Not only does the FAA define standard passenger weights they have a lengthy AC describing how you should calculate loading. You can find that AC here.
4. What should an operator consider while reading this AC?
a. Accurately calculating an aircraft’s weight and CG before flight is
essential to comply with the certification limits established for the
aircraft. These limits include both weight and CG limits. By complying
with these limits and operating under the procedures established by
the manufacturer, an operator is able to meet the weight and balance
requirements specified in the aircraft flight manual (AFM). Typically,
an operator calculates takeoff weight by adding the operational empty
weight (OEW) of the aircraft, the weight of the passenger, cargo
payload, and the weight of fuel. The objective is to calculate the
takeoff weight and CG of an aircraft as accurately as possible.
b. When using average weights for passengers and bags, the operator must
be vigilant to ensure that the weight and balance control program
reflects the reality of aircraft loading. The FAA will periodically
review the guidance in this AC and update this AC if average weights
of the traveling public should change or if regulatory requirements
for carry-on bags or personal items should change. Ultimately, the
operator is responsible for determining if the procedures described in
this AC are appropriate for use in its type of operation.
You may be particularly interested in CHAPTER 2. METHODS TO DETERMINE THE WEIGHT OF PASSENGERS AND BAGS
The opening of the chapter pretty much covers that deviations may happen
200. What should an operator consider when choosing the appropriate method?
a. For many years, operators of transport category aircraft
have used average weights for passengers and bags to calculate an
aircraft’s weight and balance, in accordance with standards and
recommended practices. This method eliminates many potential sources
of error associated with accounting for a large number of relatively
light weights. However, differences between the actual weight of
passengers and bags and the average weight of passengers and bags can
occur when using average weights.
b. Statistical probability dictates
that the smaller the sample size (i.e., cabin size), the more the
average of the sample will deviate from the average of the larger
universe. Because of this, the use of standard average passenger
weights in weight and balance programs for small and medium cabin
aircraft should be examined in greater detail.
c. The next four
sections describe four methods available to operators to determine
passenger and bag weight. They are standard average weights in Section
2; average weights based on survey results in Section 3; segmented
weights in Section 4; and actual weights in Section 5. An operator
should review the following discussion and consult Table 2-1 to
determine which method or methods are appropriate to its type of
I will refrain from quoting the rest of the document but its worth a read on covers the topic well, it should answer the main body of this question.
Also of interest may be the FAA handbook on weight and balance.
From an operations stand point there are things to consider as well. First off airplanes have a CG range, not a single CG point. This range does in fact have some room to play with. The airlines only need to be within this range for TO/LD. Check out the FAA handbook for more on CG ranges. In theory the airline can (and does on occasion) move people around, although my understanding is that this happens more so in smaller commercial planes. When I fly the Archer or Warrior I typically fly you better believe all my passengers are stepping on a scale before the flight!
Some planes may have ways to mitigate this to an extent. Depending on airframes the fuel loading can be distributed differently among the tanks but you still must be with in limits for TO/LD of course. Any large plane that may also be carrying cargo (for shipping reasons not the baggage its self) may have some ability to rearrange the pallets to keep the CG within limits but this again depends on the situation.
-- EDIT (in response to CG range question in the comments below)--
As for how the planes are designed to handle this there is a CG range. This varies from plane to plane and some have bigger ranges than others. A full break down on how that works could be the subject of its own question here. Generally speaking there is a Max Takeoff weight that must fall with in a CG envelope. Here is the CG envelope for a Piper Warrior
The charts for airliners have bigger numbers and some more boxes but its easier to describe the issue with a chart like this. As you can see at max gross weight you have 6 inches of CG play. While this may not seem like a lot its more than enough to play with when loading the plane. This leaves room to have passengers of different weights in different seats etc. For the sake of a complete answer it should be noted that these ranges are just the tested limits of the airframe if your CG falls out of these ranges the plane may still fly however it will exhibit different behaviors (high/lower cruise speed, easier entry to a stall or tougher exit etc.). There have been various accidents in history that happened as the result of a massive CG shift in flight, there is some debate as to the role that the over loading of AF4590 (the Concorde that crashed) played in the accident. You should be able to find similar CG range charts for most modern airliners out there if you want more specific ranges.