Error Analysis

The following techniques are used to determine how error propagates through an experimental procedure. This method is based upon combining the uncertainty for each step.

Calculations

 Symbols used:
  1. x; result of calculation
  2. a, b & c; numbers used for calculation
  3. sx; uncertainty in result
  4. sa, sb & sc; uncertainty in numbers used for calculation

 Operations
  1. Addition and Subtraction
    x = a + b - c
  2. Multiplication and Division
    x = a * b/c
  3. Exponentiation (assuming no uncertainty in b)
    x = ab

  4. Logarithm
    1. Base 10 x = log10 a

    2. Base e x = ln a

  5. Antilog
    1. Base 10 x = 10a a

    2. Base e x = ea

Example

  1. From a calibration curve the concentration of an unknown is 16±2 ppm.

  2. The solution was prepared by:
    • dissolving 0.0452±0.0001 g of compound
    • in 250.0±0.1 ml of water

  3. From this the weight of unknown in the compound is:

    • x = 16 ppm * 250 ml * (mg/l) * (1 g/1000 mg) * (1 l/1000 ml)
    • x = 0.0040 g of unknown in the compound

  4. The uncertainty in this weight is:

  5. Therefor the weight of the unknown is:

    0.0040±0.0005 g (or 4.0±0.5 mg)