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1.2.3Foundations in Physics

Determining Uncertainties from Graphs

5 min read 9 parts Lesson 7 of 19 1 quiz

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Lesson summary

1.2.3 Determining Uncertainties from Graphs — AQA A Level Physics 7408 notes: Use error bars and best and worst fit lines to find the gradient and intercept of a straight-line graph with uncertainties.A straight-line graph uses many measurements together, so the gradient and inte…

  • line of best fit
  • worst acceptable line
  • Worked example 1
  • g = 9.8 ± 0.5 m s−2
  • Worked example 2
  • Worked example 3

Section 1 of 9Why we use graphs

A straight-line graph uses many measurements together, so the gradient and intercept are determined more reliably than from one calculation.

Why we use graphs
To test a relationship, rearrange the equation into y = mx + c form,
  • where y and x are the plotted quantities, m is the gradient and c is the intercept.
For example, for a pendulum T2 = (4π2/g)L, plotting T2 (y) against L (x) gives a line through the origin with gradient 4π2/g.

Plot with the independent variable on the horizontal axis, label each axis with quantity and unit, choose a scale that makes the data fill more than half of the graph, and mark points with small, clear crosses.

Section 2 of 9Error bars

An error bar shows the uncertainty in a plotted value.

Error bars
A bar of length ±δy above and below the point means the true value probably lies between y − δy and y + δy.
Often the uncertainty in one variable (the one measured less precisely) is much larger,
  • and you draw bars only for that one;
    • if a question says that uncertainties in x are negligible, only vertical bars are used.

If the error bars are given as percentages, compute the absolute uncertainty for each point separately, since it varies with the value. When you plot a derived quantity such as T2, the percentage uncertainty in T2 is double that in T.

Section 3 of 9Best-fit and worst-fit lines

The line of best fit passes as near as possible to all points, with points scattered evenly on both sides, and must pass through the error bars (and through the origin only if theory says so, and the data supports it). A good best-fit line should pass through most error bars.

The worst acceptable line is the line with the greatest (or least) gradient that still passes through all the error bars. Draw two such lines: one steeper and one shallower than the best-fit line, usually pivoting about the points at the ends of the data set so that they pass through the top of the error bar at one end and the bottom at the other.

Then:

Best-fit and worst-fit lines
Gradient uncertainty = |best gradient − worst gradient|, or half the difference between the steepest and shallowest.
Percentage uncertainty in gradient = (gradient uncertainty / best gradient) × 100%.
Intercept uncertainty = |best intercept − worst intercept|, with both found where the line cuts the y-axis.

Section 4 of 9Calculating the gradient

Calculate the gradient using a large triangle: choose two points on the line (not data points, unless they lie on it) as far apart as possible, and use their coordinates read from the line.

Calculating the gradient
Gradient = Δy/Δx, including units.
A small triangle magnifies reading errors.
Then use the gradient to find the physical quantity: for T2 against L, g = 4π2/gradient,
  • and the percentage uncertainty in g equals that in the gradient.

The same method applies to V against I (gradient gives resistance, or in a cell internal resistance), force against extension (spring constant) and many more experiments. The y-intercept may have a physical meaning, such as the emf of a cell.

Section 5 of 9Checking against theory

If the accepted value of a constant lies within the range given by your gradient and its uncertainty, the experiment agrees with theory within the stated uncertainty.

Checking against theory
If the intercept is expected to be zero but the range does not include zero, a systematic error may be present.
Large scatter of points outside the error bars suggests underestimated uncertainties or a faulty method.
This topic builds on the combination of uncertainties and prepares you for the practical assessment,
  • where you must present data with uncertainties and justify conclusions.

Section 6 of 9Practical tips for drawing and reading graphs

A few habits make graph work reliable.

Practical tips for drawing and reading graphs
Use a sharp pencil and a clear rule, and plot points to within half a small square.
Do not force the line through the origin unless the physics demands it and the points support it; a line that does not pass through zero may reveal a systematic error.
Draw the best-fit line as a single thin, continuous line, not joined dots, and extend it to the y-axis if you need the intercept.

Choose the scales so that the plotted points fill at least half of the grid in both directions, and use simple scale steps such as 1, 2 or 5 units per large square, never 3 or 7. Read the coordinates of your gradient triangle to half a small square. When you state a conclusion, compare the experimental value with the accepted value and say whether it lies within the range implied by the uncertainty, which is the only valid test of agreement.

Section 7 of 9Worked examples

  1. A graph of T2 against L for a pendulum has a best-fit gradient of 4.05 s2 m−1. The worst-fit lines have gradients of 4.25 and 3.85 s2 m−1. Find g and its uncertainty.

  2. The length of a pendulum is measured as 0.800 m with a 0.2% uncertainty, and the period is 1.80 s ± 0.02 s. Find the percentage uncertainty in T2 for a graph's error bar.

  3. A V–I graph for a cell has a best-fit y-intercept of 1.52 V, and the worst-fit line meets the axis at 1.47 V. State the emf and its uncertainty.

Section 8 of 9Key terms

TermMeaning
Error barA bar through a data point showing the uncertainty in that value.
Best-fit lineThe line that passes close to all points with scatter on both sides.
Worst-fit lineThe steepest or shallowest line that still passes through all error bars.
GradientΔy/Δx of a line; units are the unit of y divided by the unit of x.
InterceptThe y value where the line crosses the y-axis.

Section 9 of 9Exam technique

1
Using data points rather than points on the line to calculate the gradient.
2
Using a small triangle for the gradient; this increases uncertainty.
3
Drawing a worst-fit line that does not pass through all error bars.
4
Not doubling the percentage uncertainty when a quantity is squared before plotting.
5
Forgetting the unit of the gradient, or taking it from the y-values without dividing by the x change.
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