_______________________________________

2

2

1) (quantitative data) 2) (qualitative data)

1.

2

- (Ungrouped data) - (Grouped data)

1.1 1.1.1 N 1 2 3 Nx , x , x ,..., x X

N

i1 2 3 N i 1

xx x x ... xX

N N=+ + + += =

N

ii 1

x NX=

=

3

1.1.2

1 2 3 Nw , w , w ,..., w

1 2 3 Nx , x , x ,..., x

N

i i1 1 2 2 3 3 N N i 1

N1 2 3 N

ii 1

w xw x w x w x ... w xX

w w w ... w w

=

=

+ + + += =+ + + +

1 4 3 1 100 3 65, 70, 80 75 70 10 1.1.3 1f 1x 2f 2x 3f 3x # kf kx

k

i i1 1 2 2 3 3 k K i 1

1 2 3 k

f xf x f x f x ... f xX

f f f ... f N=+ + + += =+ + + +

4

2

5-9 8

10-14 10 15-19 12 20-24 7 25-29 3

ix if i if x 5-9 7 8

10-14 12 10 15-19 17 12 20-24 22 7 25-29 27 3

1.1.4 1 2 3 kx , x , x ,..., x 1,2,3,,k 1,2,3,,k 1 2 3N , N , N ,..., Nk

5

N

i i1 1 2 2 3 3 k K i 1

Total N1 2 3 k

ii 1

N XN X N X N X ... N XX

N N N ... N N

=

=

+ + + += =+ + + +

3 5 50, 65 70 40, 45 35 5

1.2 (Median) 1.2.1

1. 2. N

N N 12+

N 4 5 4, 2, 3, 5, 6

6

5 8 4, 2, 3, 5, 6, 4, 7, 5 1.2.2 N N

2

= Lm

N - f2L+ I

f

= Um

Nf -2U- I

f

L U N

Lf Uf

mf I

7

6 100

(.) () 50 3

50 52 5 53 55 15 56 58 42 59 61 25

61 10

(.) () 50 3

50 52 5 53 55 15 56 58 42 59 61 25

61 10

8

1.3 (Mode)

1.3.1 1.3.2

= 11 2

dL+ Id d

+

L 1d

2d

I

7 50-52 53-55 56-58 59-61 62-64

5 18 42 27 8

9

1.4 (Geometric mean) N

0 1 2 3 Nx , x , x ,..., x

G.M. = N 1 2 3 Nx x x ...x

ix ifk

ii 1

f N=

= G.M. = 31 2 kff f fN 1 2 3 kx x x ...x 1.5 (Harmonic mean) N 1 2 3 Nx , x , x ,..., x H.M. =

1 2 N

11 1 1 1+ +...+N x x x

N

i=1 i

N1x

=

ix if

k

ii 1

f N=

=

10

H.M. = 1 2 k

1 2 k

11 1 1 1f +f +...+fN x x x

ki

i=1 i

Nfx

=

2.

4 3 10 9 100 99

rQ r(N+1)4

rD r(N+1)10

rP r(N+1)100

, , rQ rD rP

rQ rN4

rD rN10

rP rN100

, , rQ rD rP

r

L

rQ

rN - f4Q =L+ I

f

r

L

rD

rN - f10D =L+ I

f

11

r

L

rP

rN - f100P =L+ I

f

3. 2

(1) (Absolute variation) 4

1. =

2. 3 1Q -QQ.D.=2

3.

n

ii=1

X -XM.D.=

n

k

i ii=1

k

ii=1

f X -XM.D.=

f

X N n 4.

N

2i

i=1(x -)

=N

i

N2

2i=1x

= -N

N2

ii=1

(x -x)S=

n-1

i

N2 2

i=1x -nx

S=n-1

12

k2

i ii=1

f (x -)=

N

i

k2

i2i=1

f x= -

N

k2

i ii=1

f (x -x)S=

n-1

i

k2 2

ii=1

f x -nxS=

n-1

(2) (Relative variation)

1 4 1. = max min

max min

X -XX +X

2. = 3 1

3 1

Q -QQ +Q

3. = M.D.

X M.D.

4. = S

X

13

1. 1 k k

1n

2n

3n

1 2 kn n ...n

8 3 2 3 9 7 3 286 2. (Factorial) n 1 n n! n n n! n!1 2 3 n= " n 3 2 1= " 4! 4 3 2 1 24= = n = 0 0 1 0! = 1

14

3. (Permutation) 2 1. 2. 3.1 . n n = n! . n k 1 2 3 ... k

1n 2n 3n

kn

= 1 2 3 k

n!n !n !n !...n !

. n r (r < n) = n!

(n r)!

n,r n!P (n r)!= 10 5 1, 2, 3, 4, 5 5 11 MATHEMATICS

15

12 3 6 3.2 . = (n-1)! 1 n (n-1)!

. = ( )n 1 !2 7

( ) 7 1 ! 6!2 2 =

4. (Combination) n r (r < n) n!

(n r)!r!

n,r n!C (n r)!r!= nr

13 30 5

16

5. 1. (Random Experiment) 2. (Sample Space) S n(S) 1 S = {1, 2, 3, 4, 5, 6} 3. (Event) E n(E) 1 2 2 S = {, , , } E 2 E = {, } n(E) = 2 4. (Probability) P(E) 1. n(E)P(E)

n(S)=

2. P(E) 1 P(E )= P(E ) E 3. n(E )P(E )

n(S) =

17

4. P(A B) P(A) P(B) P(A B) = + 5. P(A B) P(A) P(A B) =

18

1. i X ix ix ii xz = i = 1, 2, , N N ii x Xz s

= i = 1, 2, , n X S N 14 40 25 5 20 3 27 22 x

19

2.

Histogram of x

x

Den

sity

-20 -10 0 10 20 30

0.00

0.01

0.02

0.03

0.04

20

1.

2.

3.

4. 1

x z 0 1

21

1 1. 50

() () 30 39 40 49 50 59 60 69 70 79 80 - 89

4 5 13 17 6 5

(O-NET 49) 1. 60 69 2. 50 9 3. 50 59 26% 4. 80 10%

2. ()

0 4 5 9

10 14 15 19

4 5 x 7

11 5 14 (O-NET 51) 1. 46.67 % 2. 56.67 % 3. 63.33 % 4. 73.33 %

3.

0 | 3 7 5

22

1 | 6 4 3 2 | 0 2 1 2 3 | 0 1

(O-NET 51) 1. < < 2. < < 3. < < 4. < <

4. 31 35 25 (O-NET 50) 1. 2 : 3 2. 2 : 5 3. 3 : 2 4. 3 :5

5. 4 2 2 4 45, 47.5 7 4 (O-NET 49) 1. 46 2. 47 3. 48 4. 49

6. () ()

29 31 32 34 35 37 38 40 41 - 43

1 4 5 5 5

(A-NET 51) 1. 37.35, 37.5 3 2. 37.5, 37.5 3 3. 37.35, 37.5 3.5 4. 37.5, 37.0 3

23

7. 8, a, 12, 17, 22, b, 26

17 1 10 () (A-NET 51) 1. 0.35, 0.45 2. 0.35, 0.41 3. 0.42, 0.45 4. 0.42, 0.41

8. 20 30 24.6 a b a b 0.125 0.16 4 (A-NET 51) 1. a = 22, b= -1.1 2. a = 22, b = -1 3. a = 21, b = -1.1 4. a = 21, b = -1

9. 40 ( ) if10 14 15 19 20 24 25 29 30 34 35 39

4 6 a 8 4 6

( ) 24.5 b c (A-NET 50)

3

i ii=1

f (x -)=-125

1. b = 5 c = 6.25 2. b = 6.25 c = 5 3. b = 4.5 c = 5 4. b = 5 c = 4.5

24

10. 50

() 9 13 14 18 19 23 24 28 29 33

8 22 37 47 50

26 (A-NET 49) 1. 32% 2. 34% 3. 35% 4. 37%

11. 50 () ()

156-160 6 161-165 15 166-170 21 171-175 8

a b 75% b (PAT 1 53) 1. a = 166.1 b = 168.73 2. a = 166.1 b = 169.43 3. a = 166.7 b = 168.73 4. a = 166.7 b = 169.43

12. . 3 80

75 25 70

. 5 4 a b

1 2 3 4 5x , x , x , x , x

1 2 3 4x , x , x , x

25

b 5a 2=

(PAT 1 53) 1. . .

2. . . 3. . . 4. . .

13. 60 30 . . 53 53 27 . 51-60 51-60 (PAT 1 54) 1. 3 2. 4 3. 5 4. 9

14. 10 8 2

1 3 2 (A-NET 51) 1. 3

125 2. 6

125 3. 12

125 4. 16

125

15. 10 0, 1, 2,,9

3 10 (A-NET 50) 1. 1

12 2. 1

15 3. 1

20 4. 1

30

26

16. 8 (A-NET 50) 1. 360 2. 720 3. 1080 4. 1440

17. 20 -9, -8, -7,-6,,7, 8, 9, 10 2 (A-NET 49) 1. 7

19 2. 9

19 3. 10

19 4. 13

19

18. 4 6 3

(Ent 48) 1. 0.78 2. 0.80 3. 0.82 4. 0.84

19. 100 50 40 33 5 10 5 10 12 20

. 0.15 . 0.40

(PAT 1 53) 1. . .

2. . . 3. . . 4. . .

20. 66.2

39% 56 76.4

27

0 z (A-NET 51)

z 0.40 0.51 0.85 1.23 0.1554 0.1950 0.3023 0.3907

1. 8 2. 12 3. 20 4. 25

21. 1.96 98.3 10 5

. .

(A-NET 50) 1. 2. 3. 4. 0 z

z 1.53 1.96 2.12 2.35 0.4370 0.4750 0.4830 0.4906

22. 1 10 0.2 13 8 (A-NET 50) 1. 9.19% 2. 22.55% 3. 40.81% 4. 69.19% 0 z

28

z 0.75 1 1.25 1.5

0.2734 0.3413 0.3944 0.4332

23. 30 45 5 29 1.7 (A-NET 49) 1. 36.5 2. 37 3. 37.4 4. 38

24. 5000 53 4 47 63 0 z (A-NET 49)

z 0.4 0.5 0.7 1.5 2.5 0.1554 0.1915 0.2580 0.4332 0.4938

1. 4,378 2. 4,459 3. 4,572 4. 4,635

25. z = 0 z = 1 0.3413 20,000 (Ent 48) 1. 3,413 2. 6,348 3. 6,826 4. 13,652

26. 4 50.4% 14 2 a 16 b () [14,16] a b (Ent 47)

29

z 0.01 0.99 1.01 2.65 A 0.004 0.3389 0.3488 0.496

1. a = 50.4, b = 33.78% 2. a =50.4, b = 34.29% 3. a = 99.6, b = 33.78% 4. a = 99.6, b = 34.29%

27. . . a . . c c+ 5

a

. . 78.81 (Ent 46)

Z 0.70 0.80 0.90 A 0.2580 0.2881 0.3159

1. 5 2. 4 3. 3.5 4. 2 ******************************************************************

2 1. 2 60

40 30 50 64 66 (PAT 1 53)

2. 6 2, 3, 6, 11, a, b 8 7 a b (PAT 1 53)

3. 10

(Ent 48)

30

4. 2 4 1 2 (Ent 46)

31

. .4-5-6. : . . . (2552). 2 4-6. : . . (2552). ADMISSIONS52

. : . . (2553). . : () . . (2551). .4-5-6. :

.