Understanding Relative Error Calculation for Any Relationship

Relative error is a measure of the uncertainty or error in a measurement relative to the size of the measurement itself. It is often expressed as a percentage and is an important concept in physics, engineering, and other scientific fields where precise measurements are crucial. When dealing with relationships between different quantities, understanding how to calculate relative error can help in assessing the accuracy of derived results.

What is Relative Error?

Relative error is defined as the absolute error divided by the true value of the quantity being measured. The absolute error is the difference between the measured value and the true value.

The formula for relative error is:

$$ \text{Relative Error} = \frac{\text{Absolute Error}}{\text{True Value}} $$

In many cases, the true value is not known, so the accepted value or the experimental value is used as an approximation.

Calculating Relative Error in Direct Relationships

When quantities are related directly, such as in a linear relationship, the relative error in the result can be calculated by summing the relative errors of the individual quantities.

For example, if $Q = A \cdot B$, where $A$ and $B$ are measured quantities with relative errors $\epsilon_A$ and $\epsilon_B$ respectively, the relative error in $Q$, denoted as $\epsilon_Q$, is given by:

$$ \epsilon_Q = \epsilon_A + \epsilon_B $$

Calculating Relative Error in Inverse Relationships

In inverse relationships, where one quantity is inversely proportional to another, the relative error is also additive. For instance, if $Q = \frac{A}{B}$, the relative error in $Q$ is:

$$ \epsilon_Q = \epsilon_A + \epsilon_B $$

Calculating Relative Error in Power Relationships

When dealing with power relationships, such as $Q = A^n$, the relative error in $Q$ is $n$ times the relative error in $A$:

$$ \epsilon_Q = n \cdot \epsilon_A $$

Table of Relative Error Calculations

Here's a table summarizing the relative error calculations for different types of relationships:

Relationship Type	Formula	Relative Error Calculation
Direct (Product)	$Q = A \cdot B$	$\epsilon_Q = \epsilon_A + \epsilon_B$
Inverse (Quotient)	$Q = \frac{A}{B}$	$\epsilon_Q = \epsilon_A + \epsilon_B$
Power	$Q = A^n$	$\epsilon_Q = n \cdot \epsilon_A$
Root	$Q = \sqrt[n]{A}$	$\epsilon_Q = \frac{1}{n} \cdot \epsilon_A$

Examples

Example 1: Direct Relationship

Suppose we have two measured quantities, $A = 5.0 \pm 0.1$ and $B = 3.0 \pm 0.2$. The relative errors are:

$$ \epsilon_A = \frac{0.1}{5.0} = 0.02 \quad \text{and} \quad \epsilon_B = \frac{0.2}{3.0} = 0.0667 $$

If $Q = A \cdot B$, then the relative error in $Q$ is:

$$ \epsilon_Q = \epsilon_A + \epsilon_B = 0.02 + 0.0667 = 0.0867 $$

Example 2: Inverse Relationship

For the same values of $A$ and $B$, if $Q = \frac{A}{B}$, the relative error in $Q$ is still:

$$ \epsilon_Q = \epsilon_A + \epsilon_B = 0.02 + 0.0667 = 0.0867 $$

Example 3: Power Relationship

If $Q = A^2$, where $A = 5.0 \pm 0.1$, the relative error in $Q$ is:

$$ \epsilon_Q = 2 \cdot \epsilon_A = 2 \cdot 0.02 = 0.04 $$

Example 4: Root Relationship

If $Q = \sqrt{A}$, where $A = 5.0 \pm 0.1$, the relative error in $Q$ is:

$$ \epsilon_Q = \frac{1}{2} \cdot \epsilon_A = \frac{1}{2} \cdot 0.02 = 0.01 $$

Conclusion

Understanding how to calculate relative error in different relationships is essential for accurately determining the uncertainty in derived quantities. By following the formulas and examples provided, students and professionals can ensure that their calculations reflect the precision of their measurements and the inherent uncertainties in their experimental results.