I would try to explain why this person is frustrated, but there is nothing that tells me exactly why they are. Is it because the charts contain transparency or gradient fades? Is it because they are saving them out as .jpeg and thus having a larger filsize than the .png equivalent? Or is it because they are mostly flat coloured and so .png might handle that better? I have no idea...
For those interested and keen for a little foray into the different file types, check these and ignore this pointless post, just let it fall away:
http://www.whoishostingthis.com/blog/2014/12/06/jpeg-gif-png...
Unless you're building a site where the primary focus is the photography, use a high quality JPG instead of PNG for photos, the file size increase with PNG usually isn't worth it.
That habit goes back to my experience in professional photography where top quality is paramount and "lossy" is a 4-letter word.
The compression artefacts I think you are thinking of are likely the result of someone not setting the quality parameter appropriately. For text or charts, you're best off at 100%, but you can easily get away with lower values for most photographs. Granted, not an issue with PNG, but JPEG isn't so bad when used properly. (and typically does have huge file size benefits)