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I am using ArcGIS software to create a Standard deviation of point pattern, and I know whether Ii need to expect a circle or an ellipse.

Is standard deviation in that context always circular? And if it is, what is the best / normal way to describe it (diameter? radius? and what is the Greek letter for it)? And would mean center always be the center-point of this circle?

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You should read the Gong (2010) paper "Clarifying the Standard Deviational Ellipse" which describe the geometry behind this statistic. And no, the standard deviation ellipse is not necessarily an ellipse but, rather a curve function.

Folks need to quit paying attention to ESRI and start reading the primary literature!

Gong, J., (2010) Clarifying the Standard Deviational Ellipse. Geographical Analysis 34(2):155-167

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