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1 1

2 Hitomi s English Tests

3 True Scores of 1000 Tests Density Scores of randomly chosen tests 3

4 1( )??

5 2( )??

6 1 3(1 ) 1 1 one-dimensional data 1 6

7 4( ) 2 2 two-dimensional data 1 high-dimensional data

8 5( )?? time series data ,933 72,147 83,200 89,276 93,419 98, , , , ,049 8

9 Population Year 9

10

11

12 frequency relative frequency cumulative frequency cumulative relative frequency 12

13 6( ) class frequency frequency distribution 7( ) histogram 13

14 : n k 1 + log n/ log 2 n = 373 k =

15 Frequency Score 15

16 ( ) Frequency Score 16

17 : 8( ) n ȳ = 1 n n y i i=1 = 1 n (y 1 + y y n 1 + y n ) sample mean 17

18 ( ) 9( ) order statistics: y 1,y 2,,y n 1,y n y (1) y (2) y (n 1) y (n) 10 ( ) median: n y med = y (m+1) n =2m +1 y med = y (m)+y (m+1) 2 n =2m 18

19 ( ) 11 ( ) percentile: 0 p 1 y (1) y (2) y (n 1) y (n) 100p 100p% 12 ( ) quantile: y (1) y (2) y (n 1) y (n) 4 25% 1 50% 2 75% 3 19

20 ( ) 13 ( ) mode: 14 ( ) mid-range: : y mid = y (1) + y (n) 2 20

21 : 15 ( ) variance: n Sn 2 = 1 (y i ȳ) 2 n i=1 = 1 { (y1 ȳ) 2 +(y 2 ȳ) 2 + +(y n ȳ) 2} n S 2 n = 1 n n i=1 y 2 i nȳ2 21

22 : / 16 ( ) standard deviation: : S n = S 2 n = 1 n n i=1 (y i ȳ) 2 17 ( ) coefficient of variation: : CV = = S n ȳ 22

23 y 1,,y n z 1 = y 1 ȳ S n,z 2 = y 2 ȳ, z n = y n ȳ S n S n standardization z 1,,z n (Z ) standard score z 1,,z n z 1 =10z 1 +50,,z n =10z n

24 3 2 : 24

25 2 18 (2 ) two-dimensional data 1 2 1( )??,

26 2 ( ) (x 1,y 1 ), (x 2,y 2 ),, (x n,y n ) 19 ( ) scattergram 2 (x 1,y 1 ), (x 2,y 2 ),, (x n,y n ) (x, y) n 26

27 Weight Height 27

28 2 vs C C1 28

29 x s xx = 1 n i x) n i=1(x 2 = 1 n y n s yy = 1 i ȳ) n i=1(y 2 = 1 n (x, y) ( covariance ) n n i=1 n i=1 s xy = 1 i x)(y i ȳ) = n i=1(x 1 n x 2 i x2 y 2 i ȳ2 n i=1 x i y i xȳ 29

30 ( ) 20 (correlation coefficient) 2 (x 1,y 1 ), (x 2,y 2 ),, (x n,y n ) x y r = = s xy sxx s yy ni=1 (x i x)(y i ȳ) ni=1 (x i x) 2 ni=1 (y i ȳ) 2 1( : ) 1 r 1 r>0: ; r<0: ; r =0: 30

31 r = 1 r 1 ni=1 (x i x)(y i ȳ) ni=1 (x i x) 2 n i=1 (y i ȳ) 2 a i =(x i x), b i =(y i ȳ) i =1,,n Schwarz : n i=1 a i b i 2 n a 2 n i b 2 i i=1 i=1 Schwarz t 2 n i=1 (a i + b i t)

32 : u i = ax i + b, v i = cy i + d (i =1, 2,,n) s xy sxx s yy = s uv suu s vv (ac > 0) 32

33 2 33

34 21 ( ) confounding 34

35

36 22 ( ) spurious correlation: x y 3 23 ( ) partial correlation coefficient: r xy : x y r xz : x z r yz : y z z x y r xy z = r xy r xz r yz 1 r 2 xz 1 r 2 yz 36

37 5( ) K. Pearson (1898) 50 (stature), (, femur) (, humerus), tibia ; (, radius) ( Krzanowski and Marriott, 1994, p.23) F H T R S F H T R S 1 37

38 r SR = H T R S H T R S 1 r r SR F = SR r SF r RF 1 r 2 SF 1 r 2 RF = =

39 T R S T R S 1 r SR HF = r SR F r SH F r RH F 1 r 2 SH F 1 r 2 RH F = =

2変量データの共分散・相関係数・回帰分析

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