Arithmetic Mean Suppose there is only a finite number N of items in the system of interest. Then the population arithmetic mean is

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Allison Spencer
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1 Topc : Probablty Theory Module : Descrptve Statstcs Measures of Locato Descrptve statstcs are measures of locato ad shape that perta to probablty dstrbutos The prmary measures of locato are the arthmetc mea (more commoly called the mea) ad the geometrc mea Please ote that there s a populato mea ad a sample mea We usually caot kow the former, but we ca always kow the latter Later, we wll fer the populato mea from the sample mea Rght ow, we wll smply master the basc cocepts Arthmetc Mea Suppose there s oly a fte umber of tems the system of terest The the populato arthmetc mea s x µ = Suppose we have a sample of sze draw from the populato The the sample arthmetc mea s x x = The sample mea x wll be used to draw fereces about the populato mea µ As oted, the populato mea s geerally ukow ITERPRETATIO: The populato arthmetc mea s smply a measure of the ceter of gravty of the populato, ad the sample arthmetc mea s smply a measure of the ceter of gravty of the sample I EXCEL The sample mea s foud va the AVERAGE fucto or va the Descrptve Statstcs tool uder Tools/Data Aalyss/Descrptve Statstcs

2 Geometrc Mea Suppose there s oly a fte umber of tems the system of terest The the populato geometrc mea s gm populato = x = x Suppose we have a sample of sze draw from the populato The the sample geometrc mea s gm sample = x = x ITERPRETATIO: The geometrc mea does ot have a smple terpretato The geometrc mea wll be used face to accout for cotuous compoudg I EXCEL: Excel has a GEOMEA fucto, but does ot have a geometrc mea tool Tools/Data Aalyss/Descrptve Statstcs EXAMPLE Suppose we have the data set {,, 3, 4, 5} The the sample arthmetc mea s x x = = = = 30 ad the sample geometrc mea s gm sample = x = ( 3 4 5) / 5 = (0) / 5 = PROBLEMS: Ch, #4b,

3 Topc : Probablty Theory Module : Descrptve Statstcs Measures of Varato The prmary measure of varato s the varace Aga, suppose there are oly a fte umber of tems the system of terest The the populato varace s ( x µ) σ =, where µ s the populato arthmetc mea Aga, suppose we have a sample of sze draw from the populato The the sample varace s s = ( x x), where x s the sample arthmetc mea The sample varace s wll be used to draw fereces about the populato varace σ As oted, the populato varace s geerally ukow ITERPRETATIOS: The varace s a measure of dsperso about the (arthmetc) mea, ad has may terpretatos It s used as a measure of dversty Bology ad as a measure of rsk Face Reducto a varace, over tme, ca be used as a measure of mprovemet performace (See, eg, Stephe J Gould, Full House) ote that the sample varace has a dfferet dvsor tha the populato varace We wll treat the reaso for ths dfferece later whe we treat ubased estmators I EXCEL: The sample varace s foud va the VAR fucto or va the Descrptve Statstcs tool uder Tools/Data Aalyss/Descrptve Statstcs Varaces are measured square uts Commo sese suggests that dsperso s a measure of dstace from the ceter of the system, ad therefore should be measured uts Thus, we more commoly measure dsperso va the populato stadard devato σ = ( x µ)

4 ad the sample stadard devato s = ( x x) I EXCEL: The sample stadard devato s foud va the STDEV fucto or va the Descrptve Statstcs tool uder Tools/ Data Aalyss/Descrptve Statstcs ITERPRETATIO: The stadard devato s smply a Euclda measure of the dstace from the ceter of a populato or sample to the edge of the populato or sample, respectvely MEASURES OF SHAPE There are two stadard measures of shape The stadard measure of asymmetry s the populato coeffcet of skewess 3 ( x µ ) CS = 3 σ The stadard measure of peakedess s the populato coeffcet of kurtoss 4 ( x µ ) CK = 4 σ I almost all of the aalyss doe ths course, we wll be cocered oly wth probablty dstrbutos whch ca be characterzed by the mea,µ, ad the varace, σ, ad therefore we wll ot ecouter skewess ad kurtoss aga However, you wll ecouter them more advaced courses I EXCEL: Skewess ad kurtoss are foud va the SKEW ad KURT fuctos, respectvely, or va the Descrptve Statstcs tool EXAMPLE Suppose we have the data set {,, 3, 4, 5} Recall that the sample arthmetc mea s 3 The the sample varace s

5 s = (x x ) = ( 3) + ( 3) + (3 3) + (4 3) + (5 3) 5 = 0 4 = 5 ad the sample stadard devato s s = s = 5 = PROBLEMS: Chapter, #0, b,, 3, ad 4

Econometric Methods. Review of Estimation

Econometric Methods. Review of Estimation Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators