5章 基本統計量 - heisei-u.ac.jp

1 N x1 , x2 , L , xn .. 1. (mean, average).. x . 1 1 n . x= ( x1 + x2 + L + xn ) = xi n n i=1. x x .. Excel . = average( ). 2. (median).. S S.. x .. 5-1. 1, 2, 3, 5, 7, 8, 9. 1, 2, 3, 4, 5, 7, 8, 9. 5 2 . n / 2 n / 2 + 1 (4 + 5) 2 = . Excel . = median( ).. 3. (mode).. Excel .. = mode( ). x . 1, 2, 4, 5, 6, 6 . 6 2 1 6 .. 1. (range).. R = max( X ) min( X ) X = {x1 , x2 , L , xn }.. Excel =max( ) =min( ) . 5-2.. = max( ) min( ). 2. (variance). 1 .. 2 .. s2 =. 1. n [ ] 1 n ( x1 x ) 2 + ( x2 x ) 2 + L + ( xn x ) 2 = ( xi x ) 2. n i =1. 2. s . 1 ( xi x ). 2. n .. 1 2 1 n s2 = ( x1 + x2 + L + xn ) x 2 = xi2 x 2. 2 2. n n i =1.. Excel . = varp( ). variance . 3. (unbiased estimator of variance).

2 1 . Excel . 2. u .. 1 n n 2. u2 = . n 1 i =1. ( xi x ) 2 =. n 1. s . Excel . 5-3.. = var( ). 4. (standard deviation). 2 2 . cm cm2 .. s u . s = u = . Excel standard deviation . = stdevp( ) =stdev( ). Excel .. 1. (skewness).. 1 n xi x . 3. a3 = . n i =1 s . Excel .. 2. (kurtosis).. 5-4. 1 n x x . 4. a4 = i . n i =1 s .. 2 . 1 n 1 n 2. s2 = ( xi x ) 2. n i=1. s2 = xi x 2. n i=1.. 1 n 1 n 2 1 n 2 2x n 1 n 2. s2 = i n i=1. ( x x ) 2. = i n i=1. ( x 2 x xi + x 2. ) = i n . n i =1. x . i =1. xi + x n i=1. 1 n 2 1 n 2. = i n i=1. x 2 x 2. + x 2. = xi x 2. n i =1.. 2. 1) s . 2) Excel .. (cm) 171, 181, 172, 166, 172, 175, 168, 174, 171, 170.. 1). No. x x-a (x-a)2. 1 171 -1 1.

3 2 181 9 81. 3 172 0 0. 4 166 -6 36. 5 172 0 0. 6 175 3 9. 7 168 -4 16. 8 174 2 4. 9 171 -1 1. 10 170 -2 4. 172 x a x a . 5-5. ( x a) 2 . 2). 172 15 .. ( x1 , y1 ), ( x2 , y2 ), L , ( xn , yn ). 2 . s xy r=. sx s y 1 n 1 n 1 n sx = i n i=1. ( x x ) 2. s y = i n i=1. ( y y ) 2. s xy = ( xi x )( yi y ). n i =1. s x s y x y s xy x y .. r=-1 -1<r<0 r 0 0<r<1 r=1.. x y . y = ax + b a . r = 1 0 . 0 .. 5-6.. m cm, kg g .. 1) Excel . 2) Excel . (cm) 171 181 172 166 172 175 168 174 171 170. (kg) 71 74 65 58 66 70 60 63 72 61.. 1). No. x x-a (x-a)^2 y y-b (y-b)^2 (x-a)(y-b). 1 171 -1 1 71 5 25 -5. 2 181 9 81 74 8 64 72. 3 172 0 0 65 -1 1 0. 4 166 -6 36 58 -8 64 48. 5 172 0 0 66 0 0 0.

4 6 175 3 9 70 4 16 12. 7 168 -4 16 60 -6 36 24. 8 174 2 4 63 -3 9 -6. 9 171 -1 1 72 6 36 -6. 10 170 -2 4 61 -5 25 10. 172 66 x y a , b . x a y b 2 ( x a ) , ( y b) . 2 2. ( x a )( y b) . r = / = 2).. 172, 66 , , , 15, 16 , , 5-7. [Skip OK]. 1 r 1 . c . {c( xi x ) ( yi y )} = c 2 s x2 2csxy + s y2 0. 1 n 2. n i =1. c = s xy s x . 2. s xy2 s xy2 s xy2. 2. 2 2. + s y2 = 2. + s y2 0. s x s x s x 2. s r = 1 1 r 1 . 2 xy 2 2. s s x y . y = ax + b r a . r = 1 .. yi = axi + b i = 1, L , n . 1 n a2 n s = [(axi + b) (ax + b)] = ( xi x ) 2 = a 2 s x2 s y =| a | s x 2. y 2. n i =1 n i =1. n 1 a n s xy = ( xi x )[(axi + b) (ax + b)] = ( xi x ) 2 = as x2. n i =1 n i=1. 2. s as x a r = xy = = a r = 1.

5 Sx s y sx | a | sx | a |. 5-8.