John S Allen is one of the greatest technical writers about bicycle technology. He has recently produced an amazing spreadsheet of hub gear ratios, calculated from first principles. This reveals that, in some cases, the ratios quoted by the manufacturers are incorrect.
‘I felt the exercise to be useful because almost all the information I’ve seen about ratios gives decimal numbers. The actual ratios are fractions, many of which do not reduce exactly to decimals. Some numbers are given to only two places after the decimal point, not really as accurate as would be desirable. I also found incorrect rounding and in some materials I reviewed, transcription errors.’
Here’s a link to this uniquely useful spreadsheet.