The number of each digit seems to deviate too little from the expected. For the first n digits of pi, we can model the number of occurrences of any specific digit as the sum of n independent identically distributed Bernoulli random variables each with 10% (p=0.1) probability of being one and 90% (q=0.9) probability of being zero. The average of such is np which, for n=10,000,000 (ten million) is 1,000,000. The standard deviation is sqrt(npq)=948. And yet the most deviate digit is 1 which lags its expectation by only 667. So none of the ten digits fall more than 0.71 standard deviations away from the expected value. Small sample size or a hint of some deep truth about pi?I suspect small sample size because if we back up to the first million digits, we get a standard deviation of 300 which is exceeded by 5s and 6s.
-- (August 20, 2013) on Frequency of Each Digit of Pi