1

Emil Núñez Rojas

This topic is genuinely new and discovered by the author on 08 August 2007 in

Institute of Applied Mathematics in Lima Peru - Emil Núñez studied at

Technological University of Prague (CVUT), Faculty of Engineering Physics

(Fjfi): noverbal@hotmail.com, cel: 954 638125.

LIMITS OF VARIATION COEFFICIENT FUNCTION OF FORM:

n mxyf : ; Nnm ,

It is argued that the limit of the coefficient of variation of the power function which passes through the diagonal of the table of limits of the

coefficient of variation of the function is given by the following formula:

12

)(lim

n

nx

x

n

x

And the general formula of the limits of the coefficients of variation of the table functions is given by:

Although this table was shown that Nnm , . However it holds that Rnm , .

Otherwise. If bm , then bban 2. The same property holds:

a

bxVC

bba b

x

2

.lim

TABLE LIMITS THE COEFFICIENT OF VARIATION OF THE FUNCTION OF THE FORM: n mxyf : ; Nnm ,

natural

numbers

root

Inverse

*2

*3

*4

1 1

1

1)(lim

111 2

xCVx1

1

2)(lim

221 22

xCVx2

1

3)(lim

331 32

xCVx3

1

4)(lim

441 42

xCVx4

2 1,41421356

2

1)(lim

112 2

xCVx

0,70710678

2

2)(lim

222 22

xCVx

1,41421356

2

3)(lim

332 32

xCVx

2,12132034

2

4)(lim

418 4xCVx

2,82842712

3 1,73205081

)(lim xCV

x

x

xCV )(lim 2

3

1=0,57735027

3

2)(lim

223 22

xCVx

1,15470054

3

3)(lim

333 32

xCVx

1,73205081

3

4)(lim

419 4xCVx

2,30940108

4 2

4

1)(lim

114 2

xCVx

0,5

4

2)(lim

224 22

xCVx1

4

3)(lim

334 32

xCVx

1,5

4

4)(lim

420 4xCVx2

5 2,23606798

5

1)(lim

115 2

xCVx

0,4472136

)(lim 2xCVx

)(lim 4xCVx =

5

2)(lim

225 22

xCVx

5

3)(lim

335 32

xCVx

1,34164079

5

4)(lim

421 4xCVx

1,78885438

)2()(.lim

nmn

mxVC

n m

x

2

5

2=0,89442719

6 2,44948974

6

1)(lim

116 2

xCVx

0,40824829

6

2)(lim

226 22

xCVx

0,81649658

6

3)(lim

336 32

xCVx

1,22474487

6

4)(lim

422 4xCVx

1,63299316

7 2,64575131

7

1)(lim

117 2

xCVx

0,37796447

7

2)(lim

227 22

xCVx

0,75592895

)(lim 6xCVx

)(lim 3xCVx =

7

3)(lim

337 32

xCVx

7

31,13389342

7

4)(lim

423 4xCVx

1,51185789

8 2,82842712

8

1)(lim

x

xCV

8

1)(lim

118 2

xCVx

0,35355339

8

2)(lim

228 22

xCVx

0,70710678

8

3)(lim

338 32

xCVx

1,06066017

8

4)(lim

424 4xCVx

1,41421356

9 3

9

1)(lim

119 2

xCVx

0,33333333

9

2)(lim

229 22

xCVx

0,66666667

9

3)(lim

339 32

xCVx1 )(lim 4xCVx

)(lim 8xCVx

9

4)(lim

4169 4xCVx

=1,33333333

10 3,16227766

10

1)(lim

1110 2

xCVx

0,31622777

10

2)(lim

2210 22

xCVx

0,63245553

10

3)(lim

3310 32

xCVx

0,9486833

10

4)(lim

41610 4xCVx

1,26491106

11 3,31662479

11

1)(lim

1111 2

xCVx

0,30151134

11

2)(lim

2211 22

xCVx

0,60302269

11

3)(lim

3311 32

xCVx

0,90453403

11

4)(lim

41611 4xCVx

1,20604538

12 3,46410162

12

1)(lim

1112 2

xCVx

0,28867513

12

2)(lim

2212 22

xCVx

0,57735027

12

3)(lim

3312 32

xCVx

0,8660254

12

4)(lim

41612 4xCVx

1,15470054

13 3,60555128

13

1)(lim

1113 2

xCVx

0,2773501

13

2)(lim

2213 22

xCVx

0,5547002

13

3)(lim

3313 32

xCVx

0,83205029

13

4)(lim

41613 4xCVx

1,10940039

14 3,74165739

14

1)(lim

1114 2

xCVx

0,26726124

14

2)(lim

2214 22

xCVx

0,53452248

14

3)(lim

3314 32

xCVx

0,80178373

14

4)(lim

41614 4xCVx

1,06904497

15 3,87298335

15

1)(lim

1115 2

xCVx

15

1)(lim 3

x

xCV =0,25819889

15

2)(lim

2215 22

xCVx

0,51639778

15

3)(lim

3315 32

xCVx

0,77459667

15

4)(lim

41615 4xCVx

1,03279556

16 4

16

1)(lim

1116 2

xCVx

0,25

16

2)(lim

2216 22

xCVx

0,5

16

3)(lim

3316 32

xCVx

0,75

16

4)(lim

41616 4xCVx1

17 4,12310563

17

1)(lim

1117 2

xCVx

0,24253563

17

2)(lim

2217 22

xCVx

0,48507125

17

3)(lim

3317 32

xCVx

0,72760688

17

4)(lim

41617 4xCVx

0,9701425

18 4,24264069

18

1)(lim

1118 2

xCVx

0,23570226

18

2)(lim

2218 22

xCVx

0,47140452

18

3)(lim

3318 32

xCVx

0,70710678

18

4)(lim

41618 4xCVx

0,94280904

19 4,35889894

19

1)(lim

1119 2

xCVx

0,22941573

19

2)(lim

2219 22

xCVx

0,45883147

19

3)(lim

3319 32

xCVx

0,6882472

19

4)(lim

41619 4xCVx

0,91766294

20 4,47213595

20

1)(lim

1120 2

xCVx

0,2236068

20

2)(lim

2220 22

xCVx

0,4472136

20

3)(lim

3320 32

xCVx

0,67082039

20

4)(lim

41620 4xCVx

20

4)(lim 4xCVx

0,89442719

21 4,58257569

21

1)(lim

1121 2

xCVx

0,21821789

21

2)(.lim 3 2xVC

x

21

2)(lim

2221 22

xCVx

0,43643578

21

3)(lim

3321 32

xCVx

0,65465367

21

4)(lim

41621 4xCVx

0,87287156

22 4,69041576

22

1)(lim

1122 2

xCVx

0,21320072

22

2)(lim

2222 22

xCVx

0,42640143

22

3)(lim

3322 32

xCVx

0,63960215

22

4)(lim

41622 4xCVx

0,85280287

23 4,79583152

23

1)(lim

1123 2

xCVx

0,20851441

23

2)(lim

2223 22

xCVx

0,41702883

23

3)(lim

3323 32

xCVx

0,62554324

23

4)(lim

41623 4xCVx

0,83405766

24 4,89897949

24

1)(lim

1124 2

xCVx

24

2)(lim

2224 22

xCVx

0,40824829

24

3)(lim

3324 32

xCVx

0,61237244

24

4)(lim

41624 4xCVx

0,81649658

3

24

1)(lim 4

x

xCV =0,20412415

25 5

25

1)(lim

1125 2

xCVx

0,2

25

2)(lim

2225 22

xCVx

0,4

25

3)(lim

3325 32

xCVx

0,6

25

4)(lim

41625 4xCVx

0,8

26 5,09901951

26

1)(lim

1126 2

xCVx

0,19611614

26

2)(lim

2226 22

xCVx

0,39223227

26

3)(lim

3326 32

xCVx

0,58834841

26

4)(lim

41626 4xCVx

0,78446454

27 5,19615242

27

1)(lim

1127 2

xCVx

0,19245009

27

2)(lim

2227 22

xCVx

0,38490018

27

3)(lim

3327 32

xCVx

27

3)(.lim 3 3xVC

x

0,57735027

27

4)(lim

41627 4xCVx

0,76980036

28 5,29150262

28

1)(lim

1128 2

xCVx

0,18898224

28

2)(lim

2228 22

xCVx

0,37796447

28

3)(lim

3328 32

xCVx

0,56694671

28

4)(lim

41628 4xCVx

0,75592895

29 5,38516481

29

1)(lim

1129 2

xCVx

0,18569534

29

2)(lim

2229 22

xCVx

0,37139068

29

3)(lim

3329 32

xCVx

0,55708601

29

4)(lim

41629 4xCVx

0,74278135

30 5,47722558

30

1)(lim

1130 2

xCVx

0,18257419

30

2)(lim

2230 22

xCVx

0,36514837

30

3)(lim

3330 32

xCVx

0,54772256

30

4)(lim

41630 4xCVx

0,73029674

31 5,56776436

31

1)(lim

1131 2

xCVx

0,1796053

31

2)(lim

2231 22

xCVx

0,3592106

31

3)(lim

3331 32

xCVx

0,53881591

31

4)(lim

41631 4xCVx

0,71842121

32 5,65685425

32

1)(lim

1132 2

xCVx

0,1767767

32

2)(lim

2232 22

xCVx

0,35355339

32

3)(lim

3332 32

xCVx

0,53033009

32

4)(lim

41632 4xCVx

0,70710678

33 5,74456265

33

1)(lim

1133 2

xCVx

0,17407766

33

2)(lim

2233 22

xCVx

0,34815531

33

3)(lim

3333 32

xCVx

33

3)(.lim 5 3xVC

x

0,52223297

33

4)(lim

41633 4xCVx

33

4)(.lim 3 4xVC

x

0,69631062

34 5,83095189

34

1)(lim

1134 2

xCVx

0,17149859

34

2)(lim

2234 22

xCVx

0,34299717

34

3)(lim

3334 32

xCVx

0,51449576

34

4)(lim

41634 4xCVx

0,68599434

35 5,91607978

35

1)(lim

1135 2

xCVx

35

1)(lim 5

x

xCV = 0,16903085

35

2)(lim

2235 22

xCVx

0,3380617

35

3)(lim

3335 32

xCVx

0,50709255

35

4)(lim

41635 4xCVx

0,6761234

36 6

36

1)(lim

1136 2

xCVx

0,16666667

36

2)(lim

2236 22

xCVx

0,33333333

36

3)(lim

3336 32

xCVx

0,5

36

4)(lim

41636 4xCVx

0,66666667

37 6,08276253

37

1)(lim

1137 2

xCVx

0,16439899

37

2)(lim

2237 22

xCVx

0,32879797

37

3)(lim

3337 32

xCVx

0,49319696

37

4)(lim

41637 4xCVx

0,65759595

38 6,164414

38

1)(lim

1138 2

xCVx

0,16222142

38

2)(lim

2238 22

xCVx

0,32444284

38

3)(lim

3338 32

xCVx

0,48666426

38

4)(lim

41638 4xCVx

0,64888568

39 6,244998

39

1)(lim

1139 2

xCVx

0,16012815

39

2)(lim

2239 22

xCVx

0,32025631

39

3)(lim

3339 32

xCVx

0,48038446

39

4)(lim

41639 4xCVx

0,64051262

40 6,32455532

40

1)(lim

1140 2

xCVx

0,15811388

40

2)(lim

2240 22

xCVx

0,31622777

40

3)(lim

3340 32

xCVx

102

3)(.lim 4 3xVC

x

0,47434165

40

4)(lim

41640 4xCVx

0,63245553

41 6,40312424

41

1)(lim

1141 2

xCVx

0,15617376

41

2)(lim

2241 22

xCVx

0,31234752

41

3)(lim

3341 32

xCVx

0,46852129

41

4)(lim

41641 4xCVx

0,62469505

42 6,4807407

42

1)(lim

1142 2

xCVx

0,15430335

42

2)(lim

2242 22

xCVx

0,3086067

42

3)(lim

3342 32

xCVx

0,46291005

42

4)(lim

41642 4xCVx

0,6172134

43 6,55743852

43

1)(lim

1143 2

xCVx

0,15249857

43

2)(lim

2243 22

xCVx

0,30499714

43

3)(lim

3343 32

xCVx

0,45749571

43

4)(lim

41643 4xCVx

0,60999428

44 6,63324958

44

1)(lim

1144 2

xCVx

0,15075567

44

2)(lim

2244 22

xCVx

0,30151134

44

3)(lim

3344 32

xCVx

0,45226702

44

4)(lim

41644 4xCVx

0,60302269

45 6,70820393

45

1)(lim

1145 2

xCVx

0,1490712

45

2)(lim

2245 22

xCVx

0,2981424

45

3)(lim

3345 32

xCVx

0,4472136

45

4)(lim

41645 4xCVx

0,59628479

46 6,78232998

46

1)(lim

1146 2

xCVx

0,14744196

46

2)(lim

2246 22

xCVx

0,29488391

46

3)(lim

3346 32

xCVx

0,44232587

46

4)(lim

41646 4xCVx

0,58976782

4

Data from the ages of 10 adult:

40

41

42

45

48

52

56

56

58

59

Average 49,7

standard Deviation 7,40945342

Coefficient of variation 0,14908357

EFFICIENCY E

47 6,8556546

47

1)(lim

1147 2

xCVx

0,14586499

47

2)(lim

2247 22

xCVx

0,29172998

47

3)(lim

3347 32

xCVx

0,43759497

47

4)(lim

41647 4xCVx

0,58345997

48 6,92820323

48

1)(lim

1148 2

xCVx

48

1)(lim 6

x

xCV =0,14433757

48

2)(lim

2248 22

xCVx

0,28867513

48

3)(lim

3348 32

xCVx

0,4330127

48

4)(lim

41648 4xCVx

0,57735027

49 7

49

1)(lim

1149 2

xCVx

0,14285714

49

2)(lim

2249 22

xCVx

49

2)(.lim 5xVC

x0,28571429

49

3)(lim

3349 32

xCVx

0,42857143

49

4)(lim

41649 4xCVx

0,57142857

50 7,07106781

50

1)(lim

1150 2

xCVx

0,14142136

50

2)(lim

2250 22

xCVx

0,28284271

50

3)(lim

3350 32

xCVx

0,42426407

50

4)(lim

41650 4xCVx

0,56568542

nmn

mxE n m

2

nmna

bban

bm

2

2

a

bxE

bba b

2

a

bCV

5

Example: Taking the CV of the above data:

Proof:

x 5,7817794x DESVESTA PROMEDIO CV

1 1 2 1,12736692 0,09006201 1,06368346 0,08466994

3 1,20926512 0,10545262 1,11221068 0,09481353

4 1,27095616 0,11710481 1,15189705 0,10166257

5 1,32096664 0,12649924 1,18571097 0,1066864

6 1,36328548 0,1343767 1,21530672 0,11057019

7 1,40012154 0,14116332 1,24170884 0,11368472

8 1,43283393 0,14712735 1,26559947 0,11625112

9 1,46232212 0,15244877 1,28745754 0,11841072

10 1,48921408 0,15725427 1,3076332 0,1202587

11 1,5139666 0,16163642 1,32639078 0,12186184

12 1,53692295 0,1656649 1,34393513 0,12326853

13 1,55834801 0,16939352 1,36042843 0,12451483

14 1,5784507 0,17286465 1,37600145 0,12562824

15 1,59739888 0,17611227 1,39076128 0,12663012

16 1,61532957 0,17916406 1,40479679 0,12753735

17 1,63235615 0,18204286 1,41818264 0,12836348

18 1,64857358 0,18476769 1,43098213 0,1291195

19 1,66406221 0,18735465 1,44324951 0,12981446

20 1,67889069 0,18981739 1,45503157 0,13045586

21 1,69311813 0,19216765 1,46636902 0,13104999

22 1,70679586 0,19441557 1,47729751 0,13160218

23 1,71996871 0,19656997 1,48784844 0,13211693

24 1,73267609 0,19863859 1,49804959 0,13259814

25 1,74495286 0,20062822 1,50792572 0,13304914

26 1,75682999 0,2025449 1,51749896 0,13347284

27 1,76833513 0,20439398 1,52678919 0,13387178

28 1,77949309 0,20618025 1,53581433 0,13424816

29 1,7903262 0,207908 1,5445906 0,13460395

30 1,80085464 0,20958109 1,55313273 0,13494088

31 1,81109673 0,21120303 1,56145415 0,13526047

32 1,82106911 0,21277697 1,56956712 0,13556411

33 1,830787 0,21430581 1,57748287 0,13585302

34 1,84026432 0,21579216 1,58521174 0,13612829

35 1,84951384 0,21723843 1,59276323 0,13639091

36 1,85854731 0,21864682 1,60014612 0,13664178

37 1,86737559 0,22001936 1,60736854 0,13688171

38 1,87600868 0,22135791 1,61443802 0,13711143

39 1,88445589 0,2226642 1,62136155 0,13733162

40 1,89272582 0,22393983 1,62814566 0,13754287

41 1,90082649 0,22518626 1,63479641 0,13774575

42 1,90876537 0,22640488 1,64131948 0,13794077

43 1,91654943 0,22759696 1,64772018 0,13812841

44 1,92418518 0,22876369 1,65400347 0,13830908

45 1,93167874 0,22990619 1,66017403 0,13848319

46 1,93903581 0,2310255 1,66623625 0,13865111

47 1,94626178 0,23212258 1,67219424 0,13881317

48 1,9533617 0,23319835 1,67805189 0,13896969

49 1,96034031 0,23425368 1,68381288 0,13912097

a

b0,14908357 244,9925319 ba

44,9925319

11144,9925319 2

xE

0,1490835744,9925319

15,7817794xE

6

50 1,96720212 0,23528936 1,68948067 0,13926727

51 1,97395135 0,23630615 1,69505852 0,13940884

52 1,98059201 0,23730477 1,70054955 0,13954593

53 1,98712787 0,2382859 1,70595669 0,13967875

54 1,99356252 0,23925017 1,71128272 0,13980751

55 1,99989937 0,24019819 1,7165303 0,13993239

56 2,00614164 0,24113053 1,72170193 0,14005358

57 2,01229239 0,24204772 1,72680001 0,14017125

58 2,01835453 0,24295029 1,73182681 0,14028556

59 2,02433083 0,24383871 1,7367845 0,14039664

60 2,03022394 0,24471346 1,74167516 0,14050465

30630 5,96913129 0,75849677 5,08904491 0,14904501

30631 5,96916499 0,75850106 5,08907365 0,14904502

30632 5,9691987 0,75850535 5,08910238 0,14904502

30633 5,9692324 0,75850964 5,08913111 0,14904502

30634 5,9692661 0,75851393 5,08915984 0,14904502

30635 5,96929981 0,75851822 5,08918857 0,14904502

30636 5,96933351 0,7585225 5,0892173 0,14904502

30637 5,96936721 0,75852679 5,08924603 0,14904502

30638 5,9694009 0,75853108 5,08927475 0,14904502

30639 5,9694346 0,75853537 5,08930348 0,14904503

30640 5,9694683 0,75853965 5,08933221 0,14904503

30641 5,969502 0,75854394 5,08936093 0,14904503

30642 5,96953569 0,75854823 5,08938966 0,14904503

30643 5,96956939 0,75855252 5,08941838 0,14904503

30644 5,96960308 0,7585568 5,0894471 0,14904503

30645 5,96963677 0,75856109 5,08947583 0,14904503

30646 5,96967046 0,75856538 5,08950455 0,14904503

30647 5,96970415 0,75856966 5,08953327 0,14904503

30648 5,96973784 0,75857395 5,08956199 0,14904504

30649 5,96977153 0,75857824 5,08959071 0,14904504

30650 5,96980522 0,75858252 5,08961943 0,14904504

30651 5,96983891 0,75858681 5,08964814 0,14904504

30652 5,96987259 0,7585911 5,08967686 0,14904504

30653 5,96990628 0,75859538 5,08970558 0,14904504

30654 5,96993996 0,75859967 5,08973429 0,14904504

30655 5,96997365 0,75860395 5,089763 0,14904504

30656 5,97000733 0,75860824 5,08979172 0,14904505

30657 5,97004101 0,75861252 5,08982043 0,14904505

30658 5,97007469 0,75861681 5,08984914 0,14904505

30659 5,97010837 0,75862109 5,08987785 0,14904505

30660 5,97014205 0,75862538 5,08990656 0,14904505

30661 5,97017573 0,75862966 5,08993527 0,14904505

30662 5,9702094 0,75863395 5,08996398 0,14904505

30663 5,97024308 0,75863823 5,08999269 0,14904505

30664 5,97027675 0,75864252 5,0900214 0,14904505

30665 5,97031043 0,7586468 5,09005011 0,14904506

30666 5,9703441 0,75865109 5,09007881 0,14904506

30667 5,97037777 0,75865537 5,09010752 0,14904506

30668 5,97041145 0,75865966 5,09013622 0,14904506

30669 5,97044512 0,75866394 5,09016492 0,14904506

30670 5,97047879 0,75866822 5,09019363 0,14904506

30671 5,97051246 0,75867251 5,09022233 0,14904506

30672 5,97054612 0,75867679 5,09025103 0,14904506

30673 5,97057979 0,75868108 5,09027973 0,14904507

30674 5,97061346 0,75868536 5,09030843 0,14904507

30675 5,97064712 0,75868964 5,09033713 0,14904507

30676 5,97068079 0,75869393 5,09036583 0,14904507

30677 5,97071445 0,75869821 5,09039453 0,14904507

30678 5,97074811 0,75870249 5,09042322 0,14904507

7

30679 5,97078177 0,75870678 5,09045192 0,14904507

30680 5,97081543 0,75871106 5,09048061 0,14904507

30681 5,97084909 0,75871534 5,09050931 0,14904507

30682 5,97088275 0,75871962 5,090538 0,14904508

30683 5,97091641 0,75872391 5,09056669 0,14904508

30684 5,97095007 0,75872819 5,09059538 0,14904508

30685 5,97098372 0,75873247 5,09062408 0,14904508

30686 5,97101738 0,75873675 5,09065277 0,14904508

30687 5,97105103 0,75874104 5,09068146 0,14904508

30688 5,97108469 0,75874532 5,09071014 0,14904508

30689 5,97111834 0,7587496 5,09073883 0,14904508

30690 5,97115199 0,75875388 5,09076752 0,14904509

30691 5,97118564 0,75875816 5,09079621 0,14904509

30692 5,97121929 0,75876244 5,09082489 0,14904509

Conclusion: the limit of the coefficient of variation of the data tends to infinity the number obtained by formula.

Emil Núñez Rojas: noverbal@hotmail.com

Face book: Emil Francisco Núñez Rojas