# Uncertainties

## Sources of Experimental Error

Sources of Experimental Error

### Within all experimental work, there will be sources of Error, no experiment can be performed perfectly. However, this does not have to be an issues, as long as care is taken to minimise the effect of these Errors. There are three main types of Errors within Experimental Work :-

Within all experimental work, there will be sources of Error, no experiment can be performed perfectly. However, this does not have to be an issues, as long as care is taken to minimise the effect of these Errors. There are three main types of Errors within Experimental Work :-

### 1. Random Error - Small variations for each result, giving a range of results for each variable measurement. Can be reduced by the calculation of an average value and further calculation (see below).

1. Random Error - Small variations for each result, giving a range of results for each variable measurement. Can be reduced by the calculation of an average value and further calculation (see below).

### Systematic Error - All measurements affected in same way, all are either too large or too small. If the magnitude of the Error can be found, results data can be corrected accordingly (see below).

Systematic Error - All measurements affected in same way, all are either too large or too small. If the magnitude of the Error can be found, results data can be corrected accordingly (see below).

### Reading Error - Scales on measuring equipment can only show to a certain minimum accuracy. Comes in two distinct forms; Analogue and Digital.

Reading Error - Scales on measuring equipment can only show to a certain minimum accuracy. Comes in two distinct forms; Analogue and Digital.

## Error Notation

Error Notation

### All Errors will be written in a standard form, which is shown below :-

All Errors will be written in a standard form, which is shown below :-

### Value ± Error ( with Units )

Value ± Error ( with Units )

### The Error within the above form can be given in two different ways :-

The Error within the above form can be given in two different ways :-

### 1. Absolute Error - 50 ± 2.5 cm

1. Absolute Error - 50 ± 2.5 cm

### 2. Percentage Error - 50 cm ± 5%

2. Percentage Error - 50 cm ± 5%

## Random Error

Random Error

### Any experiment that is repeated will give small variations for each result. If these are truly random, they are just as likely to be larger than the average than they are to be smaller than the average value. The magnitude of these variations gives an indication of the amount of Error in the experiment. The bigger the range of these variations , the less accurate the average will be.

Any experiment that is repeated will give small variations for each result. If these are truly random, they are just as likely to be larger than the average than they are to be smaller than the average value. The magnitude of these variations gives an indication of the amount of Error in the experiment. The bigger the range of these variations , the less accurate the average will be.

### In order to calculate the Random Error value, the following formula can be used:-

In order to calculate the Random Error value, the following formula can be used:-

## Example 1 -

Example 1 -

### The below data was gathered from an Ohm's Law experiment. Calculate the average value of the Resistance and the Error in that reading:-

The below data was gathered from an Ohm's Law experiment. Calculate the average value of the Resistance and the Error in that reading:-

### Average value = Sum of readings / Number of readings

Average value = Sum of readings / Number of readings

### Average value = (1923 + 2083 + 2000 + 1990 + 1976 + 2013)/6

Average value = (1923 + 2083 + 2000 + 1990 + 1976 + 2013)/6

### Average value = 1998 Ω

Average value = 1998 Ω

### Random uncertainty = (max - min) / Number of values

Random uncertainty = (max - min) / Number of values

### Random uncertainty = (2083 - 1923)/6

Random uncertainty = (2083 - 1923)/6

### Random uncertainty = 26.7

Random uncertainty = 26.7

### Resistance = 1998 ± 27 Ω

Resistance = 1998 ± 27 Ω

## Systematic Error

Systematic Error

### A Systematic Error is very different from a Random Error. Unlike random, all measurements affected by a Systematic Error are affected in same way, all are either too large or too small. If magnitude of this Error can be found numerically, each data point in the results can be corrected accordingly. An example of this is shown in the diagram below:-

A Systematic Error is very different from a Random Error. Unlike random, all measurements affected by a Systematic Error are affected in same way, all are either too large or too small. If magnitude of this Error can be found numerically, each data point in the results can be corrected accordingly. An example of this is shown in the diagram below:-

### In the experiment above the length of the pendulum was measured as shown (red measurement). However, the true length of the pendulum is to the centre of the bob (green measurement). By knowing that each recorded length is too short by 0.8 cm (green - red) , each result can be corrected after the practical, simply by adding 0.8 cm to each recorded length.

In the experiment above the length of the pendulum was measured as shown (red measurement). However, the true length of the pendulum is to the centre of the bob (green measurement). By knowing that each recorded length is too short by 0.8 cm (green - red) , each result can be corrected after the practical, simply by adding 0.8 cm to each recorded length.

### Not all Systematic Errors are so easily fixed. When using any measuring device, it is assumed that the values given are accurate, however, this may not be the case. These Calibration Errors occur for a variety of reasons and will introduce a Systematic Error into the measurements that is very hard to quantify.

Not all Systematic Errors are so easily fixed. When using any measuring device, it is assumed that the values given are accurate, however, this may not be the case. These Calibration Errors occur for a variety of reasons and will introduce a Systematic Error into the measurements that is very hard to quantify.

### For example, when using a metal rule, it is assumed that the markings on it give a true measurement of length. However, if the temperature of the metal rule is changed, then the rule itself will expand or contract, changing the "length" of 1 cm. As the only way to identify the value of this Error would be to measure the metal rule against a " Standard Length", which cannot be done within the school setting, this Error can be discussed in evaluation but not actually calculated.

For example, when using a metal rule, it is assumed that the markings on it give a true measurement of length. However, if the temperature of the metal rule is changed, then the rule itself will expand or contract, changing the "length" of 1 cm. As the only way to identify the value of this Error would be to measure the metal rule against a " Standard Length", which cannot be done within the school setting, this Error can be discussed in evaluation but not actually calculated.

## Reading Error

Reading Error

### As stated above, scales on measuring equipment can only show to a certain minimum accuracy. Any experimental work can only be as accurate as the measuring equipment allows. This has been seen clearly within all practical work since National 5 level, where all calculated results must have the same number of significant figures as the variables it was calculated from.

As stated above, scales on measuring equipment can only show to a certain minimum accuracy. Any experimental work can only be as accurate as the measuring equipment allows. This has been seen clearly within all practical work since National 5 level, where all calculated results must have the same number of significant figures as the variables it was calculated from.

### Reading Errors come in two different forms; Analogue and Digital reading Error. Depending on the type of measuring device used, the maximum accuracy is different. Analogue scales are actually twice as accurate as a Digital scale to the same minimum unit (eg measuring to nearest mm).

Reading Errors come in two different forms; Analogue and Digital reading Error. Depending on the type of measuring device used, the maximum accuracy is different. Analogue scales are actually twice as accurate as a Digital scale to the same minimum unit (eg measuring to nearest mm).

### Analogue scale - Accurate to ± half of the smallest unit.

Analogue scale - Accurate to ± half of the smallest unit.

### Digital scale - Accurate to ± 1 of the smallest unit.

Digital scale - Accurate to ± 1 of the smallest unit.

### This is due to the issue of rounding to the nearest unit. Both Meters below show Voltage readings; Analogue on left and Digital on right. Both have the same minimum marked unit of 1 Volt.

This is due to the issue of rounding to the nearest unit. Both Meters below show Voltage readings; Analogue on left and Digital on right. Both have the same minimum marked unit of 1 Volt.

### When taking a reading using the analogue meter, the person taking the reading will have to make a judgment call as to the position of the needle between two marking, and then decide which way to round the number. This means that the person will know that their value was within 0.5 of the minimum unit otherwise they would have rounded differently.

When taking a reading using the analogue meter, the person taking the reading will have to make a judgment call as to the position of the needle between two marking, and then decide which way to round the number. This means that the person will know that their value was within 0.5 of the minimum unit otherwise they would have rounded differently.

### When taking a reading using the Digital meter, however, the person does not know which way the reading was rounded. Was it rounded up to the value or rounded down? This means that each reading on the digital meter could be up to 0.5 below the value or up to 0.5 above the value and as such the accuracy is 1 of the smallest unit. For example, possible readings on the above meters could be:-

When taking a reading using the Digital meter, however, the person does not know which way the reading was rounded. Was it rounded up to the value or rounded down? This means that each reading on the digital meter could be up to 0.5 below the value or up to 0.5 above the value and as such the accuracy is 1 of the smallest unit. For example, possible readings on the above meters could be:-

### Analogue Meter - 27 ± 0.5 V

Analogue Meter - 27 ± 0.5 V

### Digital Meter - 27 ± 1 V

Digital Meter - 27 ± 1 V

## Overall Error Calculation

Overall Error Calculation

### Once all sources of Error have been identified, an overall Error must be identified and then applied to the final value. The overall Error is simply the source of Error with the largest percentage Error. This percentage is then applied to the final value, and an absolute overall Error can be found.

Once all sources of Error have been identified, an overall Error must be identified and then applied to the final value. The overall Error is simply the source of Error with the largest percentage Error. This percentage is then applied to the final value, and an absolute overall Error can be found.

## Example 2 -

Example 2 -

### In an experiment, the Current through a fixed Resistor was measured and the following data was collected:-

In an experiment, the Current through a fixed Resistor was measured and the following data was collected:-

### If the Voltage has a value of 6.4 ± 0.1 V and the Ammeter has a reading Uncertainty of 0.1 A, what is the overall Uncertainty in the average Resistance value ?

If the Voltage has a value of 6.4 ± 0.1 V and the Ammeter has a reading Uncertainty of 0.1 A, what is the overall Uncertainty in the average Resistance value ?

### Step 1 - Identify all sources of Error

Step 1 - Identify all sources of Error

### Voltage = 6.4 ± 0.1 V

Voltage = 6.4 ± 0.1 V

### Average Current = ( 3.20 + 3.21 + 3.18 + 3.14 + 3.22 + 3.20 ) / 6

Average Current = ( 3.20 + 3.21 + 3.18 + 3.14 + 3.22 + 3.20 ) / 6

### Average Current = 3.19 A

Average Current = 3.19 A

### Random Uncertainty in Current = ( 3.22 - 3.14 ) / 6

Random Uncertainty in Current = ( 3.22 - 3.14 ) / 6

### Random Uncertainty on Current = 0.04

Random Uncertainty on Current = 0.04

### Average Current = 3.19 ± 0.04 A

Average Current = 3.19 ± 0.04 A

### Reading Error in Current (using min value for max error) = 3.14 ± 0.1 A

Reading Error in Current (using min value for max error) = 3.14 ± 0.1 A

### Step 2 - Calculate percentage Error for each

Step 2 - Calculate percentage Error for each

### Reading Error Voltage = 6.4 ± 0.1 V

Reading Error Voltage = 6.4 ± 0.1 V

### Voltage = 6.4 ± 1.56 %

Voltage = 6.4 ± 1.56 %

### Average Current = 3.19 ± 0.04 A

Average Current = 3.19 ± 0.04 A

### Average Current = 3.19 ± 1.25 %

Average Current = 3.19 ± 1.25 %

### Reading Error Current = 3.14 ± 0.1 A

Reading Error Current = 3.14 ± 0.1 A

### Current = 3.14 ± 3.18 % this is the largest percentage Error.

Current = 3.14 ± 3.18 % this is the largest percentage Error.

### Step 3 - Calculate R and apply overall Error

Step 3 - Calculate R and apply overall Error

### R = V / I

R = V / I

### R = 6.4 / 3.19

R = 6.4 / 3.19

### R = 2.01 Ω ± 3.18 %

R = 2.01 Ω ± 3.18 %

### R = 2.01 ± 0.06 Ω

R = 2.01 ± 0.06 Ω

### Note - In the above example, when the reading Error in the Current was used, a value of 3.14 A was used. This value was selected as it is the smallest of the Current readings. This therefore has the largest percentage error, based upon a fixed error value. Due to this, a simple way to improve experimental accuracy is to measure over as large a measurement as possible, to reduce this effect.

Note - In the above example, when the reading Error in the Current was used, a value of 3.14 A was used. This value was selected as it is the smallest of the Current readings. This therefore has the largest percentage error, based upon a fixed error value. Due to this, a simple way to improve experimental accuracy is to measure over as large a measurement as possible, to reduce this effect.