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Does random variable depends on itself if squared? Suppose I measured length by ruler then squared that value. Then I measured different value not even close to the previous just another value and squared again. Is it the case when random variable is constant and therefore it depends on itself? I'm squaring the same variable $$x_1=1.2±0.1 cm$$ then square it $$x^2=1,44±?$$

What is the error of $x^2$ ?

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  • $\begingroup$ Delta rule. variance is proportional to the square of the derivative. Check out example 1.5 here $\endgroup$
    – EngrStudent
    Oct 12, 2016 at 13:45
  • $\begingroup$ The squared value is certainly a deterministic function of the original value, but if the original value was a realization from a random variable, you can consider the squared value a realization from a random variable also. $\endgroup$
    – gung - Reinstate Monica
    Oct 12, 2016 at 13:46

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Your model is not linear, but as the uncertainty is small and as you are far from x=0, you can apply Linear Uncertainty Propagation (LUP), i.e.

$$Var(x^2) = \left(\frac{d x^2}{dx}\right)^2_{x_1} Var(x) $$

then, simply

$$u_{x^2} = 2 |x_1| u_x$$

By the way, you are dealing with uncertainty, not error. The error is the difference between your measurement and the true value, that you generally don't know.

See the international reference guide:Guide to the Expression of Uncertainty in Measurement

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