How the math actually works
The average atomic mass formula is straightforward when you strip away the textbook padding. You take each isotope's mass, multiply it by its fractional abundance, and sum the results. That's it. The formula looks like this: average atomic mass = (isotope mass × fractional abundance) Where fractional abundance is just the percent abundance divided by 100. You'll see it written different ways depending on who wrote the textbook, but they all reduce to the same calculation. Don't let the Greek sigma notation scare you.
Here's what happens when I'm grading lab reports. Students will write out the percent abundances like 69.17% and 30.83% for chlorine, then forget to divide by 100 before multiplying. The math still "works" on paper if you keep the percentage units, but you get answers that are 100 times too large. I've seen it every semester for twelve years. The fix is simple: write the fractional abundances as decimals before you start multiplying. 0.6917 and 0.3083. That removes the entire class of errors in one step. I remember working through a problem once where the isotope data came from a mass spectrometry run and the abundances didn't sum to exactly 100%. They summed to 99.87%. A TA pointed it out and we spent twenty minutes debating whether to normalize or just use the raw numbers. The answer is use the raw numbers. Normalizing introduces rounding error that doesn't exist in the actual measurement. If your instrument says 45.2% and 54.7%, don't force those to add to 100. They add to what they add to. The counter-intuitive part most people miss is that average atomic mass has nothing to do with the mass number of any single isotope. It's a weighted average, not a prediction. Carbon's average atomic mass is 12.011 amu, but no carbon atom weighs 12.011 amu. Every carbon atom is either 12.000 or 13.003 or some other discrete value. The average exists only in aggregate. This trips up students who try to find "the" atomic mass of an element the way they'd find the height of a person.
Another thing that catches people: significant figures. The result should match the precision of your least precise measurement. If one isotope mass is known to four decimal places and its abundance is only two significant figures, your final answer is limited to two significant figures regardless of how many decimal places the other terms have. I see people write out 12.010789 amu for carbon as if the balance can actually measure that precisely. It can't. The abundance data limits you. When you're working with elements that have more than two isotopes, like tin with ten stable isotopes, the formula doesn't change. You just add more terms. The calculation becomes tedious by hand but mechanically identical. That's why nobody does tin by hand anymore. I still make students do it for magnesium though, because three terms is enough to show the pattern without wasting a lab period. If your isotope masses are given in grams per mole instead of amu, the calculation gives you grams per mole. The numerical value is the same. This isn't a coincidence. One amu is defined so that the molar mass in g/mol equals the atomic mass in amu numerically. You can treat them interchangeably in this formula without converting units. Just keep your answer labeled consistently.
Get the Full Details
The method breaks down when you have incomplete isotope data. Some newly synthesized elements only have mass estimates for a few isotopes and no abundance information at all. In those cases the IUPAC lists a bracketed value instead of a standard atomic weight. You can't calculate anything meaningful. Don't try to invent abundances. There's no formula for that. For practical lab work, I usually write the calculation as a simple spreadsheet. Column A gets the isotope labels, column B the masses, column C the fractional abundances, column D the product of B and C, and column E sums column D. Takes thirty seconds to set up and eliminates arithmetic errors entirely. The time saved on grading is noticeable.
