Day3Measuresofcentraltendency

Calculatingmean,medianandmodeandidentifyingwheneach isagoodchoice.

Rahim 16 28 32 28 26 31 Total161

Johann 34 30 24 26 29 26 Total169

Two car salesmen are competing for a mid-year bonus. The owner of the dealership wants to assess the better competitor. Who is the better candidate?

Monthly Sales

MeasuresofcentraltendencyThereare3waystofindthecommontrend(orcentraltendency)forasetofdata.

1) Mean (mostcommonlyreferredtoastheaverage)Tofindthemean,addupallofthenumbersinyourlistanddividebythenumberofnumbers.

Example1

Jesara isbuyingahomethatwillrequireamortgage.Thebankwantstoknowhermonthlysalary.Sheworksoncommission,soshemustcalculateheraveragesalary.Givenherincomeforthefirst6monthsoftheyear,whatisheraveragesalary?Jan--$3675, Feb--$4250,Mar--$3225, Apr--$2985, May--$3650,Jun--$4600.

2)MedianThemedianisthemiddleentryinanorderedlist.Thereareasmanydatapointsaboveitasbelowit.

Tofindthemedian,a)Ifthereisanoddnumberofdatapoints,takethemiddleone

(i.e.ifthereare13numbers,themedianisthevalueofthe7th numberwhentheyarelistedinascendingorder).

b)Ifthereisanevennumberofdatapoints,themedianistheaverageofthemiddletwonumbers.

Example2Findthemedianmarkforeachlistofstudentgrades.

a)62,64,76,89,72,54,93b)56,84,63,67,62,98

3)Mode• Themodeisthemostfrequentnumberinadataset.• Therecanbenomodeaswellasmorethanonemode.

Example3

Findthemode(s)foreachlistofnumbers.

a)5,7,9,8,6,5,4,10b)25,30,32,30,25,29

c)63,57,66,83,79,72,79,69,60,63,79,85,80

Example4

Themodesofthefollowingsetofdataare7and9.Whatmustbethevalueofy?

6,9,3,4,8,0,7,2,9,y

Whentousewhich?

Tips:

Mean—Really goodwhenthedataisfairlyclosetogether. Mostcommonlyused.

Median—Good whenthere isanoutlier(i.e.anumberthat isfarawayfromtheotherswhichwouldskewthemean).

Mode—Goodwhenthevalueofthenumber isthemostimportant information(e.g.shoesize).--Onlychoicewithcategoricaldata.

Measureofspread

Description

• MeasuresofSpread• Calculateandinterpretrangeandstandarddeviationbyhandandwithtechnology

Whatcanyouinfer,justifyandconcludeabouttheJoan’sandTaran’stestsscores?(Hint:Calculatethemean,medianandmodeforeach.Whatdotheytellyou?)Joan’sTests:76,45,83,68,64Taran’s Tests:67,70,70,62,62

Range:thedifferencebetweenthegreatestandsmallestvaluesinasetofdata

Standarddeviation:ameasureofthevariationofmeasurementsaroundtheiraveragevalue.

Mean,medianandmodeareallgoodwaystofindthecentreofyourdata.

Thisinformationismostusefulwhenthesetsofdatabeingcomparedaresimilar.

Itisalsoimportanttofindouthowmuchyourdataisspreadout.Thisgivesalotmoreinsighttodatasetsthatvaryfromeachother.

Consider the following two data sets with identical mean and median values. Why is this information misleading?Set A) 0, 2, 2, 4, 4, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 12, 12, 14, 14, 16

mean= 8med= 8

SetB)4,4,4,6,6,6,8,8,8,10,10,10,12,12,12

mean=8med=8

00.51

1.52

2.53

3.54

4.5

0 2 4 6 8 10 12 14 16

Series1

0

0.5

1

1.5

2

2.5

3

3.5

4 6 8 10 12

Series1

Sol’n: Thisinformationismisleadingbecauseonegraphisbell-shapedandtheotherisuniform,butthecalculationsmakethemappeartobesimilarwhenreallyAandBarespreadoutquitedifferently.Whatissomethingthatcanbedonetofurthercomparethesegraphs?

Lookattherange inthedatasets.

Toby 54 152 180 12 72 126 104 132

Moby 132 104 102 120 86 12 180 96

Twins, Toby and Moby, both work at a local pizza shop. Their manager has decided to give a raise to her best employee. She looks at their data.Number of Pizzas Made per Shift Who is more deserving?

Sol’n: She starts by finding the mean number of pizzas made by each and their range.

Thesestatisticsleavebothemployeesequal.ThemanagernoticesthatMoby’sdatalooksmoreconsistent,butsheneedsprooftosupportherclaim. Shedecidestocalculatethestandarddeviation foreach.

Standard Deviation ( )— best choice for measuring the spread of data

Steps for calculating

1. Find the difference between each value and the mean.

2. Square each difference.

3. Add up all of your answers from Step 2.

4. Divide this sum by the number of numbers (i.e. find the average of the

differences squared).

5. Find the square root your answer

σ

Mathematically:

( )n

xxn

ii∑

=

−= 1

2

σ

Number of Pizzas

x 54

152

180

12

72

126

104

132

Total=

Standard deviation for Toby (by hand):

Number of Pizzas

x 132

104

102

120

86

12

180

96

Total=

Standard deviation for Moby (by hand):

Findtherangeandstandarddeviationofthefollowingsetofnumbers:3,10,8,20,4,4,3,8,8,8,12

Interquartilerange

• Quartilesarethevaluesthatdividealistofnumbersintoquarters.• First putthelistofnumbersinorder• Then cutthelistintofourequalparts• TheQuartilesareatthe"cuts”• Q1isthelowerquartilerange• Q2isthemiddlequartilerangeormedian• Q3istheupperquartilerange• IQR=Q3- Q1

• Example:5,8,4,4,6,3,8FindQ1,Q2,Q3andIQR

• Example:1,3,3,4,5,6,6,7,8,8FindQ1,Q2,Q3andIQR