1.1. To appreciate that all physical readings To appreciate that all physical readings contain uncertainty or errorscontain uncertainty or errors

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 exampleA significant example

Anecdote! Reference neededAnecdote! Reference needed

Copernicus is credited with much of the early work on Copernicus 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

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

His results showed broadly circular orbits with a His results showed broadly circular orbits with a slightly elliptical characterslightly elliptical 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 observed behaviour and my nice new circular orbit law is simply experimental error.is simply experimental error.

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

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 or add from all zero (find this value and subtract or add from all readings)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

Once multiple readings have been taken we have a Once multiple readings have been taken we have a much better first hand idea of the uncertainties much better first hand idea of the uncertainties involved.... We can see it in the datainvolved.... We can see it in the data

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

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

Multiplying of dividing quantities: add percentage uncertainties

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