This is a seemingly simple question, though I'm not exactly sure where I'm going wrong (if in fact I am going wrong). To start off: you have a 2D un-normalized Gaussian function centred at the origin and with a sigma of 4. If you integrate it over a circle of radius 4 also centred at the origin, you will get a value. Let's call this value A. Then, if you integrate this same un-normalized function, but now over a circle of radius 7 once again centred at the origin, you will get another value. Let's call this value B. Now, for a regular normal distribution, within 1 standard deviation of the mean is 0.68 of all data points. Within two standard deviation is 0.95 of all data points. In this case, a circle of radius 4 should contain 0.68 of all data points, and a circle of radius 8 should contain 0.95 of all data points, right? Using this logic, since a circle of radius 7 would contain less data points than a circle of radius 8, shouldn't A/B > 0.68/0.95? I am currently doing a computation involving this same problem. I've done a few conversions to ensure the equation is in cylindrical coordinates, and there seems to be no problem. I even plug in points corresponding to the same points in space and get the same answer. I even did the integration earlier over the same spaces in X,Y and R,θ and got the same answers. Yet I am doing this exact problem and finding that A/B is less than 0.68/0.95. This has made me second guess my work, but I feel like I am not seeing the entire picture. Is there something I am overlooking here? Is normalization necessary if I am always working with the same amplitude? Any advice you have is greatly appreciated!