When one object is measured several times, the results can differ slightly. That is normal. A hand can press the tool with a different force, an eye can read the scale from a slightly different angle, and the object can shift a little.
Measurement uncertainty is an estimate of how far a measurement result may reasonably be from the value we report. For repeated measurements, uncertainty mainly reads the spread of the data.
Repeated measurements help reveal random error, meaning small changes that move up and down from one reading to another. Systematic error, such as a tool that does not start at zero or a damaged scale, still needs tool checking or calibration.
School books sometimes write repeated-data spread in the compact form below.
Δx=N1N−1N∑i=1Nxi2−(∑i=1Nxi)2
That formula is equivalent to the sample standard deviation:
Δx=sx=N−1∑i=1N(xi−xˉ)2
So the N1 in the compact form does not automatically mean the result is the formal uncertainty of the mean.
In formal metrology, NIST and the JCGM GUM describe the standard uncertainty of the mean as:
u(xˉ)=Nsx
For simple measurement practice, follow the convention used by the problem statement. If a problem uses Δx for the compact formula above, call Δx the repeated-measurement uncertainty for that problem model. The key is not to mix the two meanings without saying so.
Sometimes the measured data is diameter, but the required result is the area of the bottle cap. For a circle, the area is:
A=41πd2
Because area depends on diameter, each diameter reading gives a slightly different area. Use π≈3.14 for this example calculation, and keep guard digits during the calculation.
Do not round the area column too early and then square the shortened values. Rounding in the middle of the calculation can change the final uncertainty.
Relative uncertainty compares the uncertainty with the reported value.
εr=∣xˉ∣Δx×100%
For the bottle cap area:
εr=7.70060.0807×100%=1.05%
This means the spread of the data is about 1.05% of the reported area. That is small enough for 7.70 cm2 to be reasonable, but real enough that the uncertainty should still be written with the result.
The uncertainty references used here are NIST Technical Note 1297 for Type A evaluation and the JCGM GUM document from BIPM. NIST's Type A evaluation page can be opened through this source link, while BIPM's JCGM GUM document can be opened through this source link.
