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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?

Qmechanic
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camarman
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2 Answers2

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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.

John Darby
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The general rule-of-thumb for calculating the uncertainty of a value $z + \Delta z$ taken to some power $n$ is:

  1. convert the uncertainty to a percentage error
  2. multiply the percentage error by n
  3. Take the nth power of the value
  4. convert the scaled percentage error of the new value back to an absolute error

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

  1. With absolute error: $z = 4.480$ and $\Delta z = 0.168$
  2. Convert absolute error to percent error: $\Delta z$% = $100*0.168/4.480 = 3.75$%
  3. Scale percent error by power: $\Delta y$% = $3.75$% $*3/2 = 5.625$%
  4. Take nth power of the number: $y = 4.480^{3/2} = 9.482$
  5. Convert back to absolute error: $\Delta y$ = $9.482*5.625/100 = 0.533$
  6. 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

  1. If adding/subtracting a set of numbers, always add all the absolute errors.
  2. If multiplying/dividing a set of numbers, always add all the percentage errors.
  3. If taking power of a number, multiply the percent error by the power.
Penguino
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