Measurement and Data Processing

Accuracy: how close to the true value Precision: how close repeated measurements are to each other

Absolute Uncertainty

The ± value on a measurement (e.g. 25.0 ± 0.1 cm³).

For analogue instruments: ± half the smallest division For digital instruments: ± the smallest digit

Percentage Uncertainty

$\%\text{uncertainty} = \frac{\text{absolute uncertainty}}{\text{measured value}} \times 100$

Propagation of Uncertainties

Addition/Subtraction: add absolute uncertainties $\Delta(A \pm B) = \Delta A + \Delta B$

Multiplication/Division: add percentage uncertainties $\%\Delta(A \times B) = \%\Delta A + \%\Delta B$

Powers: multiply percentage uncertainty by the power $\%\Delta(A^n) = n \times \%\Delta A$

Rules:

• Non-zero digits are always significant
• Zeros between non-zero digits are significant
• Leading zeros are NOT significant
• Trailing zeros after decimal point ARE significant

In calculations, round to the fewest significant figures of the data.

Best Fit Lines

• Draw through as many points as possible
• Equal scatter above and below the line
• Can be straight or curved

Gradient Calculation

$\text{gradient} = \frac{\Delta y}{\Delta x}$

Use points on the line (not necessarily data points), as far apart as possible.

Uncertainty in Gradient

Draw maximum and minimum gradient lines through error bars. The uncertainty is half the range: $\Delta m = \frac{m_{max} - m_{min}}{2}$

Intercept

Where the line crosses the y-axis. Uncertainty from the range of intercepts from max/min lines.

$\%\text{error} = \frac{|\text{experimental} - \text{accepted}|}{\text{accepted}} \times 100$

If % error > total % uncertainty → systematic error present If % error ≤ total % uncertainty → random error accounts for discrepancy

Question: A student measures 25.0 ± 0.1 cm³ of solution and 10.0 ± 0.1 cm³ of another. Total volume?

Total = 35.0 cm³ Absolute uncertainty = 0.1 + 0.1 = ± 0.2 cm³ Answer: 35.0 ± 0.2 cm³

% uncertainty = (0.2/35.0) × 100 = 0.57%

• Always record raw data with units AND uncertainties
• Process data showing all working
• Include error bars on graphs
• Discuss systematic vs random errors in evaluation
• Compare % error with % uncertainty to evaluate method

• Random errors: reduce by repeating; systematic: improve method
• Uncertainty propagation: add (±), add % (×÷), multiply % (powers)
• Graphs: best fit line, gradient, error bars, max/min lines
• % error: experimental vs accepted; compare with uncertainty
• Significant figures: match the least precise data