What is X² ?

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When we hear about "least squares," it is  X� that is being minimized. What is it? And how can I relate its value to a "goodness-of-fit" that I can understand?

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If we are trying to fit some data, y, to a known function, f(x), we can write chi-squared as (See Numerical Recipes in C, Chapter. 15) 

      (1)

In this equation, is the standard deviation of measurement i, which may be different for each point measured. The problem with X� as written is that its value depends on the number of points used!  Simply duplicating all of your data points doubles the value of X�!  Consequently, I prefer to look at X�/(n-m), where m is the number of adjustable parameters in the fit, and (n-m) is the number of degrees of freedom. If the estimates for are reasonable, then X�/(n-m) should approximate unity, regardless of the number of data points, and different data sets are easily compared.

In many electrochemical experiments, the current varies widely over the course of the experiment, and it is measured by a potentiostat that is used in the autoranged mode. This is generally true of a Tafel experiment in a corrosion application, or in an EIS measurement over several decades of frequency. Many of the instrumental sources of noise (e.g., quantization noise at the ADC, or amplifier noise in the signal chain) lead to errors which are (approximately) a fixed fraction, , of the full scale current. Because the autoranging algorithm strives to keep the measured current (y_i) close to the full scale current, we may write: . Further, we see that with this assumption, weighting each of the points by y_i allows us to recast equation (1) as 

(2)

Minimizing E² is the same as minimizing X² with this weighting assumption.  However, taking the square root of E² gives the relative error in the measured current (i.e., ! ) That is a number most of us can understand! A value of E² of 1e-4 translates to an average 1% error in the measured y values. [ sqrt( E² ) = sqrt( 1E-4 ) = 0.01 ==> 1% ]
 

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