How to calculate Absolute error under square root sign? [closed]








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I was calculating standard deviation of a data set. Could someone help me find the square root of 15.13 with an absolute error of 0.005? Is there a rule for calculating absolute errors under square root sign? 




A slightly different approach to @Zhe's answer is to use the 'propagation of errors' formula. Your formula is $y=\sqrt{x}$ with $x=15.13$ and an error $\sigma\_x=0.005$ then the formula becomes $\displaystyle \sigma\_y=\frac{\mathrm dy}{\mathrm dx}\sigma\_x = \frac{\sigma\_x}{2\sqrt{x}}= 6.4\cdot 10^{-4}$.


[if you have a formula in more than one variable , say $x$ and $z$ then $y=f(x,z)$ and the equation is more complicated but still quite easy if done in steps and is $\displaystyle \sigma\_y^2=\left(\frac{\mathrm dy}{\mathrm dx} \right)^2\_z\sigma^2\_x+ \left(\frac{\mathrm dy}{\mathrm dz} \right)^2\_x\sigma^2\_z$. Notice that the square of the final error is calculated in this case.]


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Generally, you want to look at Taylor expansions when computing functions on small errors:


$$\sqrt{1+x} = 1 + \frac{x}{2} + O(x^{2})$$


This means that


$$\sqrt{15.13 \pm 0.005} = \sqrt{15.13}\sqrt{1 \pm \frac{0.005}{15.13}} = \sqrt{15.13}\left(1 \pm \frac{1}{2}\cdot\frac{0.005}{15.13}+O(0.005^{2})\right)$$


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