Combining Uncertainties I: Propagation of Errors
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An essential component of experimentation in science is to measure the response of a physical system to the applied external stimulus. These responses, in conjunction with existing theoretical models, are used to elicit properties of the system that are not readily obvious and to construct a deeper understanding of the system's nature. The validity of understanding attained through an experiment depends on the accuracy and precision of measurements made in the experiment. Various sources of errors such as systematic errors, random errors, etc. limit the accuracy and precision, hence the reliability, of measurements. We must be mindful of these errors when making inferences based on data obtained from any experiment.
Errors that creep in during measurements are fairly straightforward to figure out. It is, however, not straightforward to know how these errors propagate through calculations and how they affect the validity of the final result of calculations. Therefore, we must devise a method to keep track of these errors as we make calculations. Below, we undertake this task.
Suppose \(A\), \(B\) and \(C\) are three quantities measured in an experiment. The absolute errors in these quantities are \(\Delta A\), \(\Delta B\) and \(\Delta C\), respectively. Following figures represent ranges of quantities \(A\) and \(B\).
Let's find expressions for errors in quantities obtained by combining measured quantities using mathematical operations (addition, subtraction, multiplication, division and raising to a power).
IMPORTANT: We are assuming here that quantities \(A, \, B \mbox{ and } C\) are measured with fairly decent accuracy so that absolute errors (or uncertainty) in measured quantities are much less than the quantities themselves, i.e., \(\Delta A << A, \, \Delta B << B \mbox{ and } \Delta C << C\). This implies \(\displaystyle \frac{\Delta A}{A} << 1, \, \frac{\Delta B}{B} << 1 \mbox{ and } \frac{\Delta C}{C} << 1\). Therefore, second order quantities such as \(\displaystyle \left(\frac{\Delta A}{A}\right)^{2}\) and \(\displaystyle \frac{\Delta A}{A}\frac{\Delta B}{B}\) would be so small that we could disregard them from our calculations. All of the calculations below omit such second order quantities.
1. Error in addition of measured quantities
Let \(X = A+B\). Finding error in \(X\) due to errors in \(A\) and \(B\) entails finding the range in which the value of \(X\) is likely to lie. We represent the upper and lower bounds of this range as \((X + \Delta X)\) and \((X-\Delta X)\) respectively; here, \(\Delta X\) is the absolute error (uncertainty) in \(X\). It is obvious that the lower bound, \((X-\Delta X)\), of \(X\) can be obtained by adding the lower bounds of \(A\) and \(B\); and the upper bound. \((X+\Delta X)\), of \(X\) can be obtained by adding the upper bounds of \(A\) and \(B\). The following figure shows how these bounds can be expressed in terms of quantities \(A\) amd \(B\).
From above results, we conclude \(\Delta X = \Delta A + \Delta B\).
Alternatively, we can treat upper and lower bounds of \(X\) at once in the following way: \[ \begin{eqnarray} X \pm \Delta X & = & (A \pm \Delta A)+(B \pm \Delta B) \\ & = & A + B \pm \Delta A \pm \Delta B \\ & = & (A + B) \pm (\Delta A + \Delta B) \end{eqnarray} \] Therefore, absolute error (or uncertainty) in \(X\) is \(\Delta X = \Delta A + \Delta B \).
2. Error in subtraction of measured quantities
Let's consider the quantity \(X = A - B\). We need to be more careful with subtraction. To obtain the lower bound, \((X-\Delta X)\), of \(X\) we need to subtract the upper bound of \(B\) from the lower bound of \(A\). To obtain the upper bound, \((X+\Delta X)\), of \(X\) we need to subtract the lower bound of \(B\) from the upper bound of \(A\). The following figure illustrates this.
From above results, we conclude \(\Delta X = \Delta A + \Delta B\).
Alternatively, we can find upper and lower bounds of \(X\) in the following way: \[ \begin{eqnarray} X \pm \Delta X & = & (A \pm \Delta A)-(B \mp \Delta B) \\ & = & A + B \pm \Delta A \pm \Delta B \\ & = & (A + B) \pm (\Delta A + \Delta B) \end{eqnarray} \] Therefore, absolute uncertainty in \(X\) is \(\Delta X = \Delta A + \Delta B \). This is identical to the result we obtained when we added \(A\) and \(B\).
\[\boxed{\mbox{For } X = A \pm B, \,\, \Delta X = \Delta A + \Delta B}\]
3. Error in the product of measured quantities
In case of a quantity \(\displaystyle X = AB\), the lower bound is obtained by multiplying the lowest values of \(A\) and \(B\), and the upper bound is obtained by multiplying highest values of can be obtained \(A\) and \(B\):
We have ignored the second order term \( \displaystyle \frac{\Delta A \, \Delta B}{AB} \). This means that the values of \(X=AB\) lies between \(\displaystyle AB - AB\left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right)\) and \( \displaystyle AB + AB\left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right)\). In other words, \[ \begin{eqnarray} X \pm \Delta X & = & AB \pm AB \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right) \\ & = & X \pm X\left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right) \\ \Rightarrow \Delta X & = & X \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right) \\ \Rightarrow \frac{\Delta X}{X} & = & \frac{\Delta A}{A} + \frac{\Delta B}{B} \\ \end{eqnarray} \] We can also arrive at this result in the following manner: \[ \begin{eqnarray} X \pm \Delta X & = & (A \pm \Delta A)(B \pm \Delta B) \\ & = & AB \pm A \Delta B \pm B \Delta A + \Delta A \, \Delta B \\ & = & AB \pm AB \left(\frac{\Delta B}{B} + \frac{\Delta A}{A} + \frac{\Delta A \, \Delta B}{AB}\right) \\ & = & X \pm X \left(\frac{\Delta B}{B} + \frac{\Delta A}{A}\right) \\ \Rightarrow \Delta X & = & X \left( \frac{\Delta A}{A} + \frac{\Delta B}{B}\right) \\ \Rightarrow \frac{\Delta X}{X} & = & \frac{\Delta A}{A} + \frac{\Delta B}{B} \end{eqnarray} \] We can also express this equation as \(\displaystyle \frac{\Delta (AB)}{(AB)} = \frac{\Delta A}{A} + \frac{\Delta B}{B}\).
Therefore, the fractional error in the product of two quantities is equal to the sum of fractional errors in the individual quantities.
4. Error in division of measured quantities
We need to be careful when calculating bounds of a quantity \(X\) obtained by dividing two measured quantities. If \(\displaystyle X = \frac{A}{B}\), then lower bound of \(X\) is obtained by dividing the lowest values of \(A\) by highest value of \(B\), and the upper bound is obtained by multiplying highest values of \(A\) with the lowest value of \(B\):
\[ \begin{eqnarray} X \pm \Delta X & = & \frac{A \pm \Delta A}{B \mp \Delta B} \\ & = & \frac{A \pm \Delta A}{B \left(1 \mp \frac{\Delta B}{B}\right)} \\ & = & \left(\frac{A}{B} \pm \frac{\Delta A}{B} \right) \left(1 \mp \frac{\Delta B}{B}\right)^{-1} \\ & = & \left(\frac{A}{B} \pm \frac{\Delta A}{B} \right)\left[1 \pm \frac{\Delta B}{B} \mp \left(\frac{\Delta B}{B}\right)^{2} \pm \cdot \cdot \cdot\right] \\ & = & \left(\frac{A}{B} \pm \frac{\Delta A}{B} \right)\left(1 \pm \frac{\Delta B}{B} \right) \\ & = & \frac{A}{B} \pm \frac{\Delta A}{B} \pm \frac{A}{B} \frac{\Delta B}{B} + \frac{\Delta A}{B} \frac{\Delta B}{B} \\ & = & \frac{A}{B} \pm \frac{\Delta A}{B} \pm \frac{A}{B} \frac{\Delta B}{B} \\ & = & \frac{A}{B} \pm \frac{A}{B} \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right)\\ & = & X \pm X \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right)\\ \Rightarrow \Delta X & = & X \left( \frac{\Delta A}{B} + \frac{\Delta B}{B} \right) \\ \Rightarrow \frac{\Delta X}{X} & = & \frac{\Delta A}{A} + \frac{\Delta B}{B} \end{eqnarray} \] We can also express this equation as \(\displaystyle \frac{\Delta (A/B)}{(A/B)} = \frac{\Delta A}{A} + \frac{\Delta B}{B}\).
Therefore, the fractional error in the division of two quantities is equal to the sum of fractional errors in the individual quantities. (This is identical to the expression for error in the product of measured quantities.)
\[\boxed{\mbox{For } X = AB \mbox{ or } \frac{A}{B}, \,\, \frac{\Delta X}{X} = \frac{\Delta A}{A} + \frac{\Delta B}{B}}\]
5. Error in the \(n^{th}\) power of a measured quantity
The error in a quantity defined as \(X = A^{n}\) is \[ \begin{eqnarray} X \pm \Delta X & = & \left( A \pm \Delta A \right)^{n} \\ & = & A^{n} \left( 1 \pm \frac{\Delta A}{A} \right)^{n} \\ & = & A^{n} \left( 1 \pm n \frac{\Delta A}{A} + \cdot \cdot \cdot \right) \\ & = & A^{n} \left( 1 \pm n \frac{\Delta A}{A} \right) \\ & = & A^{n} \pm n A^{n} \frac{\Delta A}{A} \\ \Rightarrow \Delta X & = & n A^{n} \frac{\Delta A}{A} = n X \frac{\Delta A}{A} \\ \Rightarrow \frac{\Delta X}{X} & = & n \frac{\Delta A}{A} \end{eqnarray} \] Therefore, the fractional error in the \(n^{th}\) power of a measured quantity is \(n\) times the fractional error in the measured quantity.
6. Error when measured quantity is multiplied by a constant
A quantity \(X\) defined as \(X=aA\), where \(a\) is a constant, the upper and lower bounds of \(X\) are determined as \[ \begin{eqnarray} X \pm \Delta X & = & a (A \pm \Delta A) \\ & = & aA \pm a\Delta A \\ \Rightarrow \Delta X & = & a \Delta A \end{eqnarray}\]
7. Putting everything together
Suppose a quantity \(X\) is defined as \[X = \frac{A^{m} B^{n}}{C^{p}}\]. Using the rules derived above, it can be shown that the fractional error in \(X\) is given by \[\frac{\Delta X}{X} = m\frac{\Delta A}{A} + n\frac{\Delta B}{B} + p\frac{\Delta C}{C} \] Proof:
To prove this, let \(f=A^{m}, g=B^{n}\) and \(h=C^{p}\). Then, \(X = fg/h\), and it follows from above discussion that \[ \begin{eqnarray} \frac{\Delta X}{X} & = & \frac{\Delta (fg)}{(fg)} + \frac{\Delta h}{h} \\ & = & \left( \frac{\Delta f}{f} + \frac{\Delta g}{g} \right) + \frac{\Delta h}{h} \\ & = & \left( m\frac{\Delta A}{A} + n\frac{\Delta B}{B} \right) + p\frac{\Delta C}{C} \\ \end{eqnarray} \] Therefore, \[\frac{\Delta X}{X} = m\frac{\Delta A}{A} + n\frac{\Delta B}{B} + p\frac{\Delta C}{C} \]
SUMMARY
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super helpful content. Thank you!
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