【正文】 ???S 62 Calculating S using the rawscore formula ? ?NNXXS? ???22To calculate ΣX2 you square all the scores first and then sum them To calculate (ΣX)2 you sum all the scores first and then square them 63 Estimating the population standard deviation from a sample ? S, the sample standard, is usually a little smaller than the population standard deviation. Why? ? The sample mean minimizes the sum of squared deviations (SS). Therefore, if the sample mean differs at all from the population mean, then the SS from the sample will be an understimate of the SS from the population ? Therefore, statisticians alter the formula of the sample standard deviation by subtracting 1 from N 64 Population and sample variance and standard deviation ? When we have data from the entire population we use ? to pute ?X using the same formulas ? We usually need to estimate ? Variance and standard deviations of the sample are biased estimates of the population 65 Formulas for shat (estimate) ? ?1?2??? ?NXXsDefinitional formula: ? ?1?22???? ?NNXXsRawscore formula: 66 The Estimate of the Variance The standard deviation squares, or the number that you took the square root of to get the standard deviation ? ?1?22??? ?NXXs? ?NXX? ?? 22?The variance is not a very useful descriptive statistic, but it is very important value you will use in other techniques (., the analysis of variance or ANOVA) 67 For a standard normal distribution ? Sample mean is a good estimate of population mean ? The estimate of the population variance and standard deviation tells us how spread out the scores are ? 68% of the scores are within +1 and –1 SX 68 Coefficient of variation 1 0 0.. ????vc69 ? Symmetric – Left tail is the mirror image of the right tail – Examples: heights and weights of people Skewness Symmetrical distribution Relative Frequency .05 .10 .15 .20 .25 .30 .35 0 70 Skewness Asymmetrical distribution ? Moderately Skewed Left – A longer tail to the left – Example: exam scores Relative Frequency .05 .10 .15 .20 .25 .30 .35 0 71 Skewness Asymmetrical distribution ? Ine ? Populations of countries V a l u eF r e q u e n c y72 Skewness ? ?? ?sm ed i a nm ea nsem ea nnnxxxxniinii/)(3/)m o d(2)1(2/312/313??????????????3N1i3is)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。 INPUT STATE $ POP 。1 97 0 CE NS US P OP UL AT IO N IN M IL LI ON S39。 CARDS 。 VAR POP。 run 。 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(guó)慶