An arcpy-dependent function for ArcGIS 10.1+, which takes an input feature class (or layer) and the name of a weight field (e.g. population) from that feature class, and creates a new output feature class containing a single point feature at the average (weighted) coordinates of all of the input features.

create_mean_centroid.py

def create_mean_centroid(in_layer, out_fc, weight_field):
    '''
        Calculate average coordinates from all input features' centroids,
        weighted by values in weight_field.

        Output is a new feature class containing a single mean centroid point.
        It is *highly* recommended that a State Plane or UTM coordinate system
        be used, as the calculations use Euclidean geometry.

    '''
    import os
    import arcpy
    
    freq = 0
    sum_x = 0
    sum_y = 0

    with arcpy.da.SearchCursor(in_layer, ['SHAPE@TRUECENTROID', weight_field]) as cursor:
        for coords, weight in cursor:
            if weight:
                freq += weight
                sum_x += coords[0] * weight
                sum_y += coords[1] * weight

    mean_x = sum_x / freq
    mean_y = sum_y / freq
    mean_xy = (mean_x, mean_y)

    in_desc = arcpy.Describe(in_layer)
    in_sr = in_desc.spatialReference
    arcpy.CreateFeatureclass_management(os.path.dirname(out_fc), os.path.basename(out_fc), 'POINT')
    arcpy.DefineProjection_management(out_fc, in_sr)
    with arcpy.da.InsertCursor(out_fc, ['SHAPE@XY']) as cursor:
        cursor.insertRow([mean_xy])

    return mean_xy