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I often have to calculate the mass of peptide sequences using calculators such as here.

I have the option of calculating the average and monoisotopic mass, but what is the difference?

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The elements that typically make up peptides and proteins can be present in different isotopes. For example, carbon is present to 98.9% as $\ce{^12C}$, and to 1.1% as $\ce{^13C}$.

The monoisotopic mass assumes that every element is present in the most abundant isotope, the average mass uses the actual distribution of the isotopes in nature.

If you have sufficient resolution in your mass spectrum you can see multiple signals corresponding to different isotope compositions of your molecule.

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