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:

$TM= \frac{Q_1 + 2Q_2 + Q_3}{4}$

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

$TM= \frac{1}{2}\left(Q_2 + \frac{Q_1 + Q_3}{2}\right)$

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]

Efficiency

Despite its simplicity, the trimean is a remarkably efficient estimator of population mean. More precisely, for a large data set (over 100 points) from a symmetric population, the average of the 20th, 50th, and 80th percentile is the most efficient 3 point L-estimator, with 88% efficiency.[3] For context, the best 1 point estimate by L-estimators is the median, with an efficiency of 64% or better (for all n), while using 2 points (for a large data set of over 100 points from a symmetric population), the most efficient estimate is the 29% midsummary (mean of 29th and 71st percentiles), which has an efficiency of about 81%. Using quartiles, these optimal estimators can be approximated by the midhinge and the trimean. Using further points yield higher efficiency, though it is notable that only 3 points are needed for very high efficiency.