Mathematics for Chemistry/Statistics

Definition of errors

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

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

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 $\mathrm{\Delta }x$ and $\mathrm{\Delta }y$ 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

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

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

Multiplication or division

Assuming that $\mathrm{\Delta }x$ and $\mathrm{\Delta }y$ 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

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