Trimean

In statistics the trimean (TM), or Tukey's trimean, is a measure of a probability distribution's central tendency defined as a weighted average of the distribution's quartiles:

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

It gives twice as much weight to the median or second quartile ${\displaystyle Q_{2}}$ than the first and third quartiles.

This is equivalent to the arithmetic mean of the median and the midhinge:

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

The foundations of the trimean were part of Arthur Bowley's teachings, and later popularized by statistician John Tukey in his 1977 book 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 resistantL-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.