8342937121

Pertanika 14(1). 77-81(1991)

COMMUNICATION II

Detemiination of Best Straight Linę Graphs in the Undergraduate Science Laboratory

ABSTRAK

Satu huraian dibeńkan tentang kaedah penentuan kecerunan m dnu penggalan c bagi satu gańs lurus, a pa bila satu dańpada pembolehubah adalah bebas ralat dan satu lagi mempunyai ralat. Berdasarkan letni i ni. satu program

komputer ditulis untuk mengira parameter-parametergaris ini, mitu m dan c, dengan ralat mereka dan x~rut untuk kes-kes data masukan yang diketahui ralatnya. Data yang diperoleh dańpada ujikaji pzik nuklear (di mana data mempunyai ralat yang tidak tetap) digunakan untuk menguji kaedah ini, dan ia didapati brrguna dałam penentuan parameter-parc meter gańs lurus terbaik, seria ralat piawai mereka. Kegunaan kaedah ini untuk ujikaji fizik bukan nuklear di pmnghat prasiswazah (atau untuk data yang mempunyai ralat tetap) juga dibincangkan.

ABSTRACT

A descńption is given of the method of determining the gradient m and intercept c of a straight linę when one yańable is error-free, while the other is subject to error. Based on this theory, a Computer program was wńtten to compute the para-

meters oj the linę, i.e. m and c, with their standard errors, and cal for those cases where the errors oj the input data are known. Data obtainedfrom nuclear physics experiments (where data have non-constant errors) were used to test this method, which was found useful in determining the pararneters of the best straight linę, and their standard errors. The usefulness of this method for nonnuclear physics u ndergraduate expen ments (or for data with constant errors) is also discussed.

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INTRODUCTION

The increased availability of computers in undergraduate science iaboratories has madę it morę desirabie to use a rigorous procedurę lor computing the pararneters of a straight linę. preferably in conjuction with a plotdone “by band”. In most cases. the gradient of the linę will not differ appreciably between the machinęcomputation and the manuał plot. The computed value, however. can morę easily be given a standard error estimate, which is morę reliable than that detcrmined from a manuał plot. In addition, the consistency of the data - specifically, whether the deviations of the data points from the straight linę are consistenl with the estimated errors can be assessed using the

O

X' test. The mathematical principles underlying the method should be readily understood by science students.

MATHEMATICAL METHODS AND DATA

Consider an experimcnt which produces dala in the form of n pairs of readings (x, y.), i= 1,2, ...n . In this experiment, two things are known, namely that the relationship between x. andy. is a straight linę, and that there is no error in x.. Since only the -y. are liable to error, corresponding toeach y., there

must be a “theoretical” value, E, which is the value which would have been measured in a perfect experiment (one without error). The relationship

between ) and x. is

i    i

V = rnx. + r, i = 1,2,

i    i    *

where values of the constants m and c are to be found for which the theoretical values E will be as

i

close as possible to the experimental datay.. Agood generał assumption is that the each experimental value belongs to a Gaussian distribution centredon the theoretical value. The probability that a particular experimental value, e.g. y₄, should lie in

a smali interval dy. in the neighbourhood of >’ is

P(yi) dy; = const x exp where (J is the standard error in the value y.. The