STATISTICS 13. Lecture 5 Apr 7, 2010

Transcription

1 STATISTICS 13 Leture 5 Apr 7, 010

2 Revew Shape of the data -Bell shaped -Skewed -Bmodal Measures of eter Arthmet Mea Meda Mode Effets of outlers ad skewess

3 Measures of Varablt A quattatve measure that desrbes the spread or dsperso of the data alog the horzotal as of the data dstrbuto Data sets ma have the same eter (e.g., mea), but look dfferet beause the wa the umbers spread out from the eter

4 Measures of Varablt (Cot.) Eample: two data sets -data set oe: 1,,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,9 -data set two: 3,4,4,5,5,5,6,6,7 -

5 Measures of Varablt: (I) Rage Rage of a sample of measuremets s the dfferee betwee the mamum ad the mmum : R Ma M Eample : dal umber of phoe alls fve das: 5,, 14, 3, 6 ; Drawbak : depeds o ol two values amog the measuremets, ver hghl affeted b outlers

6 Measures of Varablt: (II) Varae Varae of a sample of measuremets s the sum of squared devato from the sample mea, dvded b (-1) ; It measures how far awa the measuremets are from ther mea. Eample : phoe alls -data: 5,, 14, 3, 6 ;5 -mea -devatos : -squared devatos : -varae: s ( ) 1

7 Populato vs. Sample Varae of a populato of N subjets s the averaged squared varato from the populato mea μ; σ usuall s ot alulable due to the lak of data of the whole populato µ σ N ( µ N ) Varae of a sample of measuremets s the sum of squared devato from the sample mea, dvded b (-1) s 1 1( 1 )

8 Wh Dvded b -1? The sample varae s s used to estmate the populato varae σ The dvsor (-1) stead of the defto of sample varae gves a more aurate estmate, partularl whe s small

9 Stadard Devato Varae s a sale whh s the square of the sale of the orgal measuremets Stadard devato, or the postve square root of the varae, measures the varablt the same sale as that of the measuremets Populato stadard devato : σ σ Sample stadard devato : s s

10 Calulatg Sample Varae : Method 1 ( ) Appl the defto s ( ) Sum

11 Calulatg Sample Varae : Method Sum s Appl the formula ( 1 )

12 Some Propertes Stadard devato ad varae are alwas oegatve, ad zero f ad ol f all measuremets are equal If eah measuremet s multpled b a ostat, the mea of the trasformed measuremets s tmes the orgal mea; whle stadard devato of the trasformed measuremets s tmes the orgal stadard devato If a ostat s added to eah measuremet, the mea of the trasformed measuremets s plus the orgal mea; whle the stadard devato of the trasformed measuremets equals the stadard devato of the orgal measuremets What happes to meda, IQR ad rage?

13 Propertes (Cot.) 1 ) ( s * * 1,..., 1,..., 1 Formula for sample stadard devato: for measuremets, the stadard devato s defed as : If,the ; ad If, the ; ad s s * 1 ) * * ( 1 ) ( + +,..., 1 1 s s )} ( ) {( 1 ) ( + + * *

14 Eample S X Y * X Y X + * *4 +4 * * *61 6+8

15 Eample : Temperature Do ou kow ths formula? C(F-3) 5/9 The stadard devato of the mamum temperature o the das September s 5.4 degrees Fahrehet. What s the stadard devato of the mamum temperature f the measuremets were take Cetgrade? Aswer : If s a measuremet Fahrehet, the ( 3) 5/9 5/9-35/9 s the orrespodg measuremet Cetgrade. So, stadard devato s

16 Empral Rule for Bell Shaped Dstrbutos Gve a dstrbuto of measuremets that s appromatel bell-shaped: The terval µ ± σ otas appromatel 68% of the measuremets. The terval µ ± σ otas appromatel 95% of the measuremets. The terval µ ± 3σ otas appromatel 99.7% of the measuremets.

17 Eample : Age of CEOs Hstogram of Age Peret 0 0 s Age Nearl bell-shaped

18 Empral Comparso : Age of CEOs k ± ks Iterval Proporto Iterval Empral Rule ± to /60 (.7) ± * to /60 (.93) ± 3* to /60 (1.00).997 The data dstrbuto satsfes the predtos b Empral rule losel, se t s earl bell-shaped

19 What does Empral Rule Tell Us? If data s about Bell shaped, the most of the measuremets le wth two stadard devatos from ther mea Appromato of stadard devato: whe data s about bell-shaped, the s R 4

20 Thebsheff s Theorem Gve measuremets, ad a teger k greater tha 1, the proporto of measuremets that are wth k stadard devatos from the mea of the measuremets s at least 1 1/k Apples both for a sample ad a populato ad regardless of the shape of the dstrbuto Eample: k ; at least 1-1/43/4 proporto are wth (mea s, mea + s)

21 Eample I 1999, the average tme spet ole was about 14 hours per perso per ear. Suppose that the stadard devato of the dstrbuto of -- the umber of hours per perso per ear spet ole, was 0 hours. Questos: - Wh do ou suppose the earl average for the etre U.S. populato was so low 1999? -Do ou thk the dstrbuto s relatvel bellshaped, skewed rght, or skewed left? -Wth what lmts would ou epet at least 3/4 of the measuremets to le?

22 Eample (Cot.) Aswers: -Although some people spet a large umber of hours per ear ole, a large majort of the populato ever used the Iteret Hee, there were ma zero values the populato. The dstrbuto s skewed rght. -Se the dstrbuto s ot bell-shaped, ou must use Thebsheff s Theorem to desrbe the data. -