Mathematics for Chemistry/Statistics

Definition of errors
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
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
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
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

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