Certificates (lab reports) are very important in the real world, they are detailed reports on "the grading and measure of a diamond", issued as an unbiased lab report. Makes it possible to price & evaluate stones with some measure of dependability... Unbiased and impartial, I've always considered the certification process (*Lab report) a very important part of establishing the quality of the diamond, and accurately representing it... I always suggest that a lab report (commonly termed "certificate") and a stone be matched by an independent gemologist too for your protection. Since the differences in grades are also differences in price per carat, and in value, it's very important to have this lab report and unbiased certification when considering the merits of a purchase... This is the best written documentation you can get for a diamond, and it is easily matched up by a local gemologist to verify the value & quality.
*"You see, that "unbiased accurate grading is critical" is more than just a fair statement... "
You will find that to obtain a certificate/lab report for your diamond is not quite as easy as you might think, often taking weeks and costing over a hundred dollars for a major gemological institute's evaluation and certification paperwork process... Don't make a mistake and confuse an independant certificate from GIA, EGL or AGS with local paperwork cut by an independent "graduate GIA gemologist" , I greatly value major gemological institue's lab reports, as does the majority of the thinking members of the jewelry community as well as a growing number of potential purchasers who "want to do it right the first time"... Laboratory conditions, and the completely unbiased impartial grading, are the hallmarks of gemological institutions, so I try to offer pre-certified diamonds with these original lab reports/certificates from internationally recognized gemological institutions exclussively... It's also true that math, contained within these reports, do indeed describe diamonds extremely well, giving you the protection of a detailed scientific analysis by an impartial gemological institution.
GIA certificates ( Gemological Institute of America ) are the most widely known paperwork out there, and have the much greater name recognition... The adopted standard around the world is the GIA grading system. When evaluating a diamond pay close attention to the line items on these reports/certificate, not just the color and clarity. I also highly recommend EGL as well ( European Gemological Laboratory ), and of note is that their specs on cut break down total crown depth and pavilion depth percents for you... IGI ( International Gemological Institute ) also does lab reports, and evaluate/grade many stones for the diamond industry... The important thing is to have this report/certificate/evaluation of what you are buying, with this detailed information at your fingertips you'll know what you are paying for... I think that the most complete and the most valuable certificate in the world today is the AGS certificate, for the more complete data and the grading of the cut and geometry. AGS deems a diamond as "IDEAL" when it hits a narrow range for the geometry and the cut, the polish and the symmetry are also graded "Ideal" if they are indeed perfect. The AGS lab can make the quest for perfection easier for the diamond shopper, the more complete data helps to get right to the bottom line and know that quality is not subjective at all.
A note about some of the additional comments. You will see that in all certificates there are areas for comments regarding "grain lines" and comments which may well be important to consider along with the price at which a stone is being offered. Minor grain lines are of no real concern, but pay attention to the gemologist's comments in the reports to get the whole story. I think this is so important, to have widely recognized certificates of the lab report is to established what's being offered for sale, that I prefer to "only sell" pre-certified diamonds. Comparing apples to apples is generally appreciated by the thinking, and comparison shopping depends on getting the whole story, so make sure that you predicate your purchase on an independent gemological's lab report (commonly called certificate) and comparing similar prices/stones using this information.
*You see, while these lab reports are always somewhat subjective by nature, and really can't be represented as the "Final word" their unbiased and scientific lab reports help to establish benchmarks... They are not an absolute guarantee, since humans grading stones will always be somewhat subject at the best gemological labs... I'd always recommend an independantly certified and graded diamond over one graded internally by the seller. A diamond's only job is beauty, and the lab report's stats will in large measure establish the stone's grading clearly for you... My personal opinion is that information is power, be more powerful!