# Mathematics for Chemistry/Statistics

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## 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.

### Addition or subtraction

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

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

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

### Multiplication or division

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

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