基礎生統介紹
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(2) Tabulation of Data 50. (. 244 245 246 247 248 249 250 251 252 253 254 255 256. / // // /// //// ///// //////// /////// ///// /// /// / //. 1 2 2 3 4 6 9 8 6 3 3 1 2 50. • • • •. 1 3 5 8 12 18 27 35 41 44 47 48 50. Tabulation of Data ). 2% 4% 4% 6% 8% 12% 18% 16% 12% 6% 6% 2% 4%. . 2% 6% 10% 16% 24% 36% 54% 70% 82% 88% 94% 96% 100%. 100%. •. (k) • Sturges (1926) formula: K = 1 + 3.322 log10 N •N (d) (R)∕ (K) ( ) . 5. 6. Tabulation of Data 50. 0.5 → 243.5 + 2 = 245.5 (245.5, 247.5)R = (243.5+245.5)/2 = 244.5 h :. —. . vN =50 K = 1 + 3.322 log10 50 = 1 + 3.322 ´1.7 = 6.65 @ 7 (256—244)/7 1.714 P 2 244 o 2 (244,245) (246,247)R 244 243.5 244.5 243.5. (R):. 244-245 246-247 248-249 250-251 252-253 254-255 256-257. (. 243.5-245.5 245.5-247.5 247.5-249.5 249.5-251.5 251.5-253.5 253.5-255.5 255.5-257.5. ). x. f. 244.5 246.5 248.5 250.5 252.5 254.5 256.5. 3 5 10 17 9 4 2 50. 7. cf 3 8 18 35 44 48 50. rf 6% 10% 20% 34% 18% 8% 4%. crf 6% 16% 36% 70% 88% 96% 100%. 100% 8.
(3) 3.1.3 Graph. bar chart. ，. t. (discrete variable). Ø. (bar chart) i (pie chart). R V. R 9. 10. Graph. pie chart Ø. (continuous variable) ＝？ (stem-leaf plot) Tukey(1960) S？ (leaf) R (histogram) R. 11. i. S＝ (stem). s i. n. 12.
(4) ＝？. ＝？. (stem-leaf plot). stem-leaf plot. ）. n. 50 12. n. ------------------------------------------. ＝,. n. 120 98 96 100 128 88 90 108 102 91 n 94 100 80 93 105 100 104 95 106 100 105 106 107 84 106 98 110 112 90 113 96 104 114 105 116 102 113 96 115 124 108 118 104 120 130 119 128 126 133 95 ----------------------------------------. ）. 12. n n. 3 12 18 9 6 2. ＝(stem) 8 9 10 11 12 13. ？. ＝？ ？(leaf) -. 048 001345566688 000022444555666788 023345689 004688 03. 50 13. 14. ＝？. stem-leaf plot. 6 5 4. 5. 8. 1. 4. 6. 3 0. 0. 5. 6 5. ……. 8. histogram. ？(leaf). 2. 20. 5 4. 15. 4 2. 0. 10. 0. 5. 0. 0. 0. 9 ＝(stem). 10 ＝(stem). 5 2. 24 15. 5 4. 24. 5 6. 24. 5 8. 24. 5 0. 25. 5 2. 25. 5 4. 25. 5 6. 25. 5 8. 25 16.
(5) Graph. polygon. ( ). Ø. （. (polygon) ( ) s. polygon). 20 15. （R (cumulative frequency. 0. 10 5. （ R. 0 5 2. 24. 5 4. 24. 5 6. 24. 5 8. 24. 5 0. 25. 5 2. 25. 5 4. 25. 5 6. 25. 5 8. 25. 17. 18. percentile. cumulative frequency polygon 50. k. 40. k. 30. ì ï xæç n´ k +1ö÷ ï è 100 ø Qk = í üï ï 1 ìï x x + ï 2 í æç n´ k ö÷ æç n´ k +1ö÷ ý î îï è 100 ø è 100 ø þï. 20 10 0 244.5. 246.5. 248.5. 250.5. 252.5. 254.5. 256.5 19. i R. n. : n´. k 100 n´. k 100 20.
(6) Q1 , Q2 , Q3. percentile 1 Q1. 25 R. 75. 25%. 2 )M 3. (. 25% Q1. 25% Q2. Q2. k = n´. n. 50 Q3. n. kY. n. k. P 100. , ,. (n X Q =k XQ =. ,P +1 1 ( X k + X k +1 ) 2. 25% Q3 21. v 50. ). (. , 3.2a). 22. cumulative frequency polygon. 25 = 12.5 100 = 249. P = 25%, k = 50 ´ Q1 = X 12 +1 = X 13. 50 = 25 100 1 1 Q2 = ( X 25 + X 26 ) = (250 + 250) = 250 2 2 75 P = 75%, k = 50 ´ = 37.5 100 Q3 = X 37 +1 = X 38 = 252 P = 50%, k = 50 ´. 23. 24.
(7) 5 8C. .B 3 B. Box and whisker plot. c (. (box plot)R —. V. ). Left-Skewed. —. Q1 Median Q3. R. Xsmallest Q1 Median Q3. 4. 6. 8. Symmetric Q1. Median Q3. Right-Skewed Q1 Median Q3. Xlargest. 10. 12 25. E98. DCB scatter plot. ( ). Ø. (scatter plot). R. (cholesterol) (diastolic blood pressure). • 8 f. 1. 2. 3. 4. 5. 6. 7. 8. x. 225 207 270 217 285 274 236 185. y. 76. 80. 90. 74 100 88. 78. 70 27. 28.
(8) 3.2.1. 3.2. Arithmetic Mean. Measures of Central Location. (x. R (arithmetic mean). n. (. i =1. å. R. ). åx. i =1. (median). i. = ( x1 + x 2 + !! + x n ). Summation or sigma. mu). (μ. (mode). (sample point). n. x = å xi n. R. x-bar). N. R. µ = å xi N i =1. 29. 30. sample Arithmetic Mean. Arithmetic Mean. 3.4. i. s. 20. v3.1. x. 20. xi i. xi i. xi i. x = (3264 + !! + 3106) / 20 = 63816 / 20 = 3190.8. xi. 1. 3264. 6. 3323. 11. 2580. 16. 2758. 2. 3260. 7. 3650. 12. 2845. 17. 3248. 3. 3245. 8. 3200. 13. 3585. 18. 3325. 4. 3484. 9. 3030. 14. 2480. 19. 3315. 5. 4146. 10. 2070. 15. 3542. 20. 3106. R , 31. 3087.6. 1200 , 32.
(9) Arithmetic Mean. x = 3190.8. population Arithmetic Mean. x = 3087.6. 123456 6. 3.4. i. xi i. xi i. xi i. xi. 1. 1200 3264. 6. 3323. 11. 2580. 16. 2758. 2. 3260. 7. 3650. 12. 2845. 17. 3248. 3. 3245. 8. 3200. 13. 3585. 18. 3325. 4. 3484. 9. 3030. 14. 2480. 19. 3315. 5. 4146. 10. 2070. 15. 3542. 20. 3106. µ. µ = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5. 33. 34. 3.2.2 Properties of Arithmetic Mean. (C) y=. Properties of Arithmetic Mean. R. n 1 n 1 n 1æ n ö 1æ n ö y i = å ( xi + c) = ç å xi + å c ÷ = ç å xi + nc ÷ = x + c å n i =1 n i =1 n è i =1 n è i =1 i =1 ø ø. i 1 2 3 4. 5 2 10 3 5. 5+2=7 2+2=4 10+2=12 3+2=5 5+2=7. C R. C y=. i 1 2 3 4 35. 1 n 1 n 1 n y i = å cxi = c å xi = cx å n i =1 n i =1 n i =1. 5 2 10 3 5. 5 2=10 2 2=4 10 2=20 3 2=6 5 2=10. 36.
(10) Properties of Arithmetic Mean. Properties of Arithmetic Mean. (deviation): •. Di = xi - x. 0 n. n. å D = å (x i =1. i. i =1. i 1 2 3 4. i. (sum of squares) n. n. n. i =1. i =1. SS = å ( xi - x ) 2. - x ) = å xi - å x = nx - nx = 0. i =1. Di = xi - x. 5 2 10 3 5. 5,2,10,3. 5-5=0 2-5=-3 10-5=5 3-5=-2 0. 5. SS = (5 - 5) 2 + (2 - 5) 2 + (10 - 5) 2 + (3 - 5) 2 = 38. i. a 4. 5. SS = (5 - 4) 2 + (2 - 4) 2 + (10 - 4) 2 + (3 - 4) 2 = 42 > 38 37. 38. 2 :8. &. Properties of Arithmetic Mean. `X = 5. (median). t R. ( ) 2. 3. 5. 10. !+ +. ì x n +1 ïï ( 2 ) Med = í é ù 1 ï êx n + x n ú ïî 2 ë ( 2 ) ( 2 +1) û. a =4 2. 3. 5. 10 39. h. n n. R. 40.
(11) ( Median ) v3.3 f 8,30,6,9,8,3,12,15,18. ( 103). (mode). R xY Y. 3,6,8,8,9,12,15,18,30 •. n=9. n +1 9 +1 = = 5 Þ Med = X (5) = 9 2 2 •. 8. 2B:. &(. o R Y R. n=8. n 1 8+9 = 4, Þ Med = ( X (4) + X (5) ) = = 8.5 2 2 2. v3.4 2(. 100 48). 41. 42. 3.2.5 n n n n n n n n n n. ------------------------0 2 1 15 2 48 3 26 4 6 5 2 6 1 -------------------------. V. (. )R. R R. 43. 44.
(12) r. 3.2.6. 3.2.5 (a). (b). p. n. x. (c). – –. M0. r. – –→. x Med M 0 3.8. 45. 46. n n. x1,x2, …,xn. 1. r. 2. 3. 5. 6. 7. 8. 9. 10. x = (0.8+! +3.35)/10 = 1.71. xg n. 4. 0.80 0.90 1.10 1.15 1.18 1.81 2.03 2.21 2.58 3.35. Tmax. Med = (1.18 + 1.81) / 2 = 1.50. r. xg = anti log(u ) = 10u n. f. 10 ). (. r. u. 1 n u = å log10 X i n i =1 n. ,. i , (Logarithmic Transformation),. Med. M 0 Med x. (Geometric mean). xg. u = (log10 x1 + ! + log10 xn ) / n. a. = (log10 0.8 + ! + log10 3.35) /10 = 0.18775 xg = anti log(u ) = 10u = 100.18775 = 1.54. xg = n x1 x2 ......xn. Med = 1.50 @ xg = 1.54 < x = 1.71 47. 48.
(13) 3.2.7. (Harmonic mean) v v ～. n. xH =. =. 1. 3.7. 1 1 1 1 + +! + ( ) n x1 x2 xn. f V 3.5 4 5. xH =. n 1 1 1 + +! + ( ) x1 x2 xn. ～. 3 = 4.08 1 1 1 + + 3.5 4 5 R. ( x = 4.17 ) 49. 50. 3.3. 3.2.8. (. Measures of dispersion. (v. Rv o. 3.2b) k. k. i =1. i =1. 7. x = å f i xi / 50 = i =1. xi. Y. R. x =8. x = å f i xi / å f i. fi. ). k. 6.5. 4.5. f. 10. 3 ´ 244.5 + 5 ´ 246.5 + !! + 2 ´ 256.5 50. = 250.25. 6 51. 3.9. 7 8 9 f. 10. 12. (mg% / ml). (mg% / ml) 52.
(14) 48. 8 :0. 8 B ) MD =. (range). 1 x x å n. o. R. 4.5, 6.5, 7, 10, 12. (mean deviation) (deviation). MD= =. s. Y R. 1 ( 4.5 - 8 + 6.5 - 8 + 7 - 8 + 10 - 8 + 12 - 8 ) 5. 12 = 2.4 5. 53. 3.3.2. (. Variance and Standard Deviation s. 2. s. 2. N. å (x. =. i =1. s. s =. - µ). i. N. å (x. n. S. 2. S 2 = å ( xi - x ). 2. 2. ) N N. N. - µ). i. i =1. 2. 54. µ. (variance). N. s(2. R. (n - 1). sigma square). i =1. S. S=. n. å (x - x ) i =1. 2. i. N. s = å (xi - µ )2 N 2. (n - 1). i =1. 55. 56.
(15) v. 3.8. v3.8: µ=3.5 N=6. (standard deviation). 1,2,3,4,5,6. N. SS = å ( xi - µ ) = (1 - 3.5 ) + ( 2 - 3.5 ) + !! 2. 2. 2. i =1. SD = s = s 2 =. å (x i =1. + ( 6 - 3.5 ) = 17.5 = 12 + 22 + ... + 62 - 21 / 6 2. N. i. - µ). 2. N. 2. N. s 2 = å (xi - µ )2 N = SS N = 17.5 6 = 2.9167 i =1. SD = s = s 2 = 2.9167 = 1.7078 57. 58. Y. e. R (mean square)R. S= S = 2. n. S 2 = å ( xi - x ). 2. n. å ( xi - x ). 2. (n - 1). i =1. (n - 1). i =1. 59. 60.
(16) x. Degrees of freedom (df) R l ( • •. S2 =. R. ):. =. n-1R. l. n n 1 æ n 2 ö ç å xi - 2 x å xi + å x 2 ÷ n - 1 è i =1 i =1 i =1 ø. 1 æ n 2 ö = ç å xi - 2n x 2 + nx 2 ÷ n - 1 è i =1 ø. n. å (x - x) = 0 i =1. 1 n 1 n 2 2 x x ( ) = ( xi - 2 xi x + x 2 ) å å i n - 1 i =1 n - 1 i =1. 2 ù 1 æ n 2 1 én 2 æ n ö 2ö = ç å xi - n x ÷ = êå xi - ç å xi ÷ n ú n - 1 è i =1 ø n - 1 êë i =1 è i =1 ø úû. i. 61. v. f. v. 3.9 & 3.10 t. f. 62. Y. R. (uric acid). R mg%/ml mg%/ml. f. lf. 6. 7. 7. 10 12. 8. 9. åx. i. (. = 40. åx. = 355.5. 2 i. ). S 2 = 355.5 - 40 2 5 4 = 35.5 4 = 8.875. 3.9R 4.5 6.5. 3.10. S = S 2 = 8.875 = 2.979 l. (. 10. åx. i. = 40. ). åx. 2 i. = 330. S 2 = 330 - 40 2 5 4 = 10 4 = 2.5 S = S 2 = 2.5 = 1.58 63. 64.
(17) 3.3.4 x1 , x 2 , !! , x n yi = xi + c. y1 , y 2 , !! , y n 2. s y2 =. 2. yi = cxi 2. 2. 1 n 1 n 1 n ( yi - y ) = ( xi + c - x - c ) = å å å ( xi - x ) = sx2 n - 1 i =1 n - 1 i =1 n - 1 i =1. v3.11. f. y1 , y 2 , !! , y n. s y2 =. v3.12. c=8. I. 2. 2. 1 n 1 n 1 n ( yi - y ) = ( cxi - cx ) = c 2 å å å ( xi - x ) = c 2 sx2 n - 1 i =1 n - 1 i =1 n - 1 i =1. f. c=2. i. 1. 4.5. 4.5-8=-3.5. -3.5-0=-3.5. 12.5. 1. 4.5. 4.5*2=9. -7=-3.5*2. 2. 6.5. 6.5-8=-1.5. -1.5-0=-1.5. 2.25. 49 = ( -3.5 ) ( 2 ). 2. 6.5. 6.5*2=13. -3=-1.5*2. 3. 7. 7-8=-1. -1-0=-1. 1. 9 = ( -1.5 ) ( 2 ). 3. 7. 7*2=14. -2=-1*2. 4. 10. 10-8=2. 2-0=2. 4. 4. 10. 10*2=20. 4=2*2. 16 = ( 2 ) ( 2 ). 2. 5. 12. 12-8=4. 4-0=4. 16. 8=4*2. 64 = ( 4 ) ( 2 ). 2. 8. 8-8=0. 5. 35.5. 12. 12*2=24. 8. 16=2*8. 2. 2. 4 = ( -1) ( 2 ) 2. 2. 2. 142 = 35.5 ( 2 ). 2. 2. 2. 2. 65. 3.3.5. (. 66. ). 5 8 :8D: 1DDBD. Standard Error. N. n. x. Nn. d. 2,2 2,5 2,8 5,2 5,5 5,8 8,2 8,5 8,8. R v3.13. 2,5,8 N = 3, µ = 5, s 2 = 6 n=2. 3.7R. 67. ( x - µ )2. 2 3.5 5 3.5 5 6.5 5 6.5 8. 9 2.25 0 2.25 0 2.25 0 2.25 9. 45 5 (µ ). 27 2 3 (s / n). S2 =. å (x - x ) i. (n - 1). 0 4.5 18 4.5 0 4.5 18 4.5 0 54 2 6 (s ). 2. Sb2 =. å (x - x ). 2. i. n. 0 2.25 9 2.25 0 2.25 9 2.25 0 27 3. 68.
(18) 3.3.6. µ. :. n. 9. µ x = å xi 9 = (2 + 3.5 + ! + 8) 9 = 5. µ ,s ,s 2 ,s 2 ,s _. x. i =1. s2. n. s x2 = å (x - µ )2 9 = (9 + 2.25 + !9) 9 = 3. _. x, S. i =1. s2. _. x. n. =. 6 = 3 = 1.73 2. _. x. :. n. 9. sx = s =. ). (population parameter and sample statistic). Standard Error. 2. (. 2. ,S. 69. 70. 3.4 Coefficient of Variation (CV). Y. r. v3.14. g c. X(. ):3260,3246,3324,3200,2850 _. åx. Y. i. = 15880, x = 3176, S x2 = 35178. 7. R. ( ,,. åy. i. CV =. ´100%. S ´100% x. _. )). &). &. ). = 560.14, y = 112.028, S = 43.7665 2 y. S x = 35178 = 187.5580, S y = 43.7665 = 6.6156 187.5580 ´ 100% = 5.905% 3176 6.6156 CVy = ´ 100% = 5.905% 112.028. CVx =. s CV = ´ 100% µ CV =. :. Y ,o. 72. 73.
(19) v3.16. f. 10. 36.8 62.5. 166. M. l. CV =. 36.8 ´100% = 22.17% 166. l. CV =. 12.3 ´ 100% = 19.68% 62.5. 12.3. o. Y. Y R. 10 R 74. The End. 76. f 75.
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