LEAST SQUARES, Method of. A mathematical process for treating the results of scientific observation so as to free them as far as possible from the effect of error.
The familiar idea of measurements either regards them all as perfectly exact when made, or at least sufficiently near perfect exactness. For instance, a carpenter wishing to know the width of a room simply measures with his two foot rule, and notes the size just as it came from the measurement. But in scientific work, where the highest degree of accuracy is neces sary, we must do much more than this. The carpenter's measurement will doubtless be cor rect within an and if he is a particularly careful workman, it will be correct within a fraction of an inch. But if it were necessary to know the width of that room within a hun dredth of an inch very different processes of measurement and very much more accurate tools would be essential. If it were required to attain a degree of exactness within one thousandth of an inch, it might perhaps happen that no tools or method of measurement could be devised which would accomplish the desired result.
It is obvious on reflection that since human measurements must always depend ultimately upon fallible human senses, so there must al ways exist a limit of precision beyond which it is not possible to go. It happens almost always that the degree of precision required in scientific measurements is somewhere near this unattainable limit; so that it is especially' in scientific work that we have to resort to some method adapted to diminish as far as possible the harmful effects of those small errors which thus always result from the fallibility of human senses. The Method of Least Squares was perfected for this purpose.
It is clear from the above that this method is pre-eminently a practical process; it can per haps be understood, best by approaching it from the point of view of a concrete practical exam ple. For instance, it is well known that a bar of iron changes its length with every change of temperature. If the bar is heated it becomes. longer; when cooled it becomes shorter. Now suppose an experimenter in a physical labora tory desires to determine the effect of tempera ture changes upon the length of such an iron bar. What he does is extremely simple. Ap
paratus is provided for varying the temperature of the bar by alternate heating and tooling, and additional apparatus for measuring its length very accurately at various temperatures. At the same time some form of thermOmeter is used to make certain that we know temperature of the bar at the various stages of the experiment.
Let us introduce the following notation: le- length of the bar at some assumed tem perature t., 1= length of the bar at some other tempera ture t.
Now if we are willing to assume that the length varies uniformly with changes of tem perature we may write, x= increase in length of bar per degree of heating.
We shall then have the equation, (1) + (t te) x.
The quantities appearing in this equation are of three kinds with respect to our edge of their numerical values: 1. Unknown quantities; viz.: x and h.
Z A quantity known from observation, and therefore subject to errors brought about by the fallible human senses; viz.: 1.
3. A coefficient which we may safely assume to be known, with practically complete accuracy from the thermometer readings; viz.: t to.
Let us now rewrite equation (1) in a slightly different form, and repeat it several times with subscript numbers, so as to distinguish e9ua tions depending on the successive observations at various temperatures. We have: le+ (4 te)x li 0,(2) 4+ (4-- 4)x 0, 4+(4-.-.4)x le= 0, etc. etc.
It is at once evident that as we have only two unknown quantities (le and .r) we can com pute their numerical values from any twilequa dons of the group (2) by the ordinary processes of elimination given in elementary algebras. But which two equations shall we select for this purpose? We might use the first two, for instance, or the last two, or the first equation and the last. If we try the actual process numerically upon a real example we generally find that the values of the unknowns come out different from the various combinations of equations. It was not until the Method of Least Squares was introduced that scientific men had a better way of computing their ob servations than the crude one of solving a great many different combinations and taking the average of the results so obtained.