Graphs & Data Analysis

Plotting Graphs

• The independent variable goes on the x-axis; the dependent variable goes on the y-axis.
• Label each axis with the quantity and unit in the format: Quantity / Unit (e.g. "Force / N").
• Choose scales so that at least 50% of the graph grid is used in both directions. Avoid awkward multiples like 3 or 7.
• Plot at least six data points, each accurate to within half a small square, using a sharp pencil.
• Points must not be obscured by the line of best fit.

Line or Curve of Best Fit

• There should be roughly equal numbers of points above and below the line of best fit.
• The line should not be too thick, and should not be drawn dot-to-dot.
• Not all lines will pass through the origin -- do not force the line through (0, 0) unless the data supports it.
• Anomalous resultsData points that do not fit the general trend. They should be identified, circled, and excluded from the line of best fit. should be identified and excluded, as they have a large effect on the gradient.

Calculating the Gradient

• The gradient is found using: $$\text{gradient} = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}$$
• Draw a large right-angled triangle on the line of best fit -- small triangles introduce significant error.
• Use points that lie on the line, not individual data points that may be off the line.
• Show all working, including the correct substitution of coordinate values.

The y-intercept

Linearising Relationships

• To test a non-linear relationship, rearrange the equation into the form $y = mx + c$ and plot appropriate variables.
• If a straight line through the origin is obtained, the variables are directly proportional.
• The gradient and y-intercept often have physical significance (e.g. gradient = spring constant from $F = kx$).
• Use transformations such as reciprocals, squares, or logarithmsLogarithmic plots can linearise exponential relationships. For example, plotting ln(A) against time gives a straight line for radioactive decay. to linearise data.

Key Graph Skills for A-Level

• Straight-line relationships: interpret gradient and y-intercept physically.
• Areas under graphs: calculate areas (e.g. distance from a velocity-time graph).
• Tangents to curves: draw tangents at specific points and calculate their gradients to find instantaneous rates of change.
• Logarithmic plots: use log or ln plots to linearise exponential or power-law data.
• Asymptotes: recognise where curves approach but never reach a limiting value.