A perfect definition of uncertainty, as the foundation stone of quantum mechanics, is the starting point for understanding the theory of everything. Consequently, this definition must have an entirely mathematical structure without dimensions in order to be applicable to measurement results related to all natural phenomena

Some physicists, disappointed with so called Heisenberg uncertainty principle, believe that uncertainty is that very standard deviation (Sigma). Usually they don't say anything about scientific basis of their opinion because that is an unrealistic idea and a belief only. After a long period of time I found the most important origin of this misunderstanding.

Apparently, old definition and meaning of uncertainty which has been stated by NIST (National Institute of Standards and Technology) is a source of this mistake. You will find below what has been said by NIST:

Standard Uncertainty and Relative Standard Uncertainty

The standard uncertainty u(y) of a measurement result y is the estimated standard deviation of y.

The relative standard uncertainty u_r(y) of a measurement result y is defined by u_r(y) = u(y)/|y|, where y is not equal to 0.

If the probability distribution characterized by the measurement result y and its standard uncertainty u(y) is approximately normal (Gaussian), and u(y) is a reliable estimate of the standard deviation of y, then the interval y - u(y) to y + u(y) is expected to encompass approximately 68 % of the distribution of values that could reasonably be attributed to the value of the quantity Y of which y is an estimate. This implies that it is believed with an approximate level of confidence of 68 % that Y is greater than or equal to y - u(y), and is less than or equal to y + u(y), which is commonly written as Y= y ± u(y).

http://physics.nist.gov/cgi-bin/cuu/Info/Constants/definitions.html

It is worthy of mention that there are important similarities between what has been stated by NIST and my explanation of uncertainty (see references), of course there are considerable discrepancies as well.

In my opinion, definition of standard uncertainty mentioned by NIST is not scientific and normal because there isn't any fact, observation or experimentation to verify it.

Based on my knowledge, standard uncertainty can be defined as follows:

The standard uncertainty u_x of a measurement result X is equal to six standard deviation of X, i.e. u_x = 6σ_x .

If x would be the mean value of the measurement result, then X = x ± 3σ_x.

Standard Uncertainty or Natural Tolerance = X_max – X_min = 6σ_x

I am confident that this definition of uncertainty is the fundamental common concept applicable to the micro world and also to the macro world. This concept enables us to generate a probability wave function with direct physical interpretation.

References