# 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