Chapter 2: Descriptive Statistics

Transcription

1 Chapte : Decptve Stattc Peequte: Chapte. Revew of Uvaate Stattc The cetal teecy of a oe o le yetc tbuto of a et of teval, o hghe, cale coe, ofte uaze by the athetc ea, whch efe a We ca ue the ea to ceate a evato coe,. (.), (.) o ae becaue t quatfe the evato of the coe fo the ea. Devato ofte eaue by quag, ce t equate egatve a potve evato. The u of quae evato, uually jut calle the u of quae, gve by a ( ). o (.3) Aothe etho of calculatg the u of quae wa fequetly ue ug the ea that pecee copute whe tuet woul wok wth calculatg ache, a. (.4) Regale whethe oe ue Equato (.3) o Equato (.4), the aout of evato that et aou the ea a et of coe ca be aveage ug the taa evato, o t quae, the vaace. The vaace jut a wth beg the potve quae oot of. We ca take the evato coe a taaze the, ceatg, well; taaze coe: z. (.5) Net, we efe a vey potat cocept, that of the covaace of two vaable, th cae a y. The covaace betwee a y ay be wtte Cov(, y). We have 4 Chapte

2 y y y, y (.6) whee the ae the evato coe fo the vaable, a the y ae efe aalogouly fo y. Note that wth a lttle eatc gaeahp, we ca ay that the vaace the covaace of a vaable wth telf. The pouct uually calle a co pouct. y. Mat Epeo fo Decptve Stattc I th ecto we wll etu to ou ata at,, wth obevato a vaable, { j }. We ow efe the ea vecto, uch that [ ] (.7) [ ]. You ght ote that hee we ae begg to ee oe of the avatage of at otato. Fo eaple, look at the eco le of the above equato. The pece ' epee the opeato of ag each of the colu of the at a puttg the a ow vecto. How ay oe ybol woul t take to epe th ug cala otato ug the uato opeato Σ? The ea vecto ca the be ue to ceate the evato coe at, a below. Decptve Stattc 5

3 D [ ] (.8) We woul ay of the D at that t colu-cetee, a we have ue the colu ea to cete each colu aou zeo. Now let ecoe the at '. Th at kow a the aw, o ucoecte, u of quae a co pouct at. Ofte the latte pat of th ae abbevate SSCP. We wll ue the ybol fo the aw SSCP at:. (.9) I ato, we have ee th at epee ow by ow a colu by colu Equato (.6) a (.7). The ucoecte SSCP at ca be coecte fo the ea of each vaable. Of coue, t the calle the coecte SSCP at at that pot: A DD (.) ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) A (.) ( )( ) ( )( ) ( ) Note that Equato (.) aalogou to the clac tateet of the u of quae Equato (.3) whle the eco veo Equato (.) eeble the ha calculato foula fou Equato (.4). The coecto fo the ea the foula fo the coecte SSCP at A ca be epee a vaety of othe way: 6 Chapte

4 Decptve Stattc 7 ( )( ) ( )( ). ) ( ) ( A Now, we coe to oe of the ot potat atce all of tattc, aely the vaace-covaace at, ofte jut calle the vaace at. It ceate by ultplyg the cala /(-) te A,. e. A S (.) Th the ubae foula fo S. Fo te to te we ght have occao to ee the au lkelhoo foula whch ue tea of -. The covaace at a yetc at, quae, wth a ay ow (a colu) a thee ae vaable. We ca thk of t a uazg the elatohp betwee the vaable. A uch, we ut eebe that the covaace betwee vaable a vaable the ae a the covaace betwee vaable a vaable. The at S ha ) ( + uque eleet a ) ( uque off-agoal eleet (of coue thee ae agoal eleet). We houl alo pot out that ) ( the ube of thg take two at a te. Pevouly we ha ea-cetee ug t colu ea to ceate the at D of evato coe. Now we wll futhe taaze ou vaable by ceatg Z coe. Defe Δ a the at cotg of agoal eleet of S. We efe the fucto Dag( ) fo th pupoe: ) Dag( S Δ (.3) Net, we ee to vet the Δ at, a take the quae oot of the agoal eleet. We ca ue the followg otato th cae:

5 8 Chapte / / / / Δ (.4) The oto of takg the quae oot oe ot eactly geealze to atce [ee Equato (3.38)]. Howeve, wth a agoal at, oe ca ceate a uque quae oot by takg the quae oot of all the agoal eleet. Wth o-agoal atce thee o uque way to ecopoe a at to two etcal copoet. I ay cae, the at Δ -/ wll ow pove ueful to u ceatg Z coe. Whe you potultply a at by a agoal at, you opeate o the colu of the peultplyg at. That what we wll o to D: / / / / DΔ Z (.5) whch ceate a at full of z coe. Note that jut a potultplcato by a agoal at opeate o the colu of the peultplyg at, peultplyg by a agoal at opeate o the ow of the potultplyg at. Now we ae eay to ceate the at of coelato, R. The coelato at the covaace at of the z coe, / / SΔ Δ Z Z R (.6)

6 Decptve Stattc 9 Sce the coelato of a y the ae a the coelato betwee y a, R, lke S, a yetc at. A uch we wll have occao to wte t lke R leavg off the uppe tagula pat. We ca alo o th fo S.