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Uncertainty

Learning Objectives. Uncertainty. To appreciate that all physical readings contain may contain errors and hence there is uncertainty To be able to conduct basic uncertainty calculations To be able to use key terms such as accurate, precise and reliable correctly.

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Uncertainty

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  1. Learning Objectives Uncertainty To appreciate that all physical readings contain may contain errors and hence there is uncertainty To be able to conduct basic uncertainty calculations To be able to use key terms such as accurate, precise and reliable correctly. Introductory Guide: Pages 19-23

  2. A significant historical example... Anecdote! Reference needed Kepler is credited with much of the early work on establishing the orbital paths of the planets Current views had a flat Earth with the planets revolving around it Kepler used careful observation and measurement to establish that the planets in our solar system revolved around the sun. Uncertainty

  3. His results showed broadly circular orbits with a slightly elliptical character A lesser scientist would simply have said, the planets move in circular orbits and the deviation between observed behaviour and my nice new circular orbit law is simply experimental error. However, Kepler carried out a very detailed analysis of the errors or uncertainty in his readings and found that the elliptical nature of the orbits was too significant to be explained by experimental error alone. He declared the orbits to be elliptical! Uncertainty

  4. When we take an experimental reading we hope that is it near the true value (accurate) However all of our readings will contain errors and so there is a level of uncertainty or doubt for each value Error is the difference between the measured value and the ‘true value’ of the thing being measured. Uncertainty is a quantification of the doubt about the measurement result. Taking Readings

  5. Our results may contain systematic sources of error : For example a micrometer which does not read zero at zero (find this value and subtract from all readings) A metre rule with the end worn away (use the middle of the ruler) Our results may contain many random sources of error Human misjudgements (reaction time, parallax errors) Environmental factors (change in temperature, wind) Equipment limitations Taking Readings

  6. It is always desirable to repeat readings when possible (NB. practical exam time is effectively unlimited) Once we have repeated readings we can comment upon the reliability of the data.... (the range or variance within the results) Taking “3 repeat readings” and finding the mean average has become popular. However, the number of repeat readings should reflect the spread obtained within the readings Reliable data will be broadly repeatable in terms of the values obtained Taking Readings

  7. For a single readings, we are short of additional data to comment upon the reliability of the results and so.... The uncertainty is simply the precision of the measuring device. Consider a typical meter rule with a mm scale Measuring something around foot long may return a reading of say 30cm, we can specify the uncertainty as  1mm * Our reading would be 0.300  0.001 m * Many people argue that the greatest uncertainty is half a division... i.e. 0.5mm Uncertainty : Single Readings

  8. Once multiple readings have been taken we have a much better first hand idea of the uncertainties involved.... We are considering the method as well as the precision of the measuring devices. We can see it in the data. For repeated readings the uncertainty is reflected in the range of readings obtained. For a simple analysis we may consider the uncertainty to be half of the range in the results. Consider the following resistance readings : 609; 666; 639; 661; 654; 628Ω Our mean average value is 643Ω. The Largest value is 666Ω, while the smallest is 609Ω Uncertainty : Repeated Readings

  9. The range within our readings is (666-609) = 57Ω We can estimate our uncertainty to be 57/2Ω And so we quote our measurement as 643  29Ω Note a more thorough analysis can be employed which uses “standard deviation” Uncertainty : Repeated Readings

  10. So far what we have been specifying is an absolute uncertainty. We have our reading plus or minus an absolute value. Fractional uncertainty = absolute uncertainty / value This can be multiplied by 100% to achieve a percentage. From our last example : Fractional uncertainty = 29 / 643 = 0.045 Or 4.5% Specifying Uncertainty

  11. Adding or subtracting quantities: add absolute uncertainties Multiplying of dividing quantities: add percentage uncertainties Raising to a power quantities: multiply percentage uncertainty by the power Combining Uncertainties

  12. Constants in uncertainty calculations For example if we have an estimate for the uncertainty in a radius how is this reflected in the circumference calculation? Circumference = 2π x radius The Rules Multiplying a number by a constant there are 2 rules depending on which type of uncertainty you have : Rule - Absolute Uncertainty: c(A ± ΔA) = cA ± c(ΔA) Consider: 1.5(2.0 ± 0.2) m = (3.0 ± 0.3) m Absolute Uncertainty is multiplied by the constant. Rule - Relative Uncertainty: c(A ± εA) = cA ± εA Consider: 1.5(2.0 m ± 1.0%) = (3.0 m ± 1.0%) Relative Uncertainty is not multiplied by the constant. Combining Uncertainties 2

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