Determining Uncertainties from Graphs

Calculating Gradient Uncertainty

• Step 1: Plot all data points and draw accurate error bars for each. Ensure the scale allows the graph to fill at least half of the available space for maximum precision.

• Step 2: Draw the Line of Best Fit and calculate its gradient ($m_{best}$). Use a large triangle (at least half the length of the line) to minimize calculation errors.

• Step 3: Draw the Worst-Fit Line. This is usually done by connecting the bottom of the first error bar to the top of the last error bar (steepest) or the top of the first to the bottom of the last (shallowest).

• Step 4: Calculate the gradient of the worst-fit line ($m_{worst}$). The absolute uncertainty in the gradient is then determined by the difference between the two gradients.

Uncertainty Formula: $\Delta m = |m_{best} - m_{worst}|$

• Step 5: To find the percentage uncertainty, divide the absolute uncertainty by the best gradient and multiply by 100.

Percentage Uncertainty: $\frac{|m_{best} - m_{worst}|}{m_{best}} \times 100\%$

• Absolute vs. Percentage Uncertainty: Absolute uncertainty provides the range in the same units as the measurement (e.g., $\pm 0.5$ USD), while percentage uncertainty expresses the error relative to the size of the measurement, allowing for comparison across different scales.

• Scale Selection: Always choose scales that are easy to read (multiples of 1, 2, 5, or 10). Avoid 'awkward' scales like 3 or 7, as these lead to plotting errors and lost marks.

• The 'Half-Range' Alternative: In some contexts, examiners may accept the uncertainty as half the range between the steepest and shallowest lines: $\Delta m = \frac{|m_{max} - m_{min}|}{2}$. Always check the specific requirements of your syllabus.

• Intercept Uncertainty: If asked for the uncertainty in the y-intercept, apply the same logic as the gradient. Calculate the intercept of the best-fit line ($c_{best}$) and the worst-fit line ($c_{worst}$), then find the difference: $\Delta c = |c_{best} - c_{worst}|$.

• Sanity Check: Ensure your worst-fit line actually passes through all error bars. If it misses one, it is not a valid representation of the experimental uncertainty.

• Forcing through the Origin: A common mistake is forcing the line of best fit through $(0,0)$ when the data does not support it. If there is a systematic error, the line should show a non-zero intercept even if theory predicts zero.

• Small Gradient Triangles: Using points that are too close together to calculate the gradient increases the relative error of the calculation. Always use points that are far apart on the line.

• Ignoring Error Bars: Students often draw a line of best fit but then draw a 'worst' line that is just slightly different without actually referencing the error bars. The worst-fit line must be constrained by the error bars.