Uncertainty in a Gradient

When a gradient (or intercept) gives a physical quantity, you need its uncertainty. Once error bars are drawn, two lines are used:• the line of best fit, passing through all the error bars with points evenly scattered either side;
• the worst acceptable line: the steepest or the shallowest straight line that still passes through all the error bars. It usually joins the bottom of the first error bar to the top of the last (steepest), or the top of the first to the bottom of the last (shallowest).The absolute uncertainty in the gradient is the difference between the best and worst gradients:percentage uncertainty = (|best gradient − worst gradient| ÷ best gradient) × 100%The same pair of lines gives the uncertainty in the y-intercept: the difference between the intercepts of the best and worst lines.Worked example: For a V against I graph, the line of best fit has gradient 4.0 V A⁻¹. The steepest line through all the error bars joins (0.5 A, 1.8 V) and (2.5 A, 10.2 V), so its gradient is 8.4 ÷ 2.0 = 4.2 V A⁻¹.
Absolute uncertainty = 4.2 − 4.0 = 0.2 V A⁻¹
Percentage uncertainty = 0.2 ÷ 4.0 × 100% = 5%
So R = 4.0 ± 0.2 Ω.Use large triangles (covering more than half of each line) when reading gradients, so that reading errors are small. If the gradient is then combined with other measurements, add its percentage uncertainty to theirs.