# [Cytometry] Statistics Question

Bartek Rajwa rajwa at cyto.purdue.edu
Sun Jul 5 16:56:32 EDT 2009

```John:

John wrote:
>     I am curious if the geometric mean is acceptable for calculating the 'area' of a bimodal or highly skewed (or pick an ugly distribution) population which is not easily discriminated? I acquire in log so I'm hesitant to use the arithmetic mean. Also, I use FCS Express and I'm trying to avoid exporting every gate in listmode and doing it the hard way...
>
>

Geometric and arithmetic means estimate central tendency of a
distribution, not an "area". Arithmetic mean is a reasonable measure of
central tendency for unimodal, symmetric distributions, such as Gaussian
distribution. Geometric mean estimates central tendency of log-normal
distribution. If you can demonstrate that your data can be approximated
reasonably well by log-normal distribution than you can use geometric
mean. Of course, if the intensity values are represented in logarithmic
fashion that you would have to calculate the log-average by computing
the arithmetic mean of the logarithm transformed values. Anyway, you
have to remember that mean is not a robust estimator - it  is very
sensitive to extreme values. Trimmed mean, trimean, or median are
usually better choices.

If your data shows bimodal distribution it is likely that what you see
is a combination of two underlying normal (or log-normal) distributions.
In this case reporting any single estimator is meaningless.

You may consider an alternative way of summarizing your data - instead
of reporting mean values (e.g. for treated and untreated sample), you
could report the distance between your samples and positive/negative
controls. You can use for instance modified chi-square (1), KS-distance
(2), or quadratic-form distance (3) as your measure of samples'
(dis)similarity.

(1) Roederer, Mario, Adam Treister, Wayne Moore, and Leonore A.
Herzenberg. 2001. Probability binning comparison: A metric for
quantitating univariate distribution differences. Cytometry 45, no. 1:
37-46.
(2) Brescia, Francesca, and Maurizio Sarti. 2008. Modification to the
Lampariello approach to evaluate reactive oxygen species production by
flow cytometry. Cytometry Part A 73A, no. 2: 175-179.
(3) Bernas, Tytus, Elikplimi K. Asem, J. Paul Robinson, and Bartek
Rajwa. 2008. Quadratic form: A robust metric for quantitative comparison
of flow cytometric histograms. Cytometry Part A 73A, no. 8: 715-726.

Bartek Rajwa

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|Bartlomiej Rajwa, PhD       Purdue University Cytometry Laboratories|
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