Contents

Error Calculation

Here are some tips for error calculations!

Terminology

  • Absolute error \(\delta{A}\): The error in a quantity A that has the same units as the quantity.

  • Relative error \(\frac{\delta{A}}{A}\): The error expressed as a fractional or percentage.

Caution

These terms has confused a lot of students. So please be careful!

Rules with errors

Explanation

The maximum possible error of a sum or difference is calculated by adding the absolute errors in the quantities to be added or subtracted.

\[\delta{A+B} = \delta{A} + \delta{B}\]

Explanation

The relative maximum possible error of a product or quotient is equal to the sum of the relativeerrors in the quantities being multiplied or divided.

\[ \frac{\delta{(AB)}}{AB} = \frac{\delta{(A/B)}}{A/B}= (\frac{\delta{A}}{A} + \frac{\delta{B}}{B}) \]

Explanation

If a number x is raised to some power n then the relative error in the result is \(|n|\) times the relative error in x.

\[ \frac{\delta{x^n}}{x^n} = |n| \frac{\delta{x}}{x} \]

Attention

All formula above are to calculate the maximum error associated with the calculations.

Tips for applying rules

  • Here are the helpful steps that I went through all the pains to gather:

    • Step 1: Identify the operators. This helps find which rules will be applied.

    • Step 2: Identify large component. For example, in \(X = a^3 b^{-2} c\), \(a^3\) and \(b^{-2}\) are large component while c (only itself) is a small one. In this case, c is also the simplest/smallest component.

    • Step 3: Apply the rules above with large components as if they are small.

    • Step 4: Move into each large component. Identify the smaller components and operators inside it.

    • Step 5: Apply the rules for those smaller components you just identify.

    • Step 6: Continue until the you see the smallest/simplest components in your calculation.

Challenges

  • Try to solve these formula. Assume a, b, c, d are arbitrary terms with corresponding error \(\delta{a}\), \(\delta{b}\), \(\delta{c}\), \(\delta{d}\).

    • Problem 1: \(X = abcd\)

    • Problem 2: \(X = a+bcd\)

    • Problem 3: \(X = \frac{a^3 b^-2}{c} + d\)

    • Problem 4: \(X = \frac{c+d}{a^2 b^-1} + d\)

Danger of the Internet

  • When searching on the internet, there are many interpretation of the error calculations. Most of them are nowhere near similar to that of our physics labs.

  • However, they are not wrong. The only problem is that those cannot be applied in the context of our physics labs.

Warning

Always evaluate your findings. For example, check the source website, or contact your Profs/TAs. Do not ever trust anything on first sight.

Expectations Schedule