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s z e r e g s z c z e g ó Bo w y s z e r e g p u n k t o w y s z e r e g p r z e d z i a Bo w y
k l a s y ( x 1 , x 2 ] ( x 2 , x 3 ] . . . ( x k , x k + 1 ]
x i x 1 x 2 . . . x k
m i a r y
x 1 x 2 . . . x n
n i n 1 n 2 . . . n k
o p i s o w e
n i n 1 n 2 . . . n k
[r o d k i k l a s x 1 x 2 . . . x k
n k k
[r e d n i a
1 1 1
x = x i x = x i n i x = x i n i
n n n
a r y t m e t y c z n a
i = 1 i = 1 i = 1
n k k
[r e d n i a
n n
n i i
x g = x i x g = x n x g = x n
i i
g e o m e t r y c z n a
i = 1 i = 1 i = 1
[r e d n i a
n n n
x h = x h = x h =
n
k k
h a r m o n i c z n a
1
n i n i
x i
x i x i
i = 1
i = 1 i = 1
d - n r k l a s y o n a j w i k s z e j l i c z e b n o [c i
m o d a w a r t o [ x i d l a k t ó r e j
w a r t o [ w y s t p u j c a n a j c z [c i e j
n d - n d - 1
( d o m i n a n t a ) n i j e s t n a j w i k s z a
M o = x d + | x d + 1 - x d |
2 n d - n d - 1 - n d + 1
p r z y w y z n a c z a n i u m o d y n i e b i e r z e m y p o d u w a g w a r t o [c i o r a z k l a s s k r a j n y c h
m
n
m - n r p i e r w s z e j k l a s y , d l a k t ó r e j n i
2
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m e d i a n a
2
m - 1
( k w a r t y l d r u g i )
1
n n n
M e = x + x n - p a r z y s t e
+ 1
M e = x m M e = x m + - n i | x m + 1 - x m |
2 2 2
2 n m
i = 1
m
n
m - n r p i e r w s z e j k l a s y , d l a k t ó r e j n i
w y z n a c z a s i w t e n s p o s ó b ,
4
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k w a r t y l
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m - 1
k t ó r a p o w s t a Ba p o w y z n a c z e n i u
p i e r w s z y
n
Q 1 = x m Q 1 = x m + - n i | x m + 1 - x m |
m e d i a n y , w y z n a c z a s i m e d i a n
4 n m
i = 1
m
3 n
m - n r p i e r w s z e j k l a s y , d l a k t ó r e j n i
w y z n a c z a s i w t e n s p o s ó b ,
4
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k w a r t y l
|e z d r u g i e j c z [c i z b i o r o w o [c i ,
m - 1
k t ó r a p o w s t a Ba p o w y z n a c z e n i u
t r z e c i
3 n
Q 3 = x m Q 3 = x m + - n i | x m + 1 - x m |
m e d i a n y , w y z n a c z a s i m e d i a n
4 n m
i = 1
n k k
o d c h y l e n i e
1 1 1
d = | x i - x | d = | x i - x | n i d = | x i - x | n i
n n n
p r z e c i t n e
i = 1 i = 1 i = 1
n k k
1 1 1
s 2 = ( x i - x ) 2 s 2 = ( x i - x ) 2 n i s 2 = ( x i - x ) 2 n i
n n n
i = 1 i = 1 i = 1
w a r i a n c j a
n
k k
1
1 1
]2 = ( x i - x ) 2
]2 = ( x i - x ) 2 n i ]2 = ( x i - x ) 2 n i
n - 1
n - 1 n - 1
i = 1
i = 1 i = 1
" "
o d c h y l e n i e
s = s 2 , ] = ]2
s t a n d a r d o w e
w s p ó Bc z y n n i k
s
V = · 1 0 0 [ % ]
x
z m i e n n o [c i
n k k
m o m e n t y
1 1 1
m r = x r m r = x r n i m r = x r n i
i i i
n n n
z w y k Be
i = 1 i = 1 i = 1
n k k
m o m e n t y
1 1 1
M r = ( x i - x ) r M r = ( x i - x ) r n i M r = ( x i - x ) r n i
n n n
c e n t r a l n e
i = 1 i = 1 i = 1
"
M 3 n 2 - n
Æ
s k o [n o [1 ³ = , ³ = · ³
s 3 n - 2
M 4 n 2 - 1 6
Æ
k u r t o z a 2 K = - 3 , K = K +
s 4 ( n - 2 ) ( n - 3 ) n + 1
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0 d l a x x 1 ,
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ôø 0 d l a x x 1 , ôø
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d y s t r y b u a n t a
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F n ( x ) = d l a x i <