Planning measurements and evaluating uncertainty

What you’ll learn

• Choose apparatus by its range, resolution and purpose.
• Calculate uncertainties and distinguish precision from accuracy.

Understand the idea

Start a practical plan with the quantity being changed, the response being measured, and the factors that could also affect that response. For a concentration comparison, temperature, total volume and the measurement method should be controlled. A volumetric pipette transfers one fixed volume; a burette measures a variable delivered volume; a measuring cylinder is appropriate when its larger uncertainty is acceptable. Record the reading at eye level and use the supplied apparatus uncertainty rather than assuming that resolution and uncertainty are identical.

A titre is the difference between two readings, so a maximum absolute uncertainty calculation adds the uncertainty of each reading. For products or quotients, add percentage uncertainties when the question asks for the usual maximum uncertainty estimate. Repeats help assess random variation, but cannot remove a common calibration offset. A tightly grouped set can therefore be precise while all its values are inaccurate. State the direction and mechanism of an error before suggesting a specific improvement.

A useful evaluation links the main uncertainty to an action: a larger measurable change reduces its percentage uncertainty, and a more suitable instrument may improve resolution. Neither action automatically removes systematic error. Safety reasoning should likewise connect the stated hazard to exposure: for a corrosive aqueous solution, eye protection and a supervised spill procedure address splashes; they do not justify handling an unidentified substance without the school's risk assessment.

Key formulae

$$V_{\mathrm{delivered}}=V_{\mathrm{final}}-V_{\mathrm{initial}}$$

$$u_{\Delta V}=u_{\mathrm{final}}+u_{\mathrm{initial}}$$

$$u_{\%}=\frac{u_{\mathrm{absolute}}}{|\mathrm{measured\ value}|}\times100$$

Worked example

A burette changes from $1.70$ to $20.30\ \mathrm{cm^3}$. Each reading has uncertainty $\pm0.05\ \mathrm{cm^3}$. Find the delivered volume and its maximum percentage uncertainty.

1. Subtract the readings: $V=20.30-1.70=18.60\ \mathrm{cm^3}$.
2. Two readings contribute: $u_V=0.05+0.05=0.10\ \mathrm{cm^3}$.
3. The percentage uncertainty is $100(0.10/18.60)=0.538\%$. State the volume as $(18.60\pm0.10)\ \mathrm{cm^3}$.

Answer: $18.60\ \mathrm{cm^3}$, with maximum percentage uncertainty about $0.54\%$.