What is a line of best fit?
A line of best fit (or curve of best fit) is a single line or smooth curve that passes as close as possible to all data points, with approximately equal numbers of points above and below it. It represents the underlying trend in the data, smoothing out random experimental errors.
A line of best fit does NOT have to pass through the origin unless there is a physical reason it should (e.g. V=IR passes through origin if resistance is constant and y-intercept is zero). A line of best fit also does NOT have to pass through any specific data point — it balances the scatter on both sides.
Rules for drawing best-fit lines
- 1Plot all points first — use small, neat crosses (×) or dots with circles (⊙). Never use thick blobs.
- 2Identify anomalous points — any point clearly far from the trend should be circled and excluded from the line. Do not include anomalies in best-fit line.
- 3Straight line or curve? — If the pattern is linear, use a ruler. If curved, draw a single smooth freehand curve. Never draw a dot-to-dot jagged line.
- 4Balance the points — roughly equal numbers of points above and below the line/curve along its full length.
- 5Extend the line — if asked for the y-intercept or x-intercept, extend the line to the relevant axis using a dashed line extension.
Drawing a dot-to-dot line through every point instead of a smooth best-fit. Forcing the line through the origin when it should not be. Including anomalous points in the best-fit. Drawing a thick line — use a sharp pencil.
Step-by-step gradient calculation
- 1Choose two points on the LINE — not data points, unless they happen to sit exactly on the line. Use points as far apart as possible.
- 2Draw a large right-angle triangle — mark it clearly on the graph with dotted lines.
- 3Read off Δy and Δx — using the axis scale, read the vertical and horizontal spans of your triangle.
- 4Calculate: gradient = Δy / Δx — include correct units (y-units / x-units).
- 5Check the sign — positive gradient means y increases as x increases. Negative gradient means y decreases as x increases.
The gradient of a v-t graph = acceleration (m/s²). The gradient of a distance-time graph = speed (m/s). The gradient of a V-I graph = resistance (Ω). Always state what the gradient represents and include units.
Using data points instead of points on the line. Using a small triangle (amplifies reading errors). Forgetting units. Not using the full axis scale when reading coordinates.
v-t graph gradient
Gradient = Δv/Δt = change in velocity / time = acceleration. Units: (m/s)/s = m/s². A negative gradient = deceleration.
d-t graph gradient
Gradient = Δd/Δt = distance / time = speed. Units: m/s. Steeper gradient = faster speed. Horizontal section = stationary.
V-I graph gradient
Gradient = ΔV/ΔI = resistance R. Units: V/A = Ω. For an ohmic conductor the graph is a straight line through origin — constant gradient = constant resistance.
Force-extension graph
Gradient = ΔF/Δx = spring constant k. Units: N/m or N/cm. The gradient gives the stiffness of the spring.
Why tangents are needed
When a graph is curved (not straight), the gradient is not constant — it changes at every point. To find the rate of change at a specific point, you draw a tangent: a straight line that just touches the curve at that point without crossing it (except at the point itself).
The gradient of the tangent at a point = the instantaneous rate of change of the variable at that point. This is used to find: instantaneous acceleration from a d-t graph; instantaneous rate of reaction from a volume-time curve; instantaneous speed from a position-time curve.
How to draw a tangent correctly
- 1Mark the point on the curve where you need the gradient.
- 2Place a ruler so it touches the curve at that point only — the ruler should not cut through the curve on either side of the point.
- 3Adjust the angle — rotate the ruler until the gaps between the curve and the ruler are equal on both sides of the point. The curve should "mirror" away from the ruler symmetrically.
- 4Draw a long straight line — the longer the tangent line, the easier to calculate a precise gradient. Extend it well beyond the point in both directions.
- 5Calculate the gradient of the tangent using Δy/Δx with a large triangle, as in Skill 2.
Drawing a chord (line connecting two points on the curve) instead of a tangent. Drawing the tangent correctly but then using a tiny triangle to calculate gradient. Not labelling which point the tangent is drawn at.
Straight line through origin
Meaning: Direct proportion. y ∝ x. e.g. V vs I (ohmic resistor), F vs a (Newton's 2nd Law for constant mass), extension vs force (Hooke's Law within limit).
Straight line, y-intercept ≠ 0
Meaning: Linear relationship but NOT direct proportion. e.g. Temperature vs time for a heating substance; velocity vs time with initial velocity (SUVAT: v = u + at, intercept = u).
Curve, increasing gradient
Meaning: Rate of change is increasing. e.g. Distance vs time for an accelerating object (d = ½at²); exponential growth. As x increases, y increases faster and faster.
Curve, decreasing gradient (levelling off)
Meaning: Rate of change is decreasing. e.g. Rate of reaction slowing as reactants consumed; terminal velocity being approached; radioactive decay curve. As x increases, y increases more and more slowly.
Horizontal line
Meaning: y is constant, independent of x. e.g. Constant velocity on v-t graph (zero acceleration); temperature during state change (latent heat); terminal velocity plateau.
Exponential decay curve
Meaning: y decreases by the same fraction in equal intervals of x. e.g. Radioactive decay (activity vs time). The curve approaches the x-axis but never reaches it. Half-life is read from the graph.
Anomalous results
An anomalous (outlier) result is a data point that does not follow the general trend and lies clearly away from the best-fit line. To deal with anomalous results: (1) circle the point, (2) exclude it from the best-fit line, (3) repeat the measurement if possible, (4) comment on a possible cause (e.g. timing error, contamination, misreading the scale).
Always specify the exact source: "parallax error when reading the meniscus in the burette" or "reaction time error when starting the stopwatch at the moment of mixing." "Human error" without specification earns no marks.
Reading intercepts
y-intercept: the value of y when x = 0. Extend the best-fit line to the y-axis if necessary. In v = u + at, the y-intercept = u (initial velocity). In y = mx + c, y-intercept = c.
x-intercept: the value of x when y = 0. Extend the line to the x-axis. Less commonly tested but may appear in cooling curves or charging/discharging problems.
Extrapolation: extending the best-fit line beyond the data range to estimate values. This assumes the same relationship continues — state this assumption. Extrapolation is less reliable than interpolation (reading within the data range).
(a) Initial velocity = y-intercept = 6 m/s. [1]
(b) Gradient = (30−6)/(8−0) = 24/8 = 3 m/s² (acceleration). [1]
(c) The object is moving in the positive direction throughout (velocity goes from 6 m/s upward, never crosses zero in this scenario). If instead the question described initial velocity as negative (decelerating from rest), the x-intercept gives the time at which v=0. Using v=0, 0=6+3t is not applicable here since gradient is positive. If u were −6 m/s with gradient +3: 0=−6+3t → t = 2 s. [1 for method, 1 for answer in a suitably re-worded version]