Basic statistics formulas

Transcription

1 Wth complmet of tattcmetor.com, the te for ole tattc help Set De Morga Law Bac tattc formula Meaure of Locato Sample mea (AUB) c A c B c Commutatvty & (A B) c A c U B c A U B B U A ad A B B A Aocatvty (A U B) U C A U (B U C) (A B) C A (B C) Dtrbutvty (A U B) C (A C) U (B C) (A B) U C (A U C) (B U C) Probablty p(a) 1 - p(a c ) p(a U B) p(a) + p(b) - p(a B) If evet A ad B are mutually depedet the p(a B) p(a)p(b) p(a B) p(a B)/p(B) o log a p(b)>0 p(a B) p(a B)p(B) p(b A)p(A) o log a p(a)>0 ad p(b)>0 If {B 1, B...,B k } a et of mutually excuve ad exhautve evet, the p(a) p(a B 1 )p(b 1 )+ p(a B )p(b )+...+p(a B k )p(b k ) x x Meda (for raw data.e lt of umber ugrouped) Lt the umber acedg order. Meda : (+1)/ value f odd; Mea of /th ad (+1)/th value f eve. Meaure of pread Sample varace ( x x ) 1 Sample tadard devato, Rage var ace Larget value mallet value Iterquartle rage (IQR) Upper quartle lower quartle Coeffcet of varato / x d November 009

2 Wth complmet of tattcmetor.com, the te for ole tattc help Expectato, varace, ad covarace If radom varable dcrete: E( ) x p( x ) whch calculated over all poble value of. Let g() deote a fucto of dcrete, the: Expecto rule E( g( )) g( x ) p( x ) Let,, Z deote radom varable; a, b deote cotat E1. E(a) a E. E(a) a E() E3. E(+) E() + E() Varace rule V1. var(a) 0 V. var(a) a var() V3. var( ) var() + var() *Cov(,) Covarace rule C1. Cov(a,) 0 Normal dety fucto The ormal dety fucto (aka ormal dtrbuto) ha parameter: mea ad varace. For the ormal dtrbuto: 90% of data fall wth ± % of data fall wth ± 1.95 Stadardzg (Z-core) x µ z Samplg dtrbuto of ample mea Suppoe (, ) N µ, the the dtrbuto of the ample mea N( µ, ) The tadard error of Th reult true f doe ot follow the ormal dtrbuto but large (ad the the reult follow becaue of the Cetral Lmt Theorem) C. Cov(a,b) ab*cov(,) C3. Cov(+,Z) Cov(,Z) + Cov(,Z) d November 009

3 Wth complmet of tattcmetor.com, the te for ole tattc help Cofdece terval for the mea Parameter Aumpto Formula Mea µ Data ormally dtrbuted or large (>30); x ± z α / kow Data ormally dtrbuted; mall; ukow x ± t α /, 1 Dfferece mea µ µ Cae of depedet dtrbuto Data are ormally dtrbuted;, are kow Varace ukow; Large ample ( x y) ± zα / + ( x y) ± zα / + Data are ormally dtrbuted;, are ukow but 1 1 ( x y ) ± tα /, + p + Where the etmate of the pooled varace ( 1) + ( 1) p + d November 009

4 Wth complmet of tattcmetor.com, the te for ole tattc help Hypothe tet for the mea Hypothe Aumpto Tet equato Tetg a gle mea equal a value a Ho : µ a Data ormally dtrbuted, or large ample; kow Data ormally dtrbuted ukow / x a / Z-table for crtcal value x a t-table wth df -1 for crtcal value Tetg the dfferece betwee mea equal a umber a (whch clude the cae of a0 whch a tet for o dfferece betwee mea) H : µ µ a o Cae of depedet dtrbuto Data ormally dtrbuted, or large ample; Idepedet ample;, are kow Data ormally dtrbuted, or large ample; Idepedet ample;, are ukow Data are ormally dtrbuted;, are ukow but + Z-table for crtcal value + Z-table for crtcal value 1 1 p + Where the etmate of the pooled varace ( 1) + ( 1) p + d November 009

5 Wth complmet of tattcmetor.com, the te for ole tattc help Covarace ad correlato Parameter Populato formula Sample formula Covarace betwee varable ad Pearo correlato Cov(, ) COV(,) E() E()E() ( x x)( y y) 1 xy xy Smple lear regreo Model: for 1,,, y α + β x + u OLS etmator Itercept α y β x x ( ) var( α) x x Slope β var( β ) xy xy x x x x Etmator for varace of error term u ( y y ) d November 009