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I have a series of points in space with $(x_i,y_i)$ coordinates. And I would like to represent the average/mean point of this series:

--> Should I go for an arithmetic or a geometric mean of $x$ and $y$ coordinates ?

Thanks a lot to answer to my naive question :)

danimal
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Gab
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  • What do you need the mean for? The answer to this question might make this question answerable. – Alice Ryhl Jun 11 '15 at 15:05
  • To center a standard deviational ellipse: http://resources.esri.com/help/9.3/arcgisengine/java/GP_ToolRef/spatial_statistics_tools/how_directional_distribution_colon_standard_deviational_ellipse_spatial_statistics_works.htm ...From this website I would say it's a geometric mean... – Gab Jun 11 '15 at 15:06
  • I don't know the answer, but why don't you try both and see which works best? – Alice Ryhl Jun 11 '15 at 15:07
  • I computed both, and they are very close... that's why I am asking :-) ! – Gab Jun 11 '15 at 15:10
  • Standard deviation is always computed using the arithmetic mean. –  Jun 11 '15 at 16:07

1 Answers1

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Depends on the situation. e.g the center of mass in case of uniform mass density $\rho$ would be $$ (X,Y) = \left( \frac{ \sum_{i=1}^n \rho \cdot X_i }{\sum_{i=1}^n \rho}, \frac{ \sum_{i=1}^n \rho \cdot Y_i }{\sum_{i=1}^n \rho} \right) = \left( \frac{ \sum_{i=1}^n \cdot X_i }{n}, \frac{ \sum_{i=1}^n Y_i }{n} \right) = (\bar{X}, \bar{Y}) $$ where the bar designates the arithmetic mean.

For other problems the geometric or harmonic mean (exists, if all values are positve) might be more useful.

All three are related: $$ \bar{x}_H \le \bar{x}_G \le \bar{x}_A $$

mvw
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