Four ways to average a distance

Adapted from IOAA 2018 (Beijing), theory problem 2. We have restated it in our own words and kept the original data. The solution was last checked on 16 October 2026.

An observer measures the parallaxes of many stars in a star cluster that is 5 kpc away. Random errors spread the measured parallaxes symmetrically about the true value, with a standard deviation of 0.05 mas, and there are no systematic errors. All the stars in the cluster have the same luminosity. Four students use the table of parallaxes to estimate the distance of the cluster in four different ways.

• Student A turns each parallax into a distance and takes the average of the distances, $R_{\text{A}}$.
• Student B takes the average of the parallaxes and turns it into a distance, $R_{\text{B}}$.
• Student C turns each parallax into a distance and takes the median of the distances, $R_{\text{C}}$.
• Student D takes the median of the parallaxes and turns it into a distance, $R_{\text{D}}$.

Say whether each of the following statements is true or false, and give the correct relation for each one that is false. Each part is worth 2 points, and $R$ is the true distance.

(a) If star $i$ has the smallest measured parallax and star $j$ the largest, then most likely $R_i-R>R-R_j$.

(b) $R_{\text{A}}=R$, so that A's estimate is likely to be close to the true distance.

(c) $R_{\text{B}}=R$.

(d) $R_{\text{C}}$ is less than $R$, so that C's estimate is likely to be too small.

(e) $R_{\text{D}}=R$.

AnswerThe answers are (a) true, (b) false, with $R_{\text{A}}>R$, (c) true, (d) false, with $R_{\text{C}}=R$, and (e) true. Of the four methods, only averaging the distances gives an answer that is systematically too large.