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

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

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