statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

the trimean (TM), or Tukey's trimean, is a measure of a

probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

location In geography, location or place is used to denote a region (point, line, or area) on Earth's surface. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ambiguous bou ...

defined as a weighted average of the distribution's

median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...

and its two quartiles: :

TM= \frac

This is equivalent to the

arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...

of the

and the midhinge: :

TM= \frac\left(Q_2 + \frac\right)

The foundations of the trimean were part of Arthur Bowley's teachings, and later popularized by statistician

John Tukey John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the fast Fourier Transform (FFT) algorithm and box plot. The Tukey range test, the Tukey lambda distributi ...

in his 1977 book which has given its name to a set of techniques called

exploratory data analysis In statistics, exploratory data analysis (EDA) is an approach of data analysis, analyzing data sets to summarize their main characteristics, often using statistical graphics and other data visualization methods. A statistical model can be used or ...

. Like the median and the midhinge, but unlike the

sample mean The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or me ...

, it is a statistically resistant

L-estimator In statistics, an L-estimator (or L-statistic) is an estimator which is a linear combination of order statistics of the measurements. This can be as little as a single point, as in the median (of an odd number of values), or as many as all points ...

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 (over 100 points) from a symmetric population, the average of the 18th, 50th, and 82nd percentile is the most efficient 3-point L-estimator, with 88% efficiency. 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. Using further points yield higher efficiency, though it is notable that only three points are needed for very high efficiency.

References

* {{refend

External links

Trimean
at

MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science ...

Summary statistics Means Robust statistics Exploratory data analysis

Efficiency

See also

References

External links