【正文】 3is)1(N)(s ke w n e s s???????73 Skewness V a l u eF r e q u e n c y74 Skewed Right Positive Skewness 0 100 200 300 40001020N u m b e r o f M u si c C D sFrequencyN u m b e r o f M u si c C D s o f S p r i n g 1 9 9 8 S t a t 2 5 0 S t u d e n t s75 Kurtosis ? Measures of Kurtosis – Kurtosis is a measure of the flatness or peakedness of a Distribution ? Normal Kurtosis Mesokurtic ? Flat Kurtosis Platokurtic ? Peaked Kurtosis Leptokurtic – A Measure of Kurtosis based on the 4th moment about the Mean 76 Kurtosis If less then 0 = Platokurtic More than 0 = Leptokurtic If 0 then = Mesokurtic 4N1i4is)1(N)(k u r t o s i s???????77 Kurtosis V a l u eF r e q u e n c yk 3k = 3k 378 Describing data Moment Nonmean based measure Center Mean Mode, median Spread Variance (standard deviation) Range, Interquartile range Skew Skewness Peaked Kurtosis 79 SAS sample OPTIONS LS= 75 NODATE。 DATA STATEPOP。 INPUT STATE $ POP @@。 LABEL POP = 39。1 97 0 CE NS US P OP UL AT IO N IN M IL LI ON S39。 。 CARDS 。 AL A 3. 44 AL AS KA 0 . 30 A RI Z 1. 77 AR K 1. 92 CA LI F 19 .9 5 CO LO 2 .2 1 CO NN 3 .0 3 DE L 0. 55 FL A 6. 7 9 GA 4 .5 9 HA W 0. 77 ID AH O 0. 7 1 IL L 11 .0 1 IN D 5. 1 9 IO WA 2 .8 3 KA N 2. 25 KY 3 .2 2 LA 3 .6 4 ME 0 .9 9 MD 3 .9 2 MA SS 5 .6 9 MI CH 8 .8 8 MI NN 3 .8 1 MI SS 2 .2 2 MO 4 .6 8 MO NT 0 .6 9 NE B 1. 48 NE V 0. 49 NH 0 .7 4 NJ 7 .1 7 NM 1 .0 2 NY 1 8. 24 NC 5 .0 8 ND 0 .6 2 OH IO 1 0. 65 OK LA 2 .5 6 OR E 2. 09 PA 1 1. 79 RI 0 .9 5 SC 2 .5 9 SD 0 .6 7 TE NN 3 .9 2 TE XA S 11 .2 U TA H 1. 06 VT 0 .4 4 VA 4 .6 5 WA SH 3 .4 1 W. V A 1. 74 WI S 4. 42 WY O 0. 33 PROC UNIVARIATE F RE Q PLOT NORMAL 。 VAR POP。 ID S TA TE 。 run 。 80 T h e S A S S y s t e m 1 T h e U N I V A R I A T E P r o c e d u r e V a r i a b l e : P O P ( 1 9 7 0 C E N S U S P O P U L A T I O N I N M I L L I O N S ) M o m e n t s N 5 0 S u m W e i g h t s 5 0 M e a n 4 . 0 4 7 2 S u m O b s e r v a t i o n s 2 0 2 . 3 6 S t d D e v i a t i o n 4 . 3 2 9 3 1 8 6 7 V a r i a n c e 1 8 . 7 4 3 0 0 0 2 S k e w n e s s 2 . 0 5 5 2 1 8 3 9 K u r t o s i s 4 . 5 4 5 6 1 6 7 9 U n c o r r e c t e d S S 1 7 3 7 . 3 9 8 4 C o r r e c t e d S S 9 1 8 . 4 0 7 0 0 8 C o e f f V a r i a t i o n 1 0 6 . 9 7 0 7 1 2 S t d E r r o r M e a n 0 . 6 1 2 2 5 8 1 2 SAS results 81 Basic statistics B a s i c S t a t i s t i c a l M e a s u r e s L o c a t i o n V a r i a b i l i t y M e a n 4 . 0 4 7 2 0 0 S t d D e v i a t i o n 4 . 3 2 9 3 2 M e d i a n 2 . 7 1 0 0 0 0 V a r i a n c e 1 8 . 7 4 3 0 0 M o d e 3 . 9 2 0 0 0 0 R a n g e 1 9 . 6 5 0 0 0 I n t e r q u a r t i l e R a n g e 3 . 6 9 0 0 0 82 Tests for location T e s t s f o r L o c a t i o n : M u 0 = 0 T e s t S t a t i s t i c p V a l u e S t u d e n t 39。 s t t 6 . 6 1 0 2 8 4 P r | t | . 0 0 0 1 S i g n M 2 5 P r = | M | . 0 0 0 1 S i g n e d R a n k S 6 3 7 . 5 P r = | S | . 0 0 0 1 83 Normality T e s t s f o r N o r m a l i t y T e s t S t a t i s t i c p V a l u e S h a p i r o W i l k W 0 . 7 6 2 9 6 8 P r W 0 . 0 0 0 1 K o l m o g o r o v S m i r n o v D 0 . 2 0 1 8 9 5 P r D 0 . 0 1 0 0 C r a m e r v o n M i s e s W S q 0 . 6 3 0 3 9 4 P r W S q 0 . 0 0 5 0 A n d e r s o n D a r l i n g A S q 3 . 6 1 7 6 5 4 P r A S q 0 . 0 0 5 0 84 Quantiles Q u a n t i l e s ( D e f i n i t i o n 5 ) Q u a n t i l e E s t i m a t e 1 0 0 % M a x 1 9 . 9 5 0 9 9 % 1 9 . 9 5 0 9 5 % 1 1 . 7 9 0 9 0 % 1 0 . 8 3 0 7 5 % Q 3 4 . 6 8 0 5 0 % M e d i a n 2 . 7 1 0 2 5 % Q 1 0 . 9 9 0 85 TA 莊國慶