This IDL function generates the standard deviation of a 2D array when it's compressed by an integer factor.

rebin_std.pro

function rebin_stddev, image, new_xsize, new_ysize, image_small = image_small
;+
; rebin_stddev
; It just works for making the image smaller by a integer factor (as rebin)
; 
;-
  dimensions = size(image, /dimensions)
  number_of_pixels_combined = total(dimensions / float([new_xsize, new_ysize]))
  ; Rebin image to a smaller dimenssion
  image_small = rebin(image, new_xsize, new_ysize)
  ; Make a new image with the original size with the average values 
  image_average = rebin(image_small, dimensions[0], dimensions[1], sample = 1)

  ; Difference between average and real value - Square
  difference = (image - image_average)^2

  ; Rebin the difference and multiply by a factor to get Bessel's correction ( 1/n-1)
  image_stddev = sqrt(rebin(difference, new_xsize, new_ysize)) * $
                 sqrt(number_of_pixels_combined/(number_of_pixels_combined-1))

  return, image_stddev
end