Statistics of repeated measurements

For $N$ measurements $x_i$:

$s$ describes the scatter of single measurements, while $\sigma_{\bar x}$ is the uncertainty of the mean. Use $N-1$ (the sample standard deviation), as the IOAA papers do: one degree of freedom is used up by computing $\bar x$.

In 2017 D1, five LMC Cepheids give distances with $\bar d_V = 46.2$ kpc and $s_V = 4.1$ kpc, so the mean is good to $4.1/\sqrt5 = 1.8$ kpc.

When measurements have different uncertainties $\sigma_i$, weight each by $w_i = 1/\sigma_i^2$:

A measurement twice as precise counts four times as much. If one point has a tiny error bar, it will dominate. Check that it deserves to.

After a fit, the residuals $\Delta_i = y_i - y_{\rm model}(x_i)$ measure how well the model works. The 2018 IRX–β problem defines

and finds 0.28 dex. Since IRX is a logarithm, 0.28 dex means a typical factor of $10^{0.28} = 1.9$ in flux ratio.

That scatter then sets the bar for any new object. CR7's IRX limit sits only 0.18 dex below the relation, less than one $\sigma$, so it is not a significant deviation.

Two values $a\pm\sigma_a$ and $b\pm\sigma_b$ differ significantly if

is large. A useful rule: $z<1$ means consistent, $z>2$ is suspicious, and $z>3$ is a real difference.

In 2017 D1.3, $d_V = 46.2\pm4.1$ kpc and $d_W = 48.1\pm0.9$ kpc give $z = 0.45$: not different. Their standard deviations, 4.1 and 0.9 kpc, clearly are different, so the Wesenheit band is the better distance indicator.

Do not delete points just because they spoil a fit. Reject a point only with a reason: it lies more than about $3\sigma$ from the trend, there is a physical cause (a blend, a flare, a different population), or the problem flags it (open symbols in a figure, for example). Always say which points you excluded and why, and check whether the answer changes.