Least squares fitting

For $N$ points $(x_i, y_i)$, the line $y = mx + c$ that minimises $\sum(y_i - mx_i - c)^2$ has

Equivalently, $m = \dfrac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum(x_i-\bar x)^2}$ and $c = \bar y - m\bar x$, which is less prone to rounding errors.

To get uncertainties, first estimate the scatter about the line, using $N-2$ because two parameters were fitted:

If each $y_i$ has its own uncertainty $\sigma_i$, use weights $w_i = 1/\sigma_i^2$ and replace $N\to\sum w_i$, $\sum x_i\to\sum w_ix_i$, and so on:

These errors come from the error bars, not from the scatter, so they are only honest if the error bars are.

When theory says $y = mx$ (Hubble's law, $\Delta\theta = \theta_EX$, $M_{\rm DM} = kX$), fit only the slope:

This is exactly how the 2023 solution gets $\theta_E = 4.5\pm0.4$ mas from eight epochs.

• Correlation coefficient $r$: close to $\pm1$ means a tight linear relation. It does not tell you the line is the right model: a gentle curve can have $r = 0.99$.
• Residuals $y_i - (mx_i + c)$: plot them against $x$. Random scatter is good; a systematic arch means curvature you have not modelled.
• Chi-square $\chi^2 = \sum\left(\dfrac{y_i - mx_i - c}{\sigma_i}\right)^2$: for a good model and honest errors, $\chi^2/(N-2)\approx1$. Much larger means the errors are underestimated or the model is wrong; much smaller means the errors are overestimated.

Most scientific calculators have a linear regression ("LR", "STAT", "A+BX") mode that gives $m$, $c$ and $r$ directly. They usually do not give errors, but they do give the sums $\sum x$, $\sum y$, $\sum x^2$, $\sum xy$, from which you get $\Delta$ and the formulas above. Practise this before the exam on the 2018 IRX–β data. You should recover $\mathrm{IRX} = 0.98\beta + 1.96$ in about five minutes.

Write the fitted equation on the graph, and draw the line on it as well. Several schemes give a separate mark for "best-fit equation noted on the same page".