Within a set of data one important feature are measures of location or position. The most common measurements of this kind are the first and third quartiles. These denote, respectively, the lower 25% and upper 25% of our set of data. Another measurement of position, which is closely related to the first and third quartiles, is given by the midhinge.

After seeing how to calculate the midhinge, we will see how this statistic can be used.

## Calculation of the Midhinge

The midhinge is relatively straightforward to calculate. Assuming that we know the first and third quartiles, we do not have much more to do to calculate the midhinge. We denote the first quartile by *Q*_{1} and the third quartile by *Q*_{3}. The following is the formula for the midhinge:

(*Q* _{1} + *Q* _{3}) / 2.

In words we would say that the midhinge is the mean of the first and third quartiles.

## Example

As an example of how to calculate the midhinge 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

To find the first and third quartiles we first need the median of our data. This data set has 19 values, and so the median in the tenth value in the list, giving us a median of 7. The median of the values below this ( 1, 3, 4, 4, 6, 6, 6, 6, 7 ) is 6, and thus 6 is the first quartile. The third quartile is the median of the values above the median ( 7, 8, 8, 9, 9, 10, 11, 12, 13). We find that the third quartile is 9. We use the formula above to average the first and third quartiles, and see that the midhinge of this data is ( 6 + 9 ) / 2 = 7.5.

## Midhinge and the Median

It is important to note that the midhinge differs from the median. The median is the midpoint of the data set in the sense that 50% of the data values are below the median. Due to this fact, the median is the second quartile. The midhinge may not have the same value as the median because the median may not be exactly between the first and third quartiles.

## Use of the Midhinge

The midhinge carries information about the first and third quartiles, and so there are a couple of applications of this quantity. The first use of the midhinge is that if we know this number and the interquartile range we can recover the values of the first and third quartiles without much difficulty.

For instance, if we know that the midhinge is 15 and the interquartile range is 20, then *Q*_{3} - *Q*_{1} = 20 and ( *Q*_{3} + *Q*_{1} ) / 2 = 15. From this we obtain *Q*_{3} + *Q*_{1} = 30. By basic algebra we solve these two linear equations with two unknowns and find that *Q*_{3} = 25 and *Q*_{1} ) = 5.

The midhinge is also useful when calculating the trimean. One formula for the trimean is the mean of the midhinge and median:

trimean = ( median + midhinge ) /2

In this way the trimean conveys information about the center and some of the position of the data.

## History Concerning the Midhinge

The midhinge’s name is derived from thinking of the box portion of a box and whiskers graph as being a hinge of a door. The midhinge is then the midpoint of this box. This nomenclature is relatively recent in the history of statistics, and came into widespread use in the late 1970s and early 1980s.