Trimean

In statistics the trimean (TM), or Tukey's trimean, is a measure of a probability distribution's location defined as a weighted average of the distribution's median and its two quartiles:

${\displaystyle TM={\frac {Q_{1}+2Q_{2}+Q_{3}}{4}}}$

This is equivalent to the average of the median and the midhinge:

${\displaystyle TM={\frac {1}{2}}\left(Q_{2}+{\frac {Q_{1}+Q_{3}}{2}}\right)}$

The foundations of the trimean were part of Arthur Bowley's teachings, and later popularized by statistician John Tukey in his 1977 book^[1] which has given its name to a set of techniques called exploratory data analysis.

Like the median and the midhinge, but unlike the sample mean, it is a statistically resistant L-estimator with a breakdown point of 25%. This beneficial property has been described as follows:

An advantage of the trimean as a measure of the center (of a distribution) is that it combines the median's emphasis on center values with the midhinge's attention to the extremes.

— Herbert F. Weisberg, Central Tendency and Variability^[2]