Z-score and Expected Value คะแนนมาตรฐานและค่าความคาดหวัง

1 Z-score and Expected Value . Chapter 5. 2 ? ? (Standard score or Z-score ).. (Normal Curve Normal Distribution). (Transform) .. Z ( Z-score ). ( Z-score Equation). x -x x -m Z = s = s . Z . x . x m x . s s (Sigma) ( x ). ( Z-score Equation Meaning). Z x-x = s ( Z-score ) . (Dimensionless Quantity) .. (SD S s Standard Deviation).. E (Appendix E).. 1 . SD. 1. 75 80 10. 2. 75 70 5. 3. 75 65 10.. 75. Z = (75 80)/10 , (75 - 70)/5 , (75 - 65)/10. = , 1, 1 . 2 3 1 2 3. ( z ).. 1 (x - m) 2 /2 s 2. f ( x) = e s p 2.. x = . s = Standard Deviation ( ). m = Mean ( ). p = .. ( ). e = ( ). [ Normal Curve . ]. (Normal Curve). Normal Curve x z 2 . - x ( x).

2 - Z-score ( Z). - . - x - Z = 0 Z. (Normal Curve). x x - 3s x - 2s x - s x x + s x + 2s x + 3s z -3 -2 -1 0 1 2 3. Normal Curve x, z (Normal Curve). (50%) (50%). x x+s x + 2s x + 3s z +1 +2 +3.. ( ) Normal Curve . m + s x + s Z = + 1. m + 2s x + 2s Z = + 2. m + 3s x + 3s Z = + 3. The Concept of "Six Sigma" at Motorola A Bit of Statistics .. Different numbers of Opportunities .. Sigma+ Sigma Lower spec limit upper spec limit + Manufacturing Processes 6s ppm or Zero Administrative Areas Customers or Suppliers = A structural approach to continuous improvement ( or Six steps toward excellence ). 1 - Identify the product or service you provide 4 - Define the process for doing work 2 - Identify your customers & their requirements 5 - Eliminate defect sources / optimize the process 3 - Determine your needs & suppliers 6 - Continuously improve the Sigma level Normal Distribution - Gaussian Curve s= S ( xi - x ) 2.

3 S n-1. Sigma = s = Deviation (n is sampling no.). (Square root of variance). Axis graduated in Sigma -7. -6. -5. -4. -3. -2. -1. 5. 1. 2. 3. 6. 7. 0. 4. between m + 1s %. between m + 2s %. between m + 3s %. between m + 4s %. between m + 5s %. between m + 6s %. Appendix E-1. 0 Z.. (Areas under the Standard Normal Curve form o to z). Period 2 . 2 Normal Distribution . ( ) ( ). A 3,200 250. B 3,000 175.. 1. A B .. 2. 3,200 . 2 . 3. 3,000-3,200. 2 . 4. 90% 2 . 90%. Normal Curve A Normal Curve B. 1.. A B. ( A B = 3,200 3,000 ). SD A . B ( ).. A B . (250/3,200) , (175/3,000) = , . 1 A B. Normal Curve A Normal Curve B. Z > 0 Z = 0 Z > 2. 2 3,200.

4 Z = (x m)/s A Z = (3,200 3,200)/250 = 0. P(x > 3,200) P(Z > 0) = (50%). B Z = (3,200 3,000)/175 = P(x > 3,200) P(z > ) = - = A B 3,200 . Normal Curve A Normal Curve B. Z (0 ) Z (0 ) 3. 2 3,000-3,200 . Z = (x m)/s A 3,000 Z = (3,000 - 3,200)/250 = 3,200 Z = (3,200 3,200)/250 = 0. P(3,000 < x > 3,200) P( < Z > 0) = B 3,000 Z = (3,000 3,000)/175 = 0. 3,200 Z = (3,200 3,000)/175 = P(3,000 < x > 3,200) P(0 < z > ) = B A 3,000-3,200 . Z = 4. 90% 2 . 90%. Z = (x m)/s A 90% . Z Z = 90% Z = + = (x 3,200)/250 x = 2, . = (x 3,200)/250 x = 3, . A 90% . 2, 3, . Normal Curve Z A B Z = B 90% Z = + ( 2 ). = (x 3,000)/175 x = 2, . = (x 3,000)/175 x = 3.

5 B 90% . 2, 3, . A B. 90% : . - A 2, 3, . - B 2, 3, . ( Expected Value EV). N. EV = S. i=1. pi xi S px . x = . p = . N = .. Expected Value .. EV p = 1.. EV = x = m .. m .. ( ) .. -20 - -11 -10 - -1 0 1 - 10 11 - 20 21 - 30. 20 30 5 35 40 20. (x) (f) f/N ( . = ). -20 -11 20 20/150 = -10 -1 30 30/150 = 0 5 5/150 = 1 10 35 35/150 = 11 20 40 40/150 = 21 3 0 20 20/150 = . N = 150 150/150 = .. Probability . [2] [3] [4]=[2] [3]. (x) p px f x fx 20 -310. 30 -165. 0 0 5 0 0. 35 40 620.. 20 510. 150 x = = . p = f/N.. (Relative Frequency).. (Probability).. (0-1 100 %).. (EV & x).. ( Expected Value ) . = .. x S. = N. fx f1 x1 + f2 x2 + f3 x3 f1 x1 f2 x2 f3 x3.

6 = N. = N + N + N. f f f = x1+ x2 + 3 x3. 1 2 = p1x1 + p2 x2 + p3 x3. N N N. = S px = EV ( Expected Value ).. (%) .. - 30%. 10%. 50%. ER ( Expected Return). = (-30%) + + = -9% + 5% + 10%. = 6%. 6% . 6%. + + = 1.. -30% 30%. (-30%) = -9% . 10% 50%. (10%) = 5% . 50% 20%. (50%) = 10% . 3 ( ). = 6%.. (f/N) (p) . ( Expected Value ). S (Standard Deviation) .. (f/N) (p) . s . = S N. f(x- x)2. Var. S2. S f(x- x) 2. f 2 2 f (x 1 (x1 -x) + f2 (x2 -x) + 3 3 -x)2. S2 = N = N. f (x -x). = N + N + N. 1 1. 2. f2 (x2 -x). 2. f (x 3 3 -x) 2. S 2 f1 (x1 -x). = N + N + N. 2. f2 (x2 -x). 2. f3 (x3 -x). 2. f1 (x -x) 2. = N 1 + N 2 + N 3 f2 (x -x)2 f3 (x -x)2. p (x -x).

7 = 1 1 + 2 2 + 3 3. 2. p (x -x)2. p (x -x)2. = S p(x - x) 2. = S p(x - EV)2. S = Var. = Sp(x - EV). 2. Normal Curves > 1 . ER. Expected Return of A < Expected Return of B. s of A < s of B ( ).