Statistics (Greek)

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http://www.users.auth.gr/dkugiu/Teach/ElectricEngineer/ E-mail: dkugiu@gen.auth.gr 2010

2 (..) X , fX (x) FX (x) , () E(X) () Var(X) 2 . .. ( ) .. . , .. , , . . (statistics) . , . (sampling) . (descriptive statistics). (statistical inference) . . 1 . 2 .. 3 .. () .. . : (data): .. , .. ( .. ). (population): .., .. . (sample): , .. 25 . (parameter): .. , .. . (statistic): .. , .. 25 . . .., , 2 p.

1 .

1.1 , . , , . .. X ( ) (frequency table). xi X fi , . (relative frequency) (percent) pi fi xi n

pi =

fi . n

(1.1)

(cumulative frequency) Fi xi xi ,i

Fi =j=1

fj xj xi j i.

Pii

Pi =j=1

pj xj xi j i.3

4

.. X , fX (x) . FX (x) . (bar chart), ( , ) xi . ( fX (x) ). . , (pie chart) () . 1.1 5 . . 120 1.1.

1 1 2 1 1 1 2 2 2 4

4 2 3 3 3 2 2 2 1 2

2 1 2 2 5 5 2 3 5 1

2 2 1 1 3 4 3 3 1 2

2 1 1 2 1 3 2 2 1 2

3 1 4 2 2 1 1 4 4 2

4 2 3 3 2 2 3 6 4 3

3 3 4 2 3 4 3 3 2 2

1 3 1 4 1 2 4 2 5 3

1 5 1 3 2 1 2 3 4 2

3 1 6 3 6 3 1 1 2 1

3 2 2 5 4 4 5 3 2 4

1.1: 120 . .. X . 1.1 ( 1 ). , , 1 2 , 3 4 4 . 1.1 . ; 2 3 ; .

5

xi1 2 3 4 5 6

fi28 39 27 16 7 3 120

pi0.23 0.33 0.23 0.13 0.06 0.03 1.00

Fi28 67 94 110 117 120

Pi0.23 0.56 0.78 0.92 0.97 1.00

1.2: 120 1.1 fi , pi , Fi Pi ). ( SPSS). 1.2 X ( xi i = 1, . . . , 6) , fi xi , () pi , Fi Pi . 1.1 . o 40 35 30o

25 20 15 10 5 0 1 2 3 4 5 o 6

1.1: 120 1.1. 2 , 1 3 , 4, 5 6 . .. X , X (

6

), . k (groups), , , . ri i (ri = r ). . . . () xmin () xmax

R = xmax xmin . R k r . xmin , xmax . ( ) ( ). . . (histogram). fi , () pi , Fi , Pi i . ( ) fX (x) .. X , . .. X , , . X . ( 30 ) ( ). .. X . .. X .

(stem and leaf plot) (dotplot).

7

1.2 1.3 25 ( ) ( 20 ). / 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 5.3 4.5 5.7 5.8 4.8 6.4 6.4 5.6 5.8 5.7 5.5 6.1 5.2 7.0 5.5 5.7 6.3 5.6 5.5 5.0 5.8 4.7 6.1 6.7 5.1 141.8 5.0 4.2 5.4 5.5 4.6 6.1 6.1 5.3 5.5 5.4 5.2 5.8 4.9 6.7 5.2 5.4 6.0 5.3 5.2 4.8

107.6

1.3: . X . xmin = 4.5, xmax = 7.0

R = xmax xmin = 7.0 4.5 = 2.5. 10 (k = 10)

r=

R 2.5 = = 0.25, k 10

8

4.5. . 1.4 , , , . 4.50 4.75 4.75 5.00 5.00 5.25 5.25 5.50 5.50 5.75 5.75 6.00 6.00 6.25 6.25 6.50 6.50 6.75 6.75 7.00

fi2 2 2 4 5 3 2 3 1 1 25

pi0.08 0.08 0.08 0.16 0.20 0.12 0.08 0.12 0.04 0.04 1.00

Fi2 4 6 10 15 18 20 23 24 25

Pi0.08 0.16 0.24 0.40 0.60 0.72 0.80 0.92 0.96 1.00

1.4: fi , pi , Fi Pi ). 25 ( [5.25, 6.0]). X , 1.2. Io A 6 5 4o

3 2 1 0 4

4.5

5

5.5 o

6

6.5

7

1.2: 1.3. .

9

( ) .

1.2 . .. X (summarizing or descriptive statistics). .. . : (measures of location) (variability measures) .

1.2.1 . : (sample mean value) (arithmetic mean), (average), (sample median), (sample mode). . x1 , x2 , . . . , xn , .. X . x

x1 + x2 + + xn 1 x= = n n

n

xi .i=1

(1.2)

. n . ( ) , 1.3.

10

1111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000 1111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000 0 10 20 30 40 50 60 1.3: . . x. n (n + 1)/2, n n/2 n/2 + 1,

x=

x(n+1)/2 xn/2 + xn/2+1 2

n = 2k + 1 n = 2k.(1.3)

. . , . . ( ) . , . . , . . ( ) ( ).

1.2.2 . , , . ,

11

. , . 1.4 .

1111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000 0 10 20 30 40 50 60 1111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000 1111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000 0 10 20 30 40 50 60 1.4: . : (sample range) R, (sample variance) s2 (standard deviation) s. (percentiles) (interquartile range). R = xmax xmin . . xmin xmax . ( ). . . xi xi x, 0 (1.2) n n n

i=1

(xi x) =

i=1

xi

i=1

x = n n = 0. x x

x x x. .

12

n . n 1 n s2

1 s = n12

n

i=1

(xi x)2 .

(1.4)

(1.4)

1 s = n12

n

i=1

x2 n2 x i

,

(1.5)

. s2 . s, s2 . s .. X ( ) , X . , 2 . s2 , . xi , x xi x xi . 2 n 1 n. (degrees of freedom). n n . x n n . s2 n i=1 (xi x) = 0, n1 . s2 n 1 n 1.

: n 1 n

s2

. . . p- (ppercentile) p% (0 p < 1). 50- . (quartiles). 25- (rst or lower quartile) Q1 , 75- (third or upper quartile) Q3 . ( ). , , n

13

1 n/2 n/2 + 1 n, n 1 (n + 1)/2 (n + 1)/2 n ( 1.5).

n=2k x1 1 n=2k+1 x1 1 x(n+1)/2 (n+1)/2 xn n xn/2 xn/2+1 n/2 n/2+1 xn n

1.5: . . I = Q3 Q1 (interquartile range) (). I ( R) . x, Q1 Q3 , xmin xmax , 5 (ve number summary). 5 (box plot) 1.6.

x min0

Q110

~ x20

Q330

x max40

1.6: . : , SPSS, ( (whiskers)) . xi (outlier), (extreme).

14

( Q1 Q3 ). 1.5I ( ) , 3I .

, , . : , ( ), , ( ). , ( ). . ( , ) . 1.3 . ( Bq/g) 10 . 1.5. / 1 2 3 4 5 6 7 8 9 10 0.40 0.51 0.51 0.54 0.55 0.59 0.63 0.67 0.75 2.10 7.25 0.11 0.13 0.26 0.27 0.33 0.37 0.52 0.65

2.64

1.5: ( Bq/g) .

15

X . 10 1.5

1 x= 10

10

xi =i=1

7.25 = 0.725. 10

. n

x=

xn/2 + xn/2+1 x5 + x6 0.55 + 0.59 = = = 0.57. 2 2 2

. . ( , ). xmin = 0.40 Bq/g xmax = 2.10 Bq/g. R = 1.70 Bq/g, . s2 10

i=1

x2 = 0.402 + 0.512 + + 0.752 + 2.102 = 7.44 i

(1.5)

1 s = 92

10

i=1

x2 102 x i

=

1 7.44 10 0.7252 = 0.243. 9

s=

s2 =

0.243 = 0.493,

0.5 Bq/g, ( ). 5 1.7 . , I = Q3 Q1 = 0.67 0.51 = 0.16 Bq/g. 5 . 1.8 . 2.10. , , Q3 = 0.67, 3I = 3 0.16 = 0.48.

16

x min =0.40

~ =0.57 x

x max =2.10

0.40 0.51 0.51 0.54 0.55 0.59 0.63 0.67 0.75 2.10

Q1 =0.51

Q3 =0.67

1.7: 5 10 .o o 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 1

1.8: 10 .

1.4 . 9 . 9

1 x= 9 ( n )

xi =i=1

5.15 = 0.572 9

x = x(n+1)/2 = x5 = 0.55. , . :

17

1 s2 = (5.15 9 0.5722 ) = 0.010. 8 s = 0.010 = 0.10 xmin = 0.40 xmax = 0.75 R = 0.75 0.40 = 0.35 ( {x1 , . . . , x5 }) Q1 = x3 = 0.51 ( {x5 , . . . , x9 }) Q3 = x7 = 0.63 I = 0.63 0.51 = 0.12 1.9.

o o 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 1

1.9: 9 . . 1.5 1.5 8 . 9 8 . X1 .. X2 .. . X1 . X2 . 1.6. 1.10 . ( ) . ( , ) .

18

X1 x1 = 0.572 x1 = 0.55 s2 = 0.010 1 s1 = 0.10 x1,min = 0.40 x1,max = 0.75 R1 = 0.35 Q1,1 = 0.51 Q1,3 = 0.63 I1 = 0.12

X2 x2 = 0.33 x2 = 0.30 s2 = 0.034 2 s2 = 0.18 x2,min = 0.11 x1,max = 0.65 R2 = 0.54 Q2,1 = 0.195 Q2,3 = 0.445 I2 = 0.250

1.6: 9 8 2 3 .

o o , A B 0.7 0.6 0.5 0.4 0.3 0.2 0.1 X_1 X_2

1.10: 9 8 .

2 , x s2 , , ( 2 ). . , 40 . (point estimation) (interval estimation) ...

2.1 , . , 25 25 . X .. FX (x; ) . {x1 , . . . , xn } X n. g(x1 , . . . , xn ) . (estimator) = g(x1 , . . . , xn ). {x1 , . . . , xn } n, {x1 , . . . , xn } .. {X1 , . . . , Xn }, F (x; ). ... {x1 , . . . , xn } ( .. {X1 , . . . , Xn }) ( ..). {x1 , . . . , xn } . , .. 2 Var(). E() 19

20

.. X 2 . n ,

x=

1 n

n

xi .i=1

(2.1)

2 ,

2 1 s = n12

n

i=1

(xi x)2

(2.2)

s2 =

1 n

n

i=1

(xi x)2 .

(2.3)

1 1 s2 s2 ( n1 n ). n . (2.2)

1 s = n12

n

(x2 ii=1

1 2xi x + x ) = n12

n

n

x2 ii=1

2 x

xi + n2 xi=1

x (2.1) ( (2.3) )

1 s = n12

n

i=1

x2 n2 x i

.

(2.4)

(2.4) s2 .

2.1.1 2 . . .

(unbiased) , E() = . b() = E() . x (2.1) .. X .

21

x E() = . xE() = E x

1 n

n

xii=1

1 = n

n

i=1

1 E(xi ) = n

n

= .i=1

2 .. X : 1. s2 (2.2) 2 , E(s2 ) = 2 . 2. s2 (2.3) 2 2 2 b( ) = . s n (1). .. X 2

2 = E(X 2 ) 2n

E(X 2 ) = 2 + 2 .

(2.5)

{x1 , . . . , xn } 2 X n

Vari=1

xi

=i=1

Var(xi ) = n 2

x 2 x

Var() = Var x

1 n

n

xii=1

1 = 2 Var n

n

xii=1

=

1 2 n 2 = . n2 n

(2.6)

x E() = x (2.5)2 E(2 ) = x + 2 = x x

2 + 2 . nn

s2 (2.4):

1 E(s ) = n12

n

E(x2 ) ii=1

nE( ) x

2

=

1 (n 2 + n2 2 n2 ) = 2 n1

1 = n1

i=1

( 2 + 2 ) n

2 + 2 n

s2 . (2), ( n n 1) E(2 ) = s n1 2 n

b(2 ) = E(2 ) 2 = s s

n1 2 2 2 = . n n

22

n , . (consistent)

P(| | ) 1

n ,

. . x .. X .

xd =

xmin + xmax , 2

xd .

. 1 (eective) 2 , 2 < 2 . 1 2 , x xd 2 2 2 2 x xd . x < xd x xd . (adequate) . x, .. X , , xd (xmin xmax ). x s2 2 , , . , . (minimum variance ubiased estimator), . x s2 .. X .. X . X ( ) .

2.1.2 .. X , F (x; ) f (x; ),

23

X X . {x1 , . . . , xn }. . , F (x; ) 2 . , 2 () . [a, b], a b 2 = a+b 2 = 12 . 2 ( 1 , 2 ) 2 , x s2 ( 1 , 2 ) , 2 x s2 . (method of moments). : 2 . , , , . 2.1 40 40 . 2.1 25 ( ). 2.1 , . 40 (2.1)(ba)2

1 x= 2525

25

xi =i=1

1 995.1 = 39.80 25

(2.4)

1 s = 242

i=1

x2 252 x i

=

1 (39629 25 39.802 ) = 0.854. 24

x = 39.80 2 2 2 s = 0.854 () . .. X ( 40 ) N (, 2 ), X . .. X [a, b], a b = a+b 2 = 12 . 2 2 x s2 (ba)2

x = s2 =

a+b 2 (ba)2 12

a = x 3s = x + 3s b

a = 38.20 = 41.40. b

24 / (i) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

x1i ()40.9 40.3 39.8 40.1 39.0 41.4 39.8 41.5 40.0 40.6 38.3 39.0 40.9 39.1 40.3 39.3 39.6 38.4 38.4 40.7 39.7 38.9 38.9 40.6 39.6 995.1

x2 1i1672.8 1624.1 1584.0 1608.0 1521.0 1714.0 1584.0 1722.2 1600.0 1648.4 1466.9 1521.0 1672.8 1528.8 1624.1 1544.5 1568.2 1474.6 1474.6 1656.5 1576.1 1513.2 1513.2 1648.4 1568.2 39629

x2i ()41.6 41.0 40.5 40.8 39.7 42.1 40.5 42.2 40.7 41.3 39.0 39.7 41.6 39.8 41.0 40.0 40.3 39.1 39.1 41.4

x2 2i1730.6 1681.0 1640.2 1664.6 1576.1 1772.4 1640.2 1780.8 1656.5 1705.7 1521.0 1576.1 1730....