Use of normal distribution in measurement error theory When you study an experimental results you assume that normal distribution describe random error, specially in lab courses in K12. Is this true for every random component error treatment in physics? If it is not the case, how do you know what probabilistic distribution is the one?.  
 A: The normal distribution has a very important special role in stochastics. One can prove mathematically that the distribution of the sum of many independent statistical processes is almost always a normal distribution. This is called the "Central Limit Theorem" in mathematics (there are actually several of these) and you can test it very easily yourself with dice. The distribution of a single dice is a constant (each value has equal probability). The sum of two dice thrown at the same time (i.e. values going from 1+1=2 to 6+6=12) has a triangular distribution (work out for yourself why that has to be the case, it's quite instructive). If you take three dice the value 3-18 will have a smoother bell like curve and if you keep proceeding with ever more dice, the resulting curve will converge very quickly towards a normal distribution (aka Gaussian). 
So whenever you have a process in physics that is made up of many independent events which all contribute a little bit to a measurement error, then it will most likely be Gaussian. The key word here is "statistical independence"! In the real world that is not an assumption that one can make without testing for it. 
Imagine an experiment where you are using a 1kOhm resistor. What's the error distribution on that? Is it normally distributed? Of course not. The manufacturer guarantees that the resistance will not deviate by more than e.g. 1% from the promised value. If they keep their promise this means that none of the resistors in your electronics kit will ever exceed 1.01kOhm or be less than 0.99kOhm! The manufacturer will sell those that are too large as 1.02k resistors and the ones that are too small as 0.98k ones! This means that, by design, your circuit will NOT have an error distribution that can be calculated with a normal distribution. Exactly the same thing holds for rolled steel, automobile tires, sheet glass and pretty much everything that is manufactured to specifications. All of these things will have manufacturing biases and quality cutoffs. If they are important to you, then you have to make sure that you understand the influence of the true statistical distribution on your experiments.
One can give a similar answer for instruments. The simple fact that your voltmeter has a digital readout will introduce very nasty non-Gaussian error distributions at the full resolution of the instrument. If you are doing the experiment with an analog meter and "by eye", you better understand the bias of the observer. Some people (like me!) are lazy about rounding and they have a tendency to round down instead of rounding correctly, and that, too, will introduce non-normal statistical errors. 
So how are we dealing with these? By measuring the distributions when necessary and by using either numerical statistics packages that can calculate true error distributions for the non-normal case or, and this is more commonly used in very complex bias scenarios, by using Monte-Carly simulations where we include the measured error distributions into the simulation of our experiments from the get-go and we just simulate e.g. the experiment a million times with the measured distributions for the errors. Is this complicated? Yes. It is complicated and necessary whenever you are doing a real experiment in science. 
A: The gaussian normal distribution is a good guess when you don't know anything else, due to the Law of Large Numbers.
If you know more about the process that generates the measurements, then you can choose another distribution.
For example, if there is a lower bound on measurements, (for example: concentrations of a chemical compound in blood plasma cannot be less than zero) you may want to consider taking a normal random number and exponentiating it.
That gives you a distribution called "log-normal".
Another way to modify a normal random number is to square it. That way, it cannot be negative.
If you add together some of those, it is called a "gamma" distribution.
If you know the number has both a lower bound and an upper bound, there are other distributions you can use: uniform, beta, logistic.
Long and short, the more you know about the process that generates the measurements, the more intelligently you can choose a distribution.
Alternatively, you can just take a lot of measurements, put them in a table and sort them.
Then you can use that table as an empirical distribution.
Failing any of that, you can just use a normal.
There's a good way to tell if you are using a good distribution, the Q-Q plot.
It's a really simple idea and I encourage you to get familiar with it.
