List of various methods of adding a check digit to a identification number.

No. of digits in the number + 1 check digit.
 Method of computing the check digit
 Examples
 Error  identification
10 digits
 Add all the digits, divide by 9. Remainder is the check digit
 Postal Money Order,  Traveller's Checks.
6 digits
 Divide the six digit number by 7. Remainder is the check digit
 Federal Express/UPS packages.
11 digits a1a2a3a4a5a6a7a8a9a10a11
 Total 3a1 + a2 + 3a3 + a4 + 3a5 + a6 + 3a7 + a8 + 3a9 + a10 + 3a11. The number you need to add to this sum to make it divisible by 10 is the check digit.
 UPC Code (used on grocery products).
Different UPC codes use different weights.
15 digits
 a1a2a3a4a5a6a7a8a9a10a11a12a13a14a15
 Compute  2(a1 + a3 + a5 + a7 + a9 + a11 + a13 + a15) + (no. of digits amongst a1,...,a15 which exceed 4) + (a2 + a4 + a6 + a8 + a10 + a12 + a14 ) . The check digit is the number you need to add to this sum to make it divisible by 10.
 Codabar. Used by several credit card companies.
9 digits a1a2a3a4a5a6a7a8a9 Total 10a1 + 9a2 + 8a3 + 7a4 + 6a5 + 5a6 + 4a7 + 3a8 + 2a9. The check digit is the number you need to add to make this sum divisible by 11.
 International Standard  Book Number (ISBN).


In Error detecting we talk about two kinds of errors :
1) Single digit Errors : When one of the digits is entered incorrectly.
2) Transposition Errors : Detecting error caused by rearrangement of the digits.


Note that all these different methods of coming up with a check digit didn't necessarily happen linearly.
Which is why we have different sectors using different methods, some better than others.

Weights are the different numbers which the digits are multiplied by when computing the check digit.
For example, in Postal money orders the weights are all one whereas in ISBN they use 10, 9, ..., 2  as weights.