Uncertainties

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 :-

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 effected in same way, all are either too large or too small. If magnitude 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.  

Error Notation

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

Value ± Error (with Units) 

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

Absolute Error - 50 ± 2.5 cm 

Percentage Error - 50 cm ± 5% 

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. 

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

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 :-

Average value  =  Sum of Readings / Number of Readings

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

Average value  =  1998 Ω

Random uncertainty  =  (Max - Min) / Number of Values

Random uncertainty  =  (2083 - 1923) / 6

Random uncertainty  =  ± 26.7 

Resistance  =  1998 ± 27 Ω

Systematic Error

A Systematic Error is very different from a Random Error. Unlike Random, all measurements effected 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. 

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. 

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. 

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

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. 

When taking a reading using the analogue meter, the person taking the reading will have to make a judgement 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 :-

Analogue meter - 27 ± 0.5 V 

Digital meter - 27 ± 1 V

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 within the Higher course was 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.

In the Advanced Higher course, however, a more inclusive overall error is required. Within Advanced Higher, Errors can be combined together to give an overall Error value. 

There are two main error combination methods, each dependant on the mathematical operations (+ - x /) within the formula in question :-

1. Uncertainty in a Sum or Difference (Z  =  X ± Y)  

2. Uncertainty in a Product or Quotient  (Z  =  X / Y  or  Z  =  X x Y)  

A very useful approximation to calculate overall Error is to discount any Error of less than 1/3 of biggest value. This is due to the square functions within the combination formula. If an Error is less than 1/3, the square is  less than 1/9th the value and at Advanced Higher level, this can be discounted as negligible. 

Uncertainty in a Sum or Difference Equation

When a calculation requires addition or subtraction, the absolute error in the final value can be calculated using the following formula :-

Where :-

ΔZ  =  Absolute Error in Z.

ΔX  =  Absolute Error in X.

ΔY  =  Absolute Error in Y.

Uncertainty in a Product or Quotient Equation

When a calculation requires multiplication or division, the absolute error in the final value can be calculated using the following formula :-

Where :-

ΔZ  =  Absolute Error in Z.

Z  =  Magnitude of Z.

ΔX  =  Absolute Error in X.

X  =  Magnitude of X.

ΔY  =  Absolute Error in Y.

Y  =  Magnitude of Y.

The above formula allows the calculation of the fractional Error in each value, allowing each error to be compared, even though the base values have different units. 

Example 2 - 

The Temperature of a beaker of water rises from 18.5°C to 22.0°C. If the reading uncertainty in the Thermometer is ± 0.5°C, calculate the overall uncertainty in the Temperature change.

Reading Error in T  =  0.5 °C

Tdifference  =  T2  -  T1 

Tdifference  =  22.0  -  18.5

Tdifference  =  3.5 °C

Overall Error in Tdifference  =  ( 0.52 + 0.52 )1/2

Overall Error in Tdifference  =  ± 0.7°C

Tdifference  =  3.5 ± 0.7°C

Example 3 - 

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 as a percentage?

Voltage  =  6.4 ± 0.1 V

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

Average Current  =  3.19 A 

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

Random Uncertainty on Current  =  0.04 

Average Current  =  3.19 ± 0.04 A 

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

Check by percentage to see if any Error is less than 1/3 of Largest and therefore could be discounted

Reading Error Voltage  =  6.4 ± 0.1 V

Voltage  =  6.4 ± 1.56 %

Average Current  =  3.19 ± 0.04 A 

Average Current  =  3.19 ± 1.25 %

Reading Error Current  =  3.14 ± 0.1 A

Current  =  3.14 ± 3.18 %                           

None can be discounted in this example. See below for colour coded sections of uncertainty formula

R  =  V / I 

R  =  6.4  / 3.19

R  =  2.01 Ω  

Fractional Error in R  =  ((0.1/6.4)2 + (0.04/3.19)2 + (0.1/3.14)2)1/2

Fractional Error in R  =  0.038

Overall Error in R  =  0.038 x 2.01

Overall Error in R  =  ± 0.08 Ω

R  =  2.01 ± 0.08 Ω

R  =  2.01Ω ± 4% 

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 its 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. 

Uncertainty Calculations - LINEST Function

Microsoft Excel can also be used to perform more complex uncertainties calculations than those described above. As part of the Higher and especially Advanced Higher courses, an understanding the LINEST function is valuable. 

The LINEST function allows a statistical analysis of the data used to generate a graph, and generates a set of data giving the uncertainty values for the gradient of the graph and the graph's intercept. This information can then be used to clearly show whether two variables have a directly proportional relationship or not. A directly proportional relationship is shown if the origin of a straight line graph falls within the Error margin of the intercept.  

To calculate this, the starting point required is the same as for a graph :-

To allow the calculation of a LINEST function, an additional 4x4 cell section needs to be used. The following shows the additional layout, as well as labels for each :-

To perform the necessary calculations to populate the LINEST Table, the following formulae must be added to the empty cells. 

To add the calculation, click and drag to highlight all four Cells, then type "=LINEST( Y-value cells,X-value cells,1,1)" into the selection. The two variables should be changed to show the cells giving each variable, for instance in this example the calculation will read "=LINEST(E6:E11,C6:C11,1,1)" . 

Once this calculation has been typed into the first cell, press Ctrl+shift+Enter, and the table should now show four values :-

Note - By pressing "Ctrl+Shift+Enter" at the same time, Excel treats this function as an array over all four cells. If only "Enter" is pressed it does not work. Also in the above screen shot, the function in the cells is shown to be contained within " { } ". This is how Excel denotes an array, but do not type these when inputting the LINEST function as again, this will prevent the function working. 

The values now show in the array show that :-

Gradient of the graph     =  5x10-4  ± 6.96x10-6

Y-Intercept of the graph =  -2x10-6 ± 2.71x10-5

As was stated earlier, the gradient of this graph shows 1/Resistance. If the axes are reversed (Current X axis, Voltage Y axis), the gradient will then give resistance, as shown below :-

To reverse the axes, simply change the order of the variables within the LINEST calculation (circled in red), to give the required results :-

Gradient of graph  =  1996 ± 27.77 Ω

Y-intercept  =  0.01 ± 0.05 

As the Y-intercept is 0.01 with a range of 0.06 to -0.04, the origin falls within the error range, and as such this graph shows a directly proportional relationship and the experiment "proves" Ohm's Law. 

Note - As can be seen in the above image, there are two different average values given. The arithmetic average based upon the data point values, and the LINEST average, based upon the line of best fit. In an investigation, the LINEST average should always be used.