(d) graphical treatment of errors and uncertainties; line of best fit;…

The idea

Graphical treatment means plotting data with error bars, then drawing a line of best fit through the points so that the trend is represented fairly. A worst line is the steepest or shallowest line that still passes through every error bar; the spread between worst lines gives the uncertainty in the gradient or intercept. Absolute uncertainty carries the same unit as the quantity, for example 0.5 cm, while percentage uncertainty is the absolute uncertainty divided by the measured value, multiplied by 100, for example 0.5 cm ÷ 20.0 cm × 100 = 2.5%. Percentage difference compares an experimental value with an accepted value: difference ÷ accepted value × 100. Always label axes with quantity and unit, choose sensible scales, and state the uncertainty from the graph rather than guessing.

What earns the marks

• Error bars represent the absolute uncertainty in each plotted point and must be drawn to the scale of the axes.
• The line of best fit balances points above and below it and should pass through as many error bars as possible.
• A worst line is the steepest or shallowest straight line consistent with all error bars, giving the range of possible gradients.
• Absolute uncertainty has the same unit as the measurement; percentage uncertainty = (absolute uncertainty ÷ measured value) × 100.
• Percentage difference = (difference between experimental and accepted value ÷ accepted value) × 100.
• Gradient uncertainty can be estimated as half the difference between the steepest and shallowest worst-line gradients.

Common mistakes and how to put them right

• Drawing error bars only in one direction.

  Put it right: Error bars must extend both above and below each point by the stated absolute uncertainty.

• Confusing percentage uncertainty with percentage difference.

  Put it right: Percentage uncertainty uses the measured value as the denominator, while percentage difference uses the accepted value.

• Choosing a line of best fit that passes through every point exactly.

  Put it right: Real data scatter, so the line should balance the scatter rather than join points.

• Forgetting to convert units before calculating a percentage.

  Put it right: Both quantities in the fraction must have the same unit.

Exam tips

• Label each axis with the quantity symbol, quantity name and unit, for example time t / s, before plotting.
• When estimating gradient uncertainty, calculate the steepest and shallowest gradients from the worst lines and halve their difference.
• Check that any percentage calculation uses consistent units and quote the result to a sensible number of significant figures.