Citation count and altmetric data is typically highly skewed
and the arithmetic mean is not the best measure of central tendency because of
this. The problem is that the arithmetic mean can be greatly influenced by
individual very high values and it is normal to have occasional very high
values for citation counts or altmetrics.

A simple alternative to the arithmetic mean is to use the
geometric mean. This uses the arithmetic mean of the natural logarithm of the
data instead of the raw data. This mean is then transformed back by applying
the exponential function to it, which is the inverse (reverse) of the natural
logarithm. This reduces the influence of very large values and gives a more
stable calculation. A problem with this is that uncited articles have a
citation count of zero and it is not possible to calculate the log of zero. The
simplest way to get round this problem is to add 1 to the citation counts
before taking the natural logarithm. If this step is taken then 1 should also
be subtracted after applying the exponential function.

In other words, the recommended process for
obtaining an average for any citation or altmetric count-type data is as
follows.

- Add 1 to the citation count/altmetric count data
- Take the natural logarithm of the result.
- Calculate the arithmetic mean of this transformed data.
- Calculate the exponential function of the result and then subtract 1.

Download the spreadsheet

**here**with the calculations.
Here is an example, showing the formulae used in Excel, with
some test data:

In the modification below, the citation count of only one
article has changed, becoming very large. This has had a huge impact on the
arithmetic mean, increasing it by over four times from about 2.4 to about 13.6 and a much smaller impact on
the geometric mean, increasing it by about half from about 1.5 to 2.2. This
illustrates the advantage of the geometric mean.

Here is some evidence that the geometric mean approach
works.

Fairclough, R., & Thelwall, M. (2015). More precise methods for national research citation impact comparisons.

Thelwall, M. & Fairclough, R. (2015).

*Journal of Informetrics*, 9(4), 895-906.Thelwall, M. & Fairclough, R. (2015).

__Geometric journal impact factors correcting for individual highly cited articles__.*Journal of Informetrics, 9(2)**,*263–272.
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