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Uncertainty

Physics: AQA A Level, topic 1.1.8

Uncertainty

Uncertainty is a measure of how confident you can be in a measurement. All measurements contain some amount of uncertainty. Let's say we measure a piece of string to be 10cm (to the nearest cm):

Absolute uncertainty

Absolute uncertainty

  • The absolute uncertainty is the range of possible real values for the length of the string.
  • Length of string = 10cm ± 0.5cm
    • Our absolute uncertainty is ± 0.5cm.
Fractional uncertainty

Fractional uncertainty

  • The fractional uncertainty is the absolute uncertainty divided by the measured value.
  • Fractional uncertainty = 0.5cm10cm=120\frac{0.5cm}{10cm} = \frac{1}{20}
Percentage uncertainty

Percentage uncertainty

  • The percentage uncertainty is the fractional uncertainty expressed as a percentage (i.e. multiplied by 100%).
  • Percentage uncertainty = 120×100=5%\frac{1}{20} \times 100 = 5\%

Uncertainty on Graphs

The range of possible values for a measurement can be shown as a line on an uncertainty graph. We call this line an error bar. Some points on a graph might not have error bars.

Error bars

Error bars

  • Small uncertainties are shown by small error bars.
  • Large uncertainties are shown by large error bars.
Finding the uncertainty from a graph

Finding the uncertainty from a graph

  • A systematic uncertainty can be spotted if the theory and the results do not match for when the independent variable is zero.
  • For example, when measuring the current in a circuit we would expect the current to be zero when the supply potential difference is zero.
Finding the uncertainty from a graph 2

Finding the uncertainty from a graph 2

  • If there is a systematic uncertainty, this expectation would not come true.
  • We would see that the intercept of the line wasn't zero.
  • The intercept would indicate the size of the systematic error.
  • Once the systematic error has been found, we can use the graph to find an estimate for the random error.
Using error bars

Using error bars

  • Error bars can be used to find the steepest and shallowest possible lines of best fit.
  • The steepest possible line of best fit will have a gradient g1.
  • The shallowest possible line of best fit will have a gradient g2.
  • The uncertainty in the measurement should then be quoted as g1+g22±g1−g22\frac{g_1 + g_2}{2} \pm \frac{g_1 - g_2}{2}.

Test yourself on Uncertainty

  • We measure the mass of a metal weight as 20 kg ± 2 kg. What is the absolute uncertainty?

  • We measure the mass of a plant pot as 20 kg ± 2 kg. What is the percentage uncertainty?

  • We measure the mass of a table as 20 kg ± 2 kg. What is the fractional uncertainty?

  • What feature of a graph tells us the value of a systematic error?

  • What is the equation for the uncertainty from a graph?

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