The center of a set of data can be indicated in a number of different ways. Of course the mean, median and mode are all measures of central tendency. In other words, each of these three numbers serve as a way to tell where the middle of our data is at. There are other, less well known, ways to measure the center of a dataset. One of these less popular ways is called the trimean.

Calculation of the Trimean

In order to calculate the trimean we need to know three items from the five number summary of our data.

The ingredients that go into the formula for the trimean are the median, the first quartile and the third quartile. Denoting these by m, Q₁, Q₃, respectively we have the following formula for the trimean:

(Q₁ + 2 m + Q₃) / 4.

Example

As an example we will look at the following set of data:

1, 3, 4, 4, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10, 11, 12, 13

This data set has median of 7. The median of the values below this ( 1, 3, 4, 4, 6, 6, 6, 6, 7 ) is 6, and so this is the first quartile. The median of the values above the median ( 7, 8, 8, 9, 9, 10, 11, 12, 13) is 9 and so this is the third quartile. We use the above formula and see that the trimean of this data is ( 6 + 2 x 7 + 9 ) / 4 = 7.25.

For comparison we note that the mean is (1 + 3 + 4 + 4 + 6 + 6 + 6 + 6 + 7 + 7 + 7 + 8 + 8 + 9 + 9 + 10 + 11 + 12 + 13 ) / 19 = 7.21 and the midrange is (1 + 13) / 2 = 7.

Reasons to Use the Trimean

As seen in the above example, the trimean is a measure of center for our data. The advantage of using this measurement is that we capture information about the center in a way that is resistant to outliers.

The median also indicates the center of a data set and is resistant to outliers. By incorporating the first and third quartiles in the calculation of the trimean, we include some information about the rest of the distribution of the set of data.

We see that the trimean for the following set of data is identical to that of the example above:

1, 3, 4, 4, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10, 11, 12, 1000.

Alternate Formula

An alternate way to calculate the trimean is to think of it as the mean of the median and midhinge. The midhinge is itself the mean of the first and third quartiles. Thus we have:

( median + midhinge ) /2 = ( m + (Q₁ + Q₃ ) / 2) / 2 = (2 m + Q₁ + Q₃ ) / 4,

which is equivalent to our other formula for trimean.

History Concerning the Trimean

The trimean is also known as Tukey’s trimean. This is due to the fact that statistician John Tukey included it among other measures in his 1977 book Exploratory Data Analysis. Since then the trimean has enjoyed its status as a lesser known measure of center.