Appendix A: The rms error for the zero-point data

Suppose that for star i, there are n_i > 1 zero-point data. Here one uses a standard expression to calculate

$$v_i = (n_i - 1)^{-1} \sum_{j=1}^{n_i} (\mkern2mu f_j - \langle f_j \rangle \mkern2mu )^2.$$ (A1)

f_j is the jth value of [Fe/H] for star i, and the variance v_i of the values of [Fe/H] is the square of the standard deviation per datum.

Since v_i is calculated from a finite data set, v_i itself has a finite variance. If the f_j are normally distributed, the variance of v_i is inversely proportional to $\nu_i \equiv n_i-1$ (see Keeping 1962, Eq. [5.11.14], p. 110). Inverse-variance weighting will then yield a weight which is proportional to $\nu_i$, so the expression for the mean variance becomes

$$v = \sum_{i=1}^N \nu_i v_i \Biggm/ \sum_{i=1}^N \nu_i,$$ (A2)

with N being the total number of contributing stars. The associated number of degrees of freedom is given by

$$\nu_0 = \sum_{i=1}^{N_i} n_i - N.$$ (A3)

In part, this analysis is described here because it does not appear to be common knowledge that n_i-1 weighting should be used in this context.