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: This is equivalent to the average of the median and the midhinge: 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.Despite its simplicity, the trimean is a remarkably efficient estimator of population mean.
[4] For context, the best single point estimate by L-estimators is the median, with an efficiency of 64% or better (for all n), while using two points (for a large data set of over 100 points from a symmetric population), the most efficient estimate is the 27% midsummary (mean of 27th and 73rd percentiles), which has an efficiency of about 81%.
Using quartiles, these optimal estimators can be approximated by the midhinge and the trimean.