Central tendency. mean for metric data. The mean. "I say what I means and I means what I say!."

Transcription

1 Central tendency "I say hat I means and I means hat I say!." Popeye Normal dstrbuton vdeo clp To ve an unedted verson vst: mean for metrc data mportant propertes 1) sum of devatons from the mean 0 ) sum of + devatons sum of - devatons advantages 1) more stable than other measures ) other mportant statstcs can be derved usng t Techncally t s called the arthmetc mean n x x 1 n The mean varance and standard devaton problems a) fractonal values b) cannot be computed f data s open ended c) strongly affected by extreme cases 1

2 Grouped data mean for grouped data xf x N t The eghted mean If the eghts are all equal then t s the same as the arthmetc mean x x eght assocated th th case eghts compensate for the hgher chances of selectng some cases than others Why use t? Each ndvdual data value mght actually represent a value that s used by multple people n your sample. The eght, then, s the number of people assocated th that partcular value. Your sample mght delberately over represent or under represent certan segments of the populaton. To restore balance, you ould place less eght on the over represented segments of the populaton and greater eght on the under represented segments of the populaton. dchotomous data Some values n your data sample mght be knon to be more varable (less precse) than other values. You ould place greater eght on those data values knon to have greater precson. mean for dchotomous data x p here p s the proporton of successes or cases coded 1

3 Sum of squares these measures of central tendency tell us nothng about the varablty n the data or the dsperson one ay to do ths s compare the values th the mean value the smplest ay s to subtract the mean from each value to see f t s hgher or loer f you do ths you get both + and - values f e summed them to get a sort of ndex e ould get 0 as a total, to get around ths e square the dfferences x x ths knon as the sum of squares Sum of squares or the total squared varaton about the mean from ths e can derve the varance and the standard devaton varance s the sum of the squared devatons from the mean dvded by N for the populaton and n-1 for a sample remember that sample statstcs are estmates of the populaton statstcs the sample uses n-1 because t has been shon that the use of N for a sample results n an underestmaton of the populaton varance Varance σ s N ( x x) 1 N N ( x x) 1 n 1 Standard devaton a short cut formula for the sample varance s s n n n x ( x) 1 1 nn ( 1) standard devaton s s Grouped data varance varance can also be calculated for grouped data x M f x f s ( ) N N here f frequency of classes Mgrouped mean M a large standard devaton means a large varablty n the data 3

4 A B FOR a ΣX ΣX (ΣX ) x 31/ s 6(195)-961/6(6-1)6.96 s.639 for b x 51.6 s696.6 s6.39 problem th varance and standard devaton s that for the purpose of comparson, they are senstve to the magntude of the data for example n the prevous data the varance and standard devaton of b as 10 tmes that of a to compare a and b e need to standardze s coeffcent of varaton cv x for a and b the coeffcent of varaton s.639/ or 6.39/ Measure of Spread Characterstcs of s and s Alays postve (hy) Related to mean; so can only use th mean Lke mean, large outlers exaggerate standard devaton Normal curve Specal curvature of normal curve Can be fully descrbed by mean and Standard devaton Mean tells here curve centered on number lne Standard devaton tells ho steep Normal curve Specal curvature of normal curve Can be fully descrbed by mean and Standard devaton Alays follos rule 68% of all observatons thn 1 SD of mean 95% of all observatons thn SD s of mean 99.7% of observatons thn 3 SD s of mean Normal curve formula Note you only need to kno the mean and the varance to create the curve 4

5 Normal curve rule Normal curve rule 1 SD SD % of observatons % of area under curve Value of observaton % of observatons % of area under curve Value of observaton % of observatons Normal curve rule % of area under curve 3 SD Value of observaton Normal dstrbuton For normal dstrbuton the mean s the most effcent and therefore the least subject to sample fluctuatons of all measures of central tendency. The sum of squared devatons of scores from ther mean s loer than ther squared devatons from any other number. dstrbuton statstcs for spatal dstrbutons the bvarate mean n geography the centre of an area may be of nterest, can calculate the eghted b-varate mean centre or the eghted centrod x y x y 5

6 The spatal mean? Eucldean medan Central locaton that mnmzes the unsquared dstances rather the squared ones It s methodcally complex and has to be solved teratvely ( X, Y ) mn ( X X ) + ( Y Y ) e e e e Weghted eucldean medan ( X, Y ) mn f ( X X ) + ( Y Y ) e e e e 6

7 Weghted Eucldean medan Has mportant applcatons n geography Weber locaton problem Used n publc and prvate faclty algorthms Urban fre staton Store ste for clothng store Can be extended to multple locatons to solved at one tme Neghbourhood health centers standard dstance dsperson has t counterpart n bvarate descrptve statstcs because dstances are devatons n the geographc sense, t s defned as the equvalent of a standard devaton SD ( x x) ( y y) + n 1 n 1 Standard ellpse for eghted observatons f you ant to take possblty of an ellpse rather than a crcle then e can calculate standard dstance separately for X and Y SD x x x ( ) n 1 SD ( x x) ( y y) + SD y y y ( ) n 1 ths s far too tedous to do by hand so e ould have to use a computer program 7

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