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Table 16. Methods of Averaging Data

# Table 16. Methods of Averaging Data

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OF AVERAGING D A T A (continued)

T A B L E 16.-METHODS

38

of which, as representative bracketed L 1 coefficients, we have

[atat] = alal azaz a3a3 .. .a,,a,

[acbrl= a161 a262 a3b3 . .anBn

[ a l X l l = a,X, azXz aaX3 . .anXn

+ + +

+ + +.

+ + +.

(3)

.....................................

[ktatl = klal + hzaz + k3a3+ ...knan

Solutions of equation (2) yield'the least-squares adjusted values of

Qi,

Qz...ex.

For unequally weighted values of X , that is wl, wz,. . .wnfor X , X z . . .Xn, the rrornral

equations become

+

+

[ w t a ~ a t l Q ~ [wtatbtlQ2

[ ~ i b c a < l Q i [wtDtbtlQZ

+ [zvtatctIQ3+.. . [ W C ~ ~ -~ ~[wcatXtl

I Q X =0

+ [ ~ , b t ~ i l Q+..

3 .[ ~ r b t l ~ ~ l [Q~ ~t b t X i=

l 0

(4)

.....................................................................

+

+

+.

~w~k~aclQ

[ wl l h ~ b t l Q z I W ~ ~ ~ C ..~twthtktlQkI Q ~

of which

[ w t a d = zphalal

[zv~acbtl

= walbl

+ w z ~ a+z w3a3a3+ ...wnana8,

+ zfia2bY+ w3a3b3+. . .twnanbn

IwtkcXtl = 0

(5)

............................................

[wtk+atl= wlklal + wIkza2+ w3ksar+. . .wnknan

The weights wl, m . . .w,, associated with the Xi, X Z . . . X , and with the successive observation equations are taken as inversely proportional to the squares of the probable

errors (or of the standard deviations) of the corresponding X's. It is customary to take

simple rounded numbers for the proportional values. A precise set of 28, 50, 41, and 78

may be rounded to 3, 5, 4, and 8.

As a simple application, consider the elevations of stations B, C, and D above A. Let

those elevations in order be Q1, Q2,and Q3. Let the quantities measured and the observed

elevations be such as to yield the following observation equations :

Qz - Q 3 -12 ft = A5

Qi - Q3 - 5 ft = A6

Th coefficients al, b ~ and

,

are obvious. Substitution

are seen to be 1, 0, and 0. The values of the other coefficients

equation (2) yields for the normal equations

3Qz- Q a Q36ft=O

(7)

- Qi 3 Q 2 - Q 3 - 39 ft = O

- Qi - Q z 3Q3 13 ft = 0

+

+

+

Solutions of equation ( 7 ) yield 91 ft, 174 ft, and 44 ft for the elevations of B, C, and D

above A.

P a r t 2.-Least-squares

+ bx,

equations of the type y = a

observed (x,y) values

to represent a series of

For equally weighted pairs of (x,y) of which the errors of measurement are associated

with the determinations of the y's

of which

SMITHSDNIAN PHYSICAL TABLES

T A B L E 16.-METHODS

OF A V E R A G I N G D A T A (concluded)

39

The probable errors of the a and the b of equation (8) are given by

For unequally weighted measurements of which the errors of measurement are associated with the determinations of the y's,

Z w l x i l Z w , y i - Z w ix & v i xt y i

a=

Z z w Z w l x l z - (Zzehxr ) Z

Where the erroi s of measurement are associated with the x-determination only, the corb'y can be obtained by merely

responding coefficients of an equation of the type x = a'

interchanging x and y in equation ( 8 ) .

Where the errors of measurement are associated with both the x - and the y- determinations, the expressions are complicated."

+

Worthing, A. G . , and Geffner, J., Trcatment of experimental data, p. 259, John Wiley and Sons,

New York, 1943. Used by permmion.

+

+ cxz + dx3 to

equation of the type y = a

bx

series o f observed (x, y ) values

P a r t 3.-Least-squares

represent a

For the general case involving irregularly spaced x-values, the formulae for a, b, c , etc.,

are very complex." However, for the case of equally weighted observations with errors

of measurement associated entirely with the y-values in which succeeding x-values are

equally spaced, the mechanics of the computations for least-squares constants are very

greatly simplified, thanks to tables computed by Baily and by Cox and Matuschak.Ia The

procedure requires a change of the x-variable to yield a new X-variable with a zero-value

at the midpoint of the series. I n case of an even number of terms, the shift is given by

x-x

4x

-

X,= -

(11)

of which Ax is the even spacing between successive x-values; and, if the number of terms

is odd, the shift is given by

-

x o = x4x/2

--x

(12)

The further procedure consists in determining the appropriate summations indicated in

Table 17, the appropriate k-terms given as a function of the number of terms n in Tables 19

and 20, combining the appropriate summations and k-terms, to give parameters for the

equation y = f ( X ) , and finally transferring the function to the original coordinate system

to yield y = f i ( x ) .

How to apply the simplified procedure to determine the coefficient of x2 in the leastcxz to represent the xy values of the first two columns of

squares equation y = a bx

the following tabulations is shown in the remainder of the tabulation.

+ +

X

L

(set)

(cm)

3

6

9

12

15

18

12.0

20.6

33.7

51.1

72.9

99.1

_-

x

-5

-3

-1

1

3

+

+

+5

289.4

-X2Y

(cm)

300.0

185.4

33.7

51.1

656.1

2477.5

-__

3703.8

C' = k5ZX2y - krZy

n=6

k5 = 16,741,071 X

k 4 = 19,531,250X lo-*

kJZX2y= 6.2005 cm

krZy = 5.6523 cm

c' = 0.5482 cm

Ax = 3 sec

c = 4c'i ( A x ) = 0.244 cm/sec2

67, 1YZI; Worthing, A. G . ,

I4 Birge, R . T., and Shea, J. D., Univ. California Puhl. Math., vol. 2,

and Geffner. J., Treatment of experimental data, p. 250, John Wiley and gins, New York, 1943.

Baily, J. L., Ann. Math. Statistics, vol. 2, p. 355, 1931.

'"Cox, G. C., and Matuschak, Margaret, Journ. Phys. Chem., vol. 45, p. 362, 1941.

SMITHSONIAN PHYSICAL TABLES

40

T A B L E 17.-SHOWING

T H E MAKE-UP O F T H E CONSTANTS O F T H E LEASTSQUARES EQUATION O F T H E T Y P E y = a

bx

cx2 dxS FOR EQUATIONS OF VARYING DEGREES W H E N T H E ABBREVIATED M E T H O D O F

BAILEY AND O F COX AND MATUSCHAK IS U S E D *

+ +

+

This method is applicable only when succeeding values of x have a common difference

and a r e equally weighted. T h e independent variable, changed if necessary, must have a

zero value at the midpoint of the series with succeeding values differing by unity if the

number of terms is odd and by two if even. Values for the various k's, as computed by

Cox and Matuschak, are to be found in Tables 14 and 20.

.

J'or

references, see footnotes 15 and 16, P. 39.

L U ES O F P =

TABLE 18.-VA

s

P, the probability of an observational error having a value positive or negative equal to

hZ

or less than x when h is the measure of precision, P =

e-'"''zd(hx) * I t a = (tntax')

v'T

0

where nz = no. obs. of deviation A x .

hx

0.0

.I

.2

.3

.4

0.5

.6

.7

.8

.9

1.o

.1

.2

.3

.4

1.5

.6

.7

.8

.9

2.0

.I

.2

.3

.4

2.5

.6

.7

.8

.9

0

1

4

,01128 .02256 .03384 .04511 ,05637 ,06762

.11246 ,12362 .I3476 .I4587 .I 5695 .16800 .I7901

,22270 .23352 24430 .25502 .26570 .27633 .28690

.32863 .33891 .34913 .35928 .36936 ,37938 ,38933

.42839 .43797 .44747 .45689 .46623 .47548 ,48466

7

8

9

.07886 .09008 .lo128

.I8999 .20094 .21184

,29742 ,30788 ,31828

,39921 .40901 ,41874

,49375 .SO275 51167

2

3

5

6

.52050

,60386

.67780

.74210

.79691

.52924

.61168

,68467

.74800

.80188 ,80677

3 4 9 4 .56332 ,57162

.63459 ,64203 ,64938

.70468 .71116 .71754

.76514 ,77067 .77610

,81156 .81627 32089 ,82542

,57982 ,58792 .59594

,65663 .66378 ,67084

.72382 ,73001 ,73610

,78144 ,78669 .79184

,82987 .83423 ,83851

.84270

.88021

.91031

.93401

,95229

.84681 .85084

.a353 .88679

.91296 .91553

,93606 .93807

.95385 .95538

3.5478 .85865

188997 .89308

.91805 .92051

.94002 .94191

.95686 ,95830

.86977

90200

.92751

.94731

,96237

.86244

,89612

,92290

,94376

.95970

,86614

.89910

.92524

,94556

,96105

.96611 ,96728

,97635 .97721

.98379 .98441

.98909 .98952

39279 ,99309

.96841

.97804

.98500

.98994

.99338

.96952

.97884

.98558

.99035

,99366

.97059

.97962

.98613

.99374

.99392

,99532

.99702

,99814

.99886

.99931

.99572

,99728

,99831

.99897

.99938

99591

.99741

.99839

.99902

.99941

.99609 ,99626 .99642

9 7 5 3 .99764 ,99775

,99846 .99854 ,99861

.99906 9 9 11 9 9 1 5

.99944 .99947 .99950

,99552

,99715

.99822

.99891

.99935

,97162 .97263

.98038 .98110

,98667 .98719

,99111 ,99147

.99418 ,99443

.87333

,90484

,92973

.94902

.96365

,87680

90761

,93190

.95067

.96490

,97360 ,97455 ,97546

.98181 ,98249 ,98315

.98769 .98817 .98864

,99182 .99216 ,99248

,99466 .99489 ,99511

,99658

.99785

.99867

.99920

,99952

,99673

,99795

,99874

.99924

,99955

.99688

,99805

,99880

.99928

.99957

99.59 ,99961 .99963 .99965 .99967 ,99969 .99971 .99972 .99974 .99975

99976 .99978 ,99979 .99980 .99981 .99982 ..99983 .99984 ,99985 .99986

9 9 8 7 .99987 ,99988 .99989 .99989 .99s50 . 9 W 1 ,99991 ,99992 .99992

.99992 .99993 ,99993 .99994 .99994 .99994 .99995 .99995 .99995 ,99996

99996 S9996 999% .99997 ,99997 .99997 .99997 ,99997 .99997 .99998

.99998

3.0

-

.99999

.99999 1.00000

SMITHSONIAN PHYSICAL TABLES

v)

5

4

B

T A B L E 19.-VALUES

O F T H E CONSTANTS, k,, E N T E R I N G LEAST-SQUARES SOLU TION S, U SIN G T H E A B B R E V I A T E D

M E T H O D O F B A I L Y A N D O F COX A N D M A T U S C H A K , W H E N T H E N U M B E R OF TER MS, n, IS O D D *

z

z

D

Q

5

T h e numbers in parentheses show the negative powers of 10 by which the adjacent numbers must he multiplied in order to obtain appropriate 12"'s.

To illustrate, 1 : ~for I G = 13 is 54,945,055 x lo-''.

ka

4

D

I

Im

(D

3

- 5

n 7

9

11

k4

kx

h

kG

3333 3333(8)

2000 0000

1428 5714

1111 1111

9090 9091(9)

so00 OOOO(8)

1000 0000

3571 4286(9)

1666 6667' '

9090 9091(10)

1000 OOOO(7)

4857 1429(8)

3333 3333

2554 1126

2074 5921

1000 0000(7',

1428 5714(6,

4761 9048(9)

2164 5022

1165 5012

1500 OOOO(7)

7142 8571(9)

1190 4762

3246 7532(10)

1165 5012

9027 7778(8)

2625 6614

1143 3782

6037 9435(9)

2361

3240

8277

2881

1111(8)

7407(9)

2166(10)

3779

6944 4444(9)

4629 6296(10)

7014 5903(11)

1618 7516

13

15

17

19

21

7692

6666

5882

5263

4761

5494 5055

3571 4286

2450 9803

1754 3860

1298 7013

1748 2517

1511 3122

1331 2693

1189 7391

1075 5149

6993 0070(10)

4524 8869

3095 9752

2211 4109

1634 5211

4995 0050(11)

2424 0465

1289 9897

7371 3696(12)

4457 7848

3584 6098

2304 5899

1570 2041

1118 3168

8248 9 7 0 ( 10)

1214 0637

5830 6799 ( 11)

3081 6420

1752 5617

1056 2015

4856 2549(12)

1745 7125

7166 6093(13)

3257 5497

1605 1694

23

25

27

29

31

4347 8261

4000 0000

3703 7037

3448 2759

3225 8065

9881

7692

6105

4926

4032

9813

9024

8352

7774

7270

6646(9)

1546

4904

0700

7048

1242 2360

%51 8357(11)

7662 8352

6179 7058

5056 1230

2823 2637

1858 0453

1263 1047

8828 1512(13)

6320 1537

6259

4862

3852

3104

2538

6672

4382

2974

2076

1485

8445

4692

2728

1650

1032

33

35

37

39

41

3030

2857

2702

2564

2439

3342 2460

2801 1204

2370 7918

2024 2915

1742 1603

6828

6437

6088

5775

5493

6552

3464

5061

5692

2589

4189

3510

2970

2535

2181

4620 6166

3441 1799

2605 2658

2001 6066

1558 2829

2102 4471

1760 7811

1489 3734

1271 0408

1093 4097

1084 7991

8073 4407(13)

6108 7522

4691 0081

3650 4910

6655 2091(15)

4402 0942

2979 8791

2059 2661

1449 7581

43

45

47

49

51

-

2325 5814

2222 2222

2127 6596

2040 8163

1960 7843

1510 1178

1317 5231

1156 3367

1020 4082

9049 7738(12)

5237

5004

4790

4595

4414

2849

1234

8525

0295

5960

1890 7166

1649 3485

1447 3875

1277 1066

1132 5285

1227 7380

9778 7451 (14)

7866 2362

6385 5329

5227 0545

9474

8263

7250

6396

5671

2875 1015

2289 2527

1841 0171

1494 1103

1222 7830

1037 9428

7545 3288( 16)

5561 9852

4152 6134

3136 9497

3077

6667

3529

1579

9048

3030

1429

7027

1026

0244

4229( 11)

3077

0061

1084

2581

For references. see footnotes 15 and 16. 1). 39

3590

0035

0030

3684

5961

0791

3545

7423

7316

6983

1490(11)

1159

1033

2170

3855

0719(12)

3595

5336

4076

0296

6606(14)

0337

9299

5625

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Table 16. Methods of Averaging Data

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