Trimean


In statistics the trimean , 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 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:

Efficiency

Despite its simplicity, the trimean is a remarkably efficient estimator of population mean. More precisely, for a large data set from a symmetric population, the average of the 20th, 50th, and 80th percentile is the most efficient 3 point L-estimator, with 88% efficiency. For context, the best 1 point estimate by L-estimators is the median, with an efficiency of 64% or better, while using 2 points, the most efficient estimate is the 29% midsummary, 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.