Andrew, Interesting analysis. There should indeed be a statistical bias towards one of the two possible binary digits for each of the five positions (modulo 5), but it is very slight. 16 characters share any particular digit in any particular location, so the bias will be significantly dampened. (Choose 16 characters at random, add together the probabilities, and it should result in a value rather close to 50%.) I would particularly like to see your statistical attack. As for your question, I'm sure someone on math-fun will be able to answer it. For the benefit of people on math-fun (who cannot view messages sent by people not on the list), the messages are reproduced below:
Incidentally, does anyone know how to calculate the mean value of the standard deviation of set of n values if the n values are determined by a multinomial distribution. What I mean is, given a random sting of n characters of length l, is there an easy way to work out the expected standard deviation of the frequencies of the letters? (This is somewhat relevant to my attempted attack on Adam's cipher.)
Andrew
An interesting idea. I'm not sure about the practicalities of it - are you intending to draw out a grid for every set of five letters, or do you think it is something that you could do in your head.
I think I've found one weakness already (although whether it can be put to use is another matter). There should be a positive correlation between the standard deviation of the binary digits or a character, and the standard deviation of the frequency of the character in each of the five positions mod 5. This occurs due to fact that in each of the five positions in the binary representation of a character, the two digits will usually not occur with equal probability, creating a bias towards characters with more of the more common digit, for that location in the ciphertext mod 5.
That probably sounds a bit confusing, and I still need to check the technicalities. I also intend to see whether my conclusions from this are correct. (I won't say what they are here, in case someone doesn't want to know them.)
Andrew
Sincerely, Adam P. Goucher