Least squares for different experimental cases
Vega Carrillo, Héctor René · 1989
Ve discuss the problem of fitting an experimental data sel into a linear curve when experimental uncertainties can not be overlooked. Difficulties with the standard least-squares method are pointed out. An alternative weighted least-squared method, depending on knowledge about the uncertainties ror fouT different cases.is prc. sented, and we use it to tl.nalyzc the data from a particular cxperiment. From this application it is shown that we get better results using the proper ra.se. \Ve want to point out a widespread error in the use of linear regrcssion analysis when, in experimental situations, the experimentalist wants to determine the functional relationship betwccn tIJe experimental data. The hroad use of pockct programmable ca1culators and statistical software for microcompllters makes easicr the hcavy work neccssary to obtain such relationship bet\\'cen the experimental data. Unfortunatcly, the most common analysis tcchnique available in lhe least square fitting of a line is not always lhe most proper. Experiments in physics made to determine pararneters through lhe functional relationship between valucs of x and y involvc a series of experimental Illcasurements of x and lhe corresponding y. In several cases there are not only measuremen:; errors in Yi, bul also there are measurcrnenl eTrors in Xj. Many experimentalists apply lhe slandard Ica.sl-squares mclhod, which implicitly assumes errors neilber in Xi ilOr in yi. Accordingly, this procedure affects the llnknown parameters to be obtained from the functionaI rclationship, and it givcs estimalcs of errors that are smaller than the truc errors. This paper presents a review of standard least-squarcs rnethod as applied to a slraight line, and the weighted least-squarcs method, ,here the weighting factor is related to the experimental data precision. This gives place to four different experimental cases; cach one of thcse cases are applied to the same experimental data sel and the resulls of this weighlcd lcast-square fitting method oC a straight Iine are discussed as wcll.