Error propagation for cube and then square root I have a variable $z$ and I know its error value $\Delta z$.
So $z = 4.480$ and $\Delta z = 0.168$. I need to find $y + \Delta y$ such that
$$y + \Delta y = (z+\Delta z)^{3/2}$$
So in this case, what is $y$ and $\Delta y$?
I am finding that
$$y = z^{3/2} \tag  1$$ and $$\Delta y = \frac{3}{2} z^{1/2} \Delta z \tag 2$$
Are these equations correct?
 A: I do not understand why you want to evaluate $y + \Delta y$ as $(z + \Delta z)^{3/2}$? You can evaluate the uncertainty in y(z) where y(z) is a function of the random variable z; for your case $y = z^{3/2}$.  If this is the case, the following applies.
I assume 4.480 is the mean for z and $\Delta z$  0f 0.168 is the standard deviation for z?
You can find discussions of how to evaluate the uncertainty for a function of a random variable in many statistics texts, and use that information to evaluate the uncertainty in y for your function as its standard deviation $\Delta y$.  For more complicated functions, you can to do a Taylor series expansion.  For example see Dougherty's text on Probability and Statistics or Meyer's text Data Analysis for Scientists and Engineers.
For your function, we have $\Delta y = y_{mean} (m^2 {\Delta z^2/z_m}^2)^{1/2}$ where $m = 3/2$, and  $y_{mean}$ = $z_{mean}^{3/2}$.  With this I calculate $y_{mean}$ of 9.48 and $\Delta y$ of 0.53; same result as Penguino provides in his answer.
Then you can express your answer for y with uncertainty as $y_{mean} \pm \Delta y$; 9.48 $\pm$ 0.53.
A: The general rule-of-thumb for calculating the uncertainty of a value $z + \Delta z$ taken to some power $n$ is:

*

*convert the uncertainty to a percentage error

*multiply the percentage error by n

*Take the nth power of the value

*convert the scaled percentage error of the new value back to an absolute error

For example: calculating $y = z^{3/2}$ with error limits

*

*With absolute error: $z = 4.480$ and $\Delta z = 0.168$

*Convert absolute error to percent error: $\Delta z$% = $100*0.168/4.480 = 3.75$%

*Scale percent error by power: $\Delta y$% = $3.75$% $*3/2 = 5.625$%

*Take nth power of the number: $y = 4.480^{3/2} = 9.482$

*Convert back to absolute error: $\Delta y$ = $9.482*5.625/100 = 0.533$

*Round off if appropriate: $y = 9.48$ and $\Delta y = 0.53$
If you compare manual calculations of the absolute upper and lower limits for your value of z: $(4.480-0.168)^{3/2} = 8.954$, $(4.480+0.168)^{3/2} = 10.021$, then they are a very close match to the results obtained from the above algorithm $8.949$ and $10.015$
The more general rules are

*

*If adding/subtracting a set of numbers, always add all the absolute errors.

*If multiplying/dividing a set of numbers, always add all the percentage errors.

*If taking power of a number, multiply the percent error by the power.

