Mathematics for Chemistry/Statistics

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Definition of errors[edit]

For a quantity x the error is defined as \Delta x. Consider a burette which can be read to ±0.05 cm3 where the volume is measured at 50 cm3.

  • The absolute error is \pm \Delta x, \pm 0.05~\text{cm}^3
  • The fractional error is \pm \frac{\Delta x}{x}, \pm \frac{0.05}{50} = \pm 0.001
  • The percentage error is \pm 100 \times \frac{\Delta x}{x} = \pm 0.1%

Combination of uncertainties[edit]

In an experimental situation, values with errors are often combined to give a resultant value. Therefore, it is necessary to understand how to combine the errors at each stage of the calculation.

Addition or subtraction[edit]

Assuming that \Delta x and \Delta y are the errors in measuring x and y, and that the two variables are combined by addition or subtraction, the uncertainty (absolute error) may be obtained by calculating

\sqrt{(\Delta x)^2 + (\Delta y)^2}

which can the be expressed as a relative or percentage error if necessary.

Multiplication or division[edit]

Assuming that \Delta x and \Delta y are the errors in measuring x and y, and that the two variables are combined by multiplication or division, the fractional error may be obtained by calculating

\sqrt{(\frac{\Delta x}{x})^2 + (\frac{\Delta y}{y})^2}