.Then,wecalculatethesumofthe
productsofthemidpointandfrequency,dividedbythetotalfrequency.
So,
120+130/2=125...soon
Then,
=14350/100
=143.5mmHg
Blood
Pressure
Range
(mm/Hg)
Mid point
value
(Xi)
Frequency
of patient
(Wi)
Mid pointX
Frequency
120-130 125 15 1875
130-140 135 25 3375
140-150 145 30 4350
150-160 155 20 3100
160-170 165 10 1650
Wi=100 Wi.Xi=14350
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15. Measure of central tendency (Median)
Median:Themedianofadatasetisthevaluethatliesexactlyinthe
middle.
Tocalculatethemedian:
•Createanorderedarrayofvalues.
•Locatethepositionofthemedianwhichdependsonthenumberof
observationsandasfollows:
•Foroddnumberofobservations:(n+1/2)
•Forevennumberofobservations:Twopositions;(n/2)+(n/2)+1
•Thevalueofthemedianwillbethevalueinthemiddleforodd
numberandtheaverageofthetwovaluesforevennumbers.
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16. Measure of central tendency
Properties median:
Advantagesofmedian:
•Itisasinglevalue,simple,easytocomputeeasytounderstand,
unaffectedbyextremevalues.
Disadvantagesofmean:
•Itprovidesnoinformationaboutallvalues(observations).
•Itislessamenablethanthemeantotestsofstatisticalsignificance.
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17. Measure of central tendency
•ComputingMedianforungroupeddata
•Whenthescoresareungrouped,thesearearrangedinascendingor
descendingorder.Mediancanbefoundbylocatingthecentralobservationor
valueinthearrangedseries.
•Thecentralvaluemaybelocatedfromeitherendoftheseriesarrangedin
ascendingordescendingorder
•ComputingMedianforGroupedData
•Whenthescoresaregrouped,wehavetofindthevalueofthepointwherean
individualorobservationiscentrallylocatedinthegroup.
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18. Measure of central tendency
Examples for median of ungrouped data:
Problem: A set of drug prices (in dollars) for a certain medication: 10,
15, 20, 25, 30, 35, 40 find the median of this ungrouped data.
•Solution: arrange the data in ascending order: 10, 15, 20, 25, 30, 35, 40
•According to the formula for odd numbers: (n+1)/2= (7+1)/2=4
•As the 4
th
(fourth) value of arranged dataset is 25, then median of this
dataset would be 25.
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19. Measure of central tendency
Example of median for grouped data:
A set of drug prices for another medication but grouped into intervals,
along with their respective frequencies:
Solution: Here as data is given in ungrouped form we first need to
calculate the cumulative frequency of the given data..
Pricerange Frequency(F)
10-20 4
20-30 6
30-40 5
40-50 3
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20. Measure of central tendency
So,
Median falls in the 10th position because 10 is half of the total frequency (n = 20).
Since the median falls within the interval 20-30, and we have the lower limit (L) of the
interval, cumulative frequency of the preceding interval (cf), frequency of the interval
containing the median (f), and the interval width (h), we can use the following formula
to find the median:
FORMULA: Median = L + ((n/2 -cf) / f) * h
Pricerange Frequency(F) CumulativeFrequency
(CF)
10-20 4 4
20-30 6 10
30-40 5 15
40-50 5 18
F=20 CF=47
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21. Measure of central tendency
•Where: L = Lower limit of the median interval (20)
n = Total frequency (20)
cf= Cumulative frequency of the preceding interval (4)
f = Frequency of the interval containing the median (6)
h = Interval width (10)
Plugging in the values:
•Median = 20 + ((10 -4) / 6) * 10
= 20 + (6/6) * 10
= 20 + 10
= 30
So, the median price for this grouped data is $30.
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22. Measure of central tendency
•Example for median of ungrouped data (even number):
•Problem: Suppose we have the following data representing the prices of a
certain medication: 10 , 15 , 20 , 25 , 30 , 35 , 40 , 45 10,15,20,25,30,35,40,45
•Solution: To find the median of this ungrouped data, first, we arrange the data
in ascending order: 10,15,20,25,30,35,40,45
•Since there are 8 data points, the median will be the average of the two middle
values, which are the 4th and 5th values:
•So, Median: 25+30/2= 55/2= 27.5
•So, the median price for this ungrouped data with an even number of
observations is $27.5.
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23. Measure of central tendency (Mode)
The Mode:
It is the value which occurs most frequently in the data.
Data distribution with one mode is called unimodal.
If all values are different there is no mode or non-modal.
Sometimes, there are more than one mode: two modes is called
bimodal; more than two is called multimodal distribution.
Normally, the mode is used for categorical data where we wish to know
which is the most common category.
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24. Measure of central tendency
Properties of Mode:
Advantage of mode:
•Sometimes gives a clue about the etiologyof the disease.
Disadvantages:
•Withsmallnumberofobservations,theremaybenomode
•Itislessamenabletotestsofstatisticalsignificance.Otherpropertiesof
mode:
•Sometimes,itisnotunique
•Itmaybeusedfordescribingqualitativedata
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25. Measure of central tendency
ExamplesofModeUngroupedData:
Problem:Adatasetrepresentingthenumberoftimesdifferent
medicationswereprescribedbyadoctorinamonth:10,15,20,20,
25,30,30,30,35
Solution:Inthisdataset,thevalue30appearsmostfrequently(3
times),makingitthemode.So,themodeforthisdatasetis30,
indicatingthatthemedicationprescribed30timesisthemostcommon.
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26. Measure of central tendency
ExamplesofModeGroupedData:
Example:
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27. Measure of central tendency
ExamplesofModeGroupedData:
Formula:
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28. References:
1.KothariCR,ResearchMethodology:MethodsandTechniques,2nd
Edition,NewAgeInternational(P)Ltd,NewDelhi.
2.MalhotraNK,BirksDF,MarketingResearchanAppliedApproach,4th
Edition,PrenticeHall,NewDelhi.
3.A.M.Goon,M.K.GuptaandB.Dasgupta,FundamentalsofStatistics
Vol.1,2008,WorldPressOrganization(P)Ltd,India.
4.A.K.Sharma,TextBookofElementaryStatistics,2005,Discovery
PublishingHouse,NewDelhi
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