# 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.

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