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.
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]
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.
trimean, statistics, trimean, tukey, trimean, measure, probability, distribution, location, defined, weighted, average, distribution, median, quartiles, displaystyle, frac, this, equivalent, average, median, midhinge, displaystyle, frac, left, frac, right, fou. 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 T M Q 1 2 Q 2 Q 3 4 displaystyle TM frac Q 1 2Q 2 Q 3 4 This is equivalent to the average of the median and the midhinge T M 1 2 Q 2 Q 1 Q 3 2 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 Contents 1 Efficiency 2 See also 3 References 4 External linksEfficiency EditDespite 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 See also EditTruncated mean Interquartile meanReferences Edit Tukey John Wilder 1977 Exploratory Data Analysis Addison Wesley ISBN 0 201 07616 0 Weisberg H F 1992 Central Tendency and Variability Sage University ISBN 0 8039 4007 6 p 39 Evans 1955 Appendix G Inefficient statistics pp 902 904 Evans Robley Dunglison 1955 The Atomic Nucleus International series in pure and applied physics McGraw Hill pp 972 ISBN 0 89874414 8 External links EditTrimean at MathWorld Retrieved from https en wikipedia org w index php title Trimean amp oldid 1029934022, wikipedia, wiki, book, books, library,
