It is impossible to know a measured quantity exactly, to an infinite number of significant figures.

All measurements of physical quantities are liable to uncertainty, which should be expressed in absolute or percentage form.

Absolute uncertainties should be given to the same number of decimal places as the quantity they are paired with, significant figures notwithstanding.

A percentage uncertainty is calculated by taking the absolute uncertainty for a value over the best estimate of that value and then multiplying by a hundred.

These are three types of uncertainties in measured quantities:

• scale reading uncertainties

  → The scale reading uncertainty is an indication of how precisely an instrument scale can be read. — when measuring bigger quantities the scale reading uncertainty will be smaller

  <aside> 🧮 The general rule for calculating scale uncertainties is that for analogue instruments the scale uncertainty is $\pm$ 0.5 of the smallest division on the scale, and for digital instruments it is taken as $\pm$1 of the smallest interval on display.

  </aside>

• random uncertainties

  → Random uncertainties arise when measurements are repeated and slight variations occur.

  → Random uncertainties may be reduced by increasing the number of repeated measurements.

  <aside> 🧮 When mean values are used, the approximate random uncertainty should be calculated.

  The following relationship can be used to determine the approximate random uncertainty in a value using repeated measurements.

  $$ \Delta R=\dfrac{R_{max}-R_{min}}{n} $$

  Always exclude outliers from calculations of the experimental mean and the random uncertainty.

  </aside>

• systematic uncertainties

  → Systematic uncertainties occur when readings taken are either all too small or all too large.

  → Systematic uncertainties can arise due to measurement techniques or experimental design.

  → e.g. a clock which runs slowly or a meter that is not zeroed correctly.

The mean of a set of repeated measurements is the best estimate of the ‘true’ value of the quantity being measured.

→Always exclude outliers from this calculation.

When systematic uncertainties are present in an experiment, the mean value of a set of repeated measurements will be offset.

When an experiment is being undertaken and more than one physical quantity is measured, the quantity with the largest percentage uncertainty should be identified and this may often be used as a good estimate of the percentage uncertainty in the final numerical result of an experiment.

Where there is a measured quantity with both a reading uncertainty and a random uncertainty, both of these should be compared and the greater of the two taken as the final uncertainty value.

The numerical result of an experiment should be expressed in the form $final \space value \space ± \space uncertainty$.