## Uncertainties

It is really important that you get to grips with the uncertainty section. You will need this information for your Assignment and it could well form a question on the exam paper.

The key is remembering that ANY measurement is liable to uncertainty. Get that and you’re half way there!

#### CONTENT ASSOCIATED WITH UNCERTAINTIES

Random and systematic uncertainty

Uncertainties and data analysis

• All measurements of physical quantities are liable to uncertainty, which should be expressed in absolute or percentage form. Random uncertainties occur when an experiment is repeated and slight variations occur. Scale reading uncertainty is a measure of how well an instrument scale can be read. Random uncertainties can be reduced by taking repeated measurements.Systematic uncertainties occur when readings taken are either all too small or all too large. They can arise due to measurement techniques or experimental design.
• The mean of a set of readings is the best estimate of a ‘true’ value of the quantity being measured. When systematic uncertainties are present, the mean value of measurements will be offset. When mean values are used, the approximate random uncertainty should be calculated. When an experiment is being undertaken and more than one physical quantity is measured, the quantity with the largest percentage uncertainty should be identified and this may often be used as a good estimate of the percentage uncertainty in the final numerical result of an experiment. The numerical result of an experiment should be expressed in the form final value ±uncertainty.

#### UNCERTAINTIES NOTES

Whenever you do an experiment there will be uncertainties.

There are three types of uncertainty and effects to look out for at Higher.

##### Systematic Effects

Here the problem lies with the design of the experiment or apparatus. It includes zero errors. Sometimes they show up when you plot a graph but they are not easy to recognise, as they are not deliberate. Systematic effects include slow running clocks, zero errors, warped metre sticks etc. The best way to ensure that these are spotted is to acknowledge their existence and go looking for them. Where accuracy is of the utmost importance, the apparatus would be calibrated against a known standard. Note that a systematic effect might also be present if the experimenter is making the same mistake each time in taking a reading.

##### Random Uncertainties

These uncertainties cannot be eliminated. They cannot be pinpointed. examples include fluctuating temperatures, pressure and friction. Their effect can be reduced by taking several readings and finding a mean.

These occur because we cannot be absolutely certain about our readings when taking measurements from scales. Use scales with mirrors where possible, good scales and repeat all measurements.

Repeat all experiments to reduce the reading and random uncertainties. Systematic effects are not improved by taking lots of results.

Which experiment has the best design?

## Quantifying Uncertainties

##### 1.Find the mean

This is the best estimate of the “true” value but not necessary the “true” value.

##### 2. Find the approximate random uncertainty in the mean (absolute uncertainty)

This can be written as  and it is sometimes referred to as average deviation or absolute uncertainty.

##### 3. Find the percentage uncertainty.

or

This value indicates how well an instrument scale can be read.

An estimate of reading uncertainty for an analogue scale is generally taken as:

± half the least division of the scale.

Note: for widely spaced scales, this can be a little pessimistic and a reasonable estimate should be made.

For a digital scale it is taken as

± 1 in the least significant digit displayed.

### Overall final Uncertainty

When comparing uncertainties, it is important to take the percentage in each.

In an experiment, where more than one physical quantity has been measured, spot the quantity with the largest percentage uncertainty. This percentage uncertainty is often a good estimate of the percentage uncertainty in the final numerical result of the experiment.

eg if one measurement has an uncertainty of 3% and another has an uncertainty of 5%, then the overall percentage uncertainty in this experiment should be taken as 5%

###### learning outcomes
1. To review the work completed so far
2. To practice uncerts and practical experiments
3. To practice risk assessments

1. Starting on approximately p14 of the introduction notes complete tutorial 1 & 2
2. Make notes on uncerts and quantifying them from chapter 4
3. Risk assessment -Go through the powerpoint on the network (higher physics-> intro-> on risk assessment)
4. In your classwork jotter answer the questions as you go through the power point
5. Complete the practical below and write it up, including hazards, risks and controls.

Aim:      To find the average speed of a trolley moving down a slope, estimating the uncertainty in the final value.

Apparatus: 1 ramp, 1 metre stick, 1 trolley, 1 stop clock.

Instructions:

1. Set up a slope and mark two points 85 cm apart.
2. Note the scale reading uncertainty.
3. Calculate the percentage uncertainty in the distance.
4. Ensuring the trolley starts from the same point each time, measure how long it takes the trolley to pass between the two points.
5. Repeat 5 times, calculate the mean time and estimate the random uncertainty.
6. Note the scale reading uncertainty in the time.
7. Calculate the percentage uncertainty in the time.
8. Calculate the average speed and associated uncertainty.
9. Express your result in the form:

(speed ± absolute uncertainty) m s-1