1.3 Uncertainty in Measurement

1.3 Uncertainty in Measurement

You can count the number of pages of a book exactly but measurement of its length needs some measuring instrument. Every instrument is calibrated to a certain smallest division mark on it and this fact puts a limit regarding its accuracy. When you take a reading with one instrument, its limit of measurement is the smallest division or graduation on its scale. Hence, every measured quantity has some uncertainty about its value.

When a measurement is made, it is taken to the nearest graduation or marking on the scale. You can estimate the maximum uncertainty as being one smallest division of the instrument. This is called absolute uncertainty. It is one millimetre on a metre rule that is graduated in millimetres.

For example, if one edge of the book coincides with $10.0\,\mathrm{cm}$ mark and the other with $33.5\,\mathrm{cm}$, then the length with uncertainty is given by:

It means that the true length of the book is between $23.4\,\mathrm{cm}$ and $23.6\,\mathrm{cm}$. Hence, the maximum uncertainty is $\pm 0.05\,\mathrm{cm}$, which is equivalent to an uncertainty of $0.1\,\mathrm{cm}$. In fact, it is equal to least count of the metre rule.

Uncertainty may be recorded as:

Fractional uncertainty = Absolute uncertainty / Measured value

or

Percentage uncertainty = (Absolute uncertainty / Measured value) $\times 100\%$