1.1. To appreciate that all physical readings To appreciate that all physical readings contain may contain errors and hence there contain may contain errors and hence there is uncertaintyis uncertainty

2.2. To be able to conduct basic uncertainty To be able to conduct basic uncertainty calculationscalculations

3.3. To be able to use key terms such as accurate, To be able to use key terms such as accurate, precise and reliable correctly.precise and reliable correctly.

Introductory Guide: Pages 19-23Introductory Guide: Pages 19-23

A significant historical example...A significant historical example...

Anecdote! Reference neededAnecdote! Reference needed

Kepler is credited with much of the early work on Kepler is credited with much of the early work on establishing the orbital paths of the planetsestablishing the orbital paths of the planets

Current views had a flat Earth with the planets Current views had a flat Earth with the planets revolving around itrevolving around it

Kepler used careful observation and measurement to Kepler used careful observation and measurement to establish that the planets in our solar system revolved establish that the planets in our solar system revolved around the sun.around the sun.

His results showed broadly circular orbits with a slightly His results showed broadly circular orbits with a slightly elliptical characterelliptical character

A lesser scientist would simply have said, the planets A lesser scientist would simply have said, the planets move in circular orbits and the deviation between move in circular orbits and the deviation between observed behaviour and my nice new circular orbit law is observed behaviour and my nice new circular orbit law is simply experimental error.simply experimental error.

However, Kepler carried out a very detailed analysis of However, Kepler carried out a very detailed analysis of the errors or uncertainty in his readings and found that the errors or uncertainty in his readings and found that the elliptical nature of the orbits was too significant to the elliptical nature of the orbits was too significant to be explained by experimental error alone.be explained by experimental error alone.

He declared the orbits to be elliptical!He declared the orbits to be elliptical!

When we take an experimental reading we When we take an experimental reading we hopehope that that is it is it near the true value (accurate)near the true value (accurate)

However all of our readings However all of our readings willwill contain errors and so contain errors and so there is a level of uncertainty or doubt for each valuethere is a level of uncertainty or doubt for each value

Error is the difference between the measured value Error is the difference between the measured value and the ‘true value’ of the thing beingand the ‘true value’ of the thing beingmeasured.measured.

Uncertainty is a quantification of the doubt about the Uncertainty is a quantification of the doubt about the measurement result.measurement result.

Our results may contain systematic sources of error :Our results may contain systematic sources of error :

For example a micrometer which does not read zero at For example a micrometer which does not read zero at zero (find this value and subtract from all readings)zero (find this value and subtract from all readings)

A metre rule with the end worn away (use the middle A metre rule with the end worn away (use the middle of the ruler)of the ruler)

Our results may contain many random sources of errorOur results may contain many random sources of error

Human misjudgements (reaction time, parallax errors)Human misjudgements (reaction time, parallax errors)

Environmental factors (change in temperature, wind)Environmental factors (change in temperature, wind)

Equipment limitationsEquipment limitations

It is always desirable to repeat readings when possible It is always desirable to repeat readings when possible (NB. practical exam time is effectively unlimited)(NB. practical exam time is effectively unlimited)

Once we have repeated readings we can comment Once we have repeated readings we can comment upon the upon the reliabilityreliability of the data.... (the range or of the data.... (the range or variance within the results)variance within the results)

Taking “3 repeat readings” and finding the mean Taking “3 repeat readings” and finding the mean average has become popular. However, the number of average has become popular. However, the number of repeat readings should reflect the spread obtained repeat readings should reflect the spread obtained within the readingswithin the readings

Reliable data will be broadly repeatable in terms of Reliable data will be broadly repeatable in terms of the values obtainedthe values obtained

For a For a singlesingle readings, we are short of additional data readings, we are short of additional data to comment upon the to comment upon the reliabilityreliability of the results and of the results and so.... so....

The uncertainty is simply the precision of the The uncertainty is simply the precision of the measuring device. Consider a typical meter rule with a measuring device. Consider a typical meter rule with a mm scalemm scale

Measuring something around foot long may return a Measuring something around foot long may return a reading of say 30cm, we can specify the uncertainty as reading of say 30cm, we can specify the uncertainty as 1mm * 1mm *

Our reading would be 0.300 Our reading would be 0.300 0.001 m 0.001 m

* Many people argue that the greatest uncertainty is * Many people argue that the greatest uncertainty is half a division... i.e. 0.5mmhalf a division... i.e. 0.5mm

Once multiple readings have been taken we have a much Once multiple readings have been taken we have a much better first hand idea of the uncertainties involved.... We better first hand idea of the uncertainties involved.... We are considering the method as well as the precision of the are considering the method as well as the precision of the measuring devices. We can see it in the data.measuring devices. We can see it in the data.

For repeated readings the uncertainty is reflected in the For repeated readings the uncertainty is reflected in the range of readings obtained.range of readings obtained.

For a simple analysis we may consider the uncertainty to For a simple analysis we may consider the uncertainty to be half of the range in the results.be half of the range in the results.

Consider the following resistance readings : 609; 666; Consider the following resistance readings : 609; 666; 639; 661; 654; 628Ω639; 661; 654; 628Ω

Our mean average value is 643Ω. The Largest value is Our mean average value is 643Ω. The Largest value is 666Ω, while the smallest is 609Ω666Ω, while the smallest is 609Ω

The range within our readings is (666-609) = 57The range within our readings is (666-609) = 57ΩΩ

We can estimate our uncertainty to be 57/2We can estimate our uncertainty to be 57/2ΩΩ

And so we quote our measurement as And so we quote our measurement as

643 643 29 29ΩΩ

Note a more thorough analysis can be employed which Note a more thorough analysis can be employed which uses “standard deviation”uses “standard deviation”

So far what we have been specifying is an absolute So far what we have been specifying is an absolute uncertainty. We have our reading plus or minus an uncertainty. We have our reading plus or minus an absolute value.absolute value.

Fractional uncertainty = absolute uncertainty / valueFractional uncertainty = absolute uncertainty / value

This can be multiplied by 100% to achieve a This can be multiplied by 100% to achieve a percentage.percentage.

From our last example :From our last example :

Fractional uncertainty Fractional uncertainty = 29 / 643= 29 / 643

= 0.045= 0.045

Or 4.5%Or 4.5%

Adding or subtracting quantities: Adding or subtracting quantities: add absolute add absolute uncertaintiesuncertainties

Multiplying of dividing quantities: Multiplying of dividing quantities: add percentage add percentage uncertaintiesuncertainties

Raising to a power quantities:Raising to a power quantities: multiply percentage multiply percentage uncertainty by the poweruncertainty by the power

ConstantsConstants in uncertainty calculations in uncertainty calculations

For example if we have an estimate for the uncertainty in a For example if we have an estimate for the uncertainty in a radius how is this reflected in the circumference calculation? radius how is this reflected in the circumference calculation?

Circumference = Circumference = 2π2π x radius x radius

The RulesThe RulesMultiplying a number by a constant there are Multiplying a number by a constant there are 2 2 rules depending rules depending on which type of uncertainty you have : on which type of uncertainty you have :

Rule - Absolute Uncertainty:Rule - Absolute Uncertainty: c(A ± ΔA) = cA ± c(ΔA) c(A ± ΔA) = cA ± c(ΔA) Consider: 1.5(2.0 ± 0.2) m = (3.0 ± 0.3) m Consider: 1.5(2.0 ± 0.2) m = (3.0 ± 0.3) m AbsoluteAbsolute Uncertainty Uncertainty isis multiplied by the constant. multiplied by the constant.

Rule - Relative Uncertainty:Rule - Relative Uncertainty: c(A ± εA) = cA ± εA c(A ± εA) = cA ± εA Consider: 1.5(2.0 m ± 1.0%) = (3.0 m ± 1.0%) Consider: 1.5(2.0 m ± 1.0%) = (3.0 m ± 1.0%) Relative Uncertainty is Relative Uncertainty is notnot multiplied by the constant. multiplied by the constant.