Revise Errors and Uncertainties in Paper 5 for Cambridge 9702 (1.3)

Every measurement you take is subject to errors. These are not 'mistakes' but inherent variations in the measurement process. They are broadly classified into two types: systematic errors and random errors.

These two terms are often used interchangeably in everyday language, but in physics, they have distinct meanings.

• Accuracy is how close a measured value is to the true or accepted value.
• Precision is how close repeated measurements are to each other.

Imagine an archer shooting at a target:

• Accurate and Precise: All arrows are in the bullseye.
• Precise but not Accurate: All arrows are clustered together, but far from the bullseye. This suggests a systematic error (e.g., the bow's sight is misaligned).
• Accurate but not Precise: The arrows are scattered all over the target, but their average position is the bullseye. This suggests significant random errors.
• Neither Accurate nor Precise: The arrows are scattered all over the target, and their average is not in the bullseye.

When you use measured quantities to calculate a new quantity, the uncertainties in your measurements 'propagate' to the final result. You need to know the rules for combining them.

In Paper 5, you'll often need to calculate uncertainties for values in a table, especially for quantities that are reciprocals or logarithms of your measurements.

For a reciprocal, $y = 1/x$: The fractional uncertainty rule gives $\frac{\Delta y}{y} = \frac{\Delta x}{x}$. Rearranging for the absolute uncertainty $\Delta y$ gives $\Delta y = y \frac{\Delta x}{x} = \frac{1}{x} \frac{\Delta x}{x} = \frac{\Delta x}{x^2}$.

For a logarithm, $y = \lg(x)$: The simplest method is to find the range: $\Delta y = \lg(x_{max}) - \lg(x) = \lg(x + \Delta x) - \lg(x)$. Alternatively, you can use $\Delta y = \lg(x) - \lg(x - \Delta x)$. Both are acceptable.

A core task in Paper 5 is to determine the gradient and y-intercept of a graph and find the uncertainties in these values. This is done by drawing a line of best fit (LOBF) and a worst acceptable line (WAL).

1. Plot error bars: For each point, draw a vertical bar representing its absolute uncertainty.
2. Draw the LOBF: Draw a single straight line that passes as close as possible to all data points, with a balance of points above and below the line.
3. Draw the WAL: This is the line with the steepest or shallowest gradient that still passes through all the error bars. You only need to draw one WAL (either steepest or shallowest). To draw the steepest WAL, pivot the line around the lowest error bar until it touches the highest error bar. For the shallowest, pivot around the highest error bar until it touches the lowest.