The *Interquartile Mean* (IQM) is a statistical
measure of central tendency, much like
the mean (in more popular terms called the average), the median,
and the mode.

The IQM is very similar to the scoring method used in
sports that are evaluated by a panel of judges: *discard the
lowest and the highest scores; calculate the mean value of the remaining
scores*.

In calculation of the IQM, the lowest 25% and the highest 25% of the
scores are discarded. These points are called the first and third
quartiles, hence the name of the IQM. (Note that the
*second* quartile is also called the median). The method is best
explained with an example:

Consider the following dataset:

5, 8, 4, 38, 8, 6, 9, 7, 7, 3, 1, 6

First sort the list from lowest-to-highest:

1, 3, 4, 5, 6, 6, 7, 7, 8, 8, 9, 38

There are 12 observations (datapoints) in the dataset, thus we have 4 quartiles of 3 numbers. Discard the lowest and the highest 3 values:

~~1, 3, 4~~, 5, 6, 6, 7, 7, 8, ~~8, 9, 38~~

We now have 6 of the 12 observations remaining; next, we calculate the average of the

mean of these numbers:

x_{IQM} = (5 + 6 + 6 + 7 + 7 + 8) / 6 = 6.5

The Interquartile Mean shares some properties from both the

mean as well as
the

median:

- Like the median, the IQM insensitive to outliers; in the
example given, the highest value (38) was an obvious outlier of the
dataset, but its value is not used in the calculation of the IQM. On the
other hand, the common average (the arithmetic mean) is sensitive to these
outliers: x
_{mean} = 8.5.
- Like the mean, the IQM is a discrete parameter, based on a large
number of observations from the dataset. The median is always equal to
*one* of the observations in the dataset (assuming an odd number of
observations). The mean can be equal to *any* value between the
lowest and highest observation, depending on the value of *all* the
other observations. The IQM can be equal to *any* value between the
first and third quartiles, depending on *all* the observations in the
interquartile range.

The above example consisted of 12 observations in the dataset, which
made the determination of the quartiles very easy. Of course, not all
datasets have a number of observations that is divisible by 4. We can
adjust the method of calculating the IQM to accommodate this. Ideally we
want to have the IQM equal to the mean for symmetric distributions,
*e.g.*:

1, 2, 3, 4, 5

has a mean value x

_{mean} = 3, and since it is a symmetric
distribution, x

_{IQM} = 3 would be desired.

We can solve this by using a weighted average of the quartiles and
the interquartile dataset:

Consider the following dataset of 9 observations:

1, 3, 5, 7, 9, 11, 13, 15, 17

There are 9/4 = 2.25 observations in each quartile, and 4.5 observations
in the interquartile range. Truncate the fractional quartile size, and
remove this number from the 1st and 3rd quartiles (2.25 observations in
each quartile, thus the lowest 2 and the highest 2 are removed).

~~1, 3~~, (5), 7, 9, 11, (13), ~~15, 17~~

Thus, there are 3

*full* observations in the interquartile range,
and 2 fractional observations. Since we have a total of 4.5 observations
in the interquartile range, the two fractional observations each count
for 0.75 (and thus 3x1 + 2x0.75 = 4.5 observations).
The IQM is now calculated as follows:

x_{IQM} = {(7 + 9 + 11) + 0.75 x (5 + 13)} / 4.5 = 9

In the above example, the mean has a value x

_{mean} = 9. The
same as the IQM, as was expected. The method of calculating the IQM for
any number of observations is analogous; the fractional
contributions to the IQM can be either 0, 0.25, 0.50, or 0.75.

*
Note: I worked out a method similar to Tom Rook, but didn't bother to write it down. This one is easier since it uses/corrects the built in trimmean function in Excel. It works with 65536 values (or even more if adapted to multiple columns) One also does not need to sort the data. It goes as follows:
*

- Put the data in column A
- Define a name for a range in the sheet: (insert... Name... Define...) e.g. name it "IQM"
- In the reference field put: "=Sheet1!$A:$A;Sheet1!$A:$A;Sheet1!$A:$A;Sheet1!$A:$A". Thus, the name "IQM" should refer to 4 times the column A.
- The interquartile mean can be calculated with this function: "=TRIMMEAN(IQM;0.5)"

I think this one is almost winning the beauty contest, except for using a name range. Unfortunately one cannot use:
TRIMMEAN(Sheet1!$A:$A;Sheet1!$A:$A;Sheet1!$A:$A;Sheet1!$A:$A;0.5)

If someone knows a method to put it all in one function without using a name range, let me know.

Update 20070103: DutchDemon writes me that it is actually possible to put everything into one formula as follows: **=TRIMMEAN((A:A;A:A;A:A;A:A);0.5)**. Magic.