# Mathematics for Chemistry/Statistics

## Definition of errors

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

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

## 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.

Assuming that ${\displaystyle \Delta x}$  and ${\displaystyle \Delta y}$  are the errors in measuring ${\displaystyle x}$  and ${\displaystyle y}$ , and that the two variables are combined by addition or subtraction, the uncertainty (absolute error) may be obtained by calculating
${\displaystyle {\sqrt {(\Delta x)^{2}+(\Delta y)^{2}}}}$
Assuming that ${\displaystyle \Delta x}$  and ${\displaystyle \Delta y}$  are the errors in measuring ${\displaystyle x}$  and ${\displaystyle y}$ , and that the two variables are combined by multiplication or division, the fractional error may be obtained by calculating
${\displaystyle {\sqrt {({\frac {\Delta x}{x}})^{2}+({\frac {\Delta y}{y}})^{2}}}}$