in the summation (21) should equal, on average, ~1a , but note the a in the denominator; the deviations are expressed in units of a. One such term should, on average, thus be about unity. Summation over the n terms yields a x 2 value of about n. Thus if there were 20 data points, we might expect x 2 to lie somewhere in a broad band of values about x 2 ~ 20 if the model and our error bars are both correct. If, instead, we find x 2 ~ 40, the deviations of our data would be unreasonably large. If x2 ~ 5, the deviations would be unreasonably small.

It is actually the number of degrees of freedom, f = n — p, that comes into play here. Suppose we are fitting a straight line, with its two parameters, a and b, and have only two data points. In this case the best-fit line would go exactly through both points, and we would have x 2 = 0. It is only a third data point that, with the first two, begins to yield a non-zero x 2. Thus in our example, we would have f = 20 — 2 = 18, and we would expect x2 to lie in a broad band of values about 18, more or less. This is in accord with the probability limits 0.1 to 0.9 for satisfactory data in Fig. 11.

Astronomers often use the reduced chi square xV = x 2/f, a value expected to be near unity for the correct model. How close to unity depends critically on f. For satisfactory fits, namely probabilities in the range 0.1-0.9, one requires 0.49 <xV <1.6 if f = 10, but if f = 200, one requires 0.87 <xV < 1.13. If there are a lot of data points, one expects x2 to be very close to unity. In either case, if your data do not fall within the prescribed probability limits, try another model or check your assumptions on systematic errors.

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