# Trimean (Tukey’s Trimean)

A trimean is a number that represents the general tendency of a set of numbers or data set. Like the mean, median, and mode, it is a measure of central tendency. It is defined to be the weighted average of the median and upper and lower quartiles. What that means is that:

Trimean=(Q1+2Q2+Q3)/4

where Q1 And Q3 are the upper and lower quartiles (also known as hinges) and Q2 is the mean.

The two quartiles in the formula make the trimean more likely to be representative of the data than the median. The formula’s weighted middle results in a lower sensitivity to outliers.

## Example

Let’s say you wanted to find the trimean for the set 1, 2, 3, 4, 5, 6, 7.
Your median is 4, Q1 is 2 and Q3 is 6, so your trimean will be:
(4 · 2 + 2 + 6) / 4, or 16 / 4 = 4.
The data was uniformly spaced and symmetric, so the median and trimean are equal.

This is not always the case, however. For example, take the set 1, 3, 5, 6, 7, 9, 12, 15, 18, 19, 90.
The median is digit 6 — the number 9.
Q1 is 5 and Q3 is 18, so your trimean is (9 · 2 + 5 + 18) / 4 = 41 / 4 = 10.25. This is different from the median (9).

The set 1, 2, 3, 15, 1000, 2000, 3000 will have a trimean of (30+2+2000)/4, or 508.

## Why Use the Trimean

The trimean is easy to calculate, and a surprisingly good estimate of the arithmetical mean. It is considered ‘resistant’ or ‘robust’ because it is not very effected by outliers. The inclusion of the weighted median give it a strong emphasis on the center, but the two quartiles also bring in significant representation from the edges. Maybe that’s why the Trimean has been also named the BES (Best Easy Systematic) Estimate.

## Sources

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