This section presents the uncertainties and uncertainty calculations to be carried out in a laboratory. However, the content required may vary according to the teaching method used or the student's level of education.
Uncertainty represents the margin of error associated with the values measured or determined during an experiment.
In all experiments, there is a degree of imprecision involved in taking measurements. A laboratory reading is always the most accurate under the circumstances. However, the uncertainty describes the dispersion of the value, i.e. the interval within which the exact value lies.
Uncertainty may be associated with the measuring instrument used, the lack of rigour shown by the person taking the measurement or the difficulty of interpreting a measurement on a given scale.
Uncertainty, whether absolute or relative, is always written with a single significant figure.
Absolute uncertainty is the maximum error that can be made in determining a measurement on an instrument.
Any experimental result lies between a minimum value and a maximum value. This result, which we can call |x|, therefore lies between a minimum value called |x_{min}| and a maximum value |x_{max}|. The range of possible values for the |x| measure could therefore be described as |\left[x_{min}, \: x_{max}\right]|.
To simplify writing the uncertainty, we write the measurement with its uncertainty as follows: |x \pm \Delta x|.
A ruler is used to measure a book. The measurement obtained, with its absolute uncertainty, is |(21.90 \pm 0.05) \: \text {cm}|. This means that the smallest value the book could have |\left( x_{min} \right)| would be |21.85 \: \text {cm}|, while the largest value |\left( x_{max} \right)| would be |21.95 \: \text {cm}|.
The experimental result gives a value |x|, which is the best possible estimate of the result of the reading, and a value |Delta x|, which represents the absolute uncertainty associated with this value.
Case 1: analogue measuring instruments
Analogue measuring instruments are devices fitted with a pointer indicating the value of the quantity measured on a scale, or with a scale indicating the value of the quantity measured.
A needle-type voltmeter, a graduated cylinder or an alcohol thermometer are all analogue instruments, because they all consist of scales on which a reading must be taken to read the value of the quantity being measured.
The reading uncertainty associated with an analogue measuring instrument corresponds to half the smallest graduation on the instrument.
The absolute uncertainty of a ruler graduated in millimetres is therefore: |\displaystyle \frac {1: \text {mm}}{2} = 0.5 \: \text {mm}|.
Uncertainty could also be calculated in centimetres: |\displaystyle \frac {0,1 \: \text {cm}}{2} = 0.05 \: \text {cm}|
The absolute uncertainty of an alcohol thermometer whose smallest graduation is the degree would be: |displaystyle \frac {1 \: ^{circ}\text {C}}{2} = 0.5 \: ^{circ}\text {C}|.
Case 2: digital measuring instruments
Digital measuring instruments are devices which give a direct reading in the form of a numerical value.
A multimeter and a stopwatch are examples of digital instruments, because these devices allow you to obtain a reading directly by observing the device.
The reading uncertainty associated with a digital measuring instrument corresponds to the equivalent of one unit of the smallest graduation of the instrument.
The absolute uncertainty of a stopwatch accurate to one hundredth of a second will be one hundredth of a second |\left( {0.01 \: \text {s}}\right)|.
The absolute uncertainty of a multimeter measuring the resistance of a resistor to within one unit will be one ohm |\left( {1 \: \Omega} \right)|.
Case 3: theoretical values
The uncertainty associated with a theoretical value corresponds to the equivalent of one unit on the last figure.
Since the boiling temperature of water is |100 \: ^{circ} \text {C}|, the uncertainty will be |pm 1 \: ^{\circ} \text {C}|.
Knowing that the density of water is |1.00: \text {g/ml}|, the uncertainty will be |\pm 0.01 \: \text {g/ml}|.
Sometimes, in addition to the absolute uncertainty of an instrument, we have to add the uncertainty of the measurement by the observer. In these cases, the uncertainty related to the measurement is often equal to the sum of the uncertainties on each reading.
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The parallax effect: When two lines must be matched to interpret a measurement, such as the pointer of an analogue instrument and the scale below it, the reading can vary from one observer to another depending on the position of the eye in relation to these lines.
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Reflex time: There is an uncertainty linked to the observer's reflexes. For example, if a person has to time the time it takes for an object to fall, the time between the object actually hitting the ground and the moment when the thumb presses the stopwatch button has to be taken into account.
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The meniscus: When measuring the volume of a liquid, a particular phenomenon must always be taken into account, namely the presence of a curved line formed by the liquid in the graduated cylinder. This curve, known as the meniscus, may be concave or convex. Reading the volume therefore involves a degree of uncertainty. To reduce this uncertainty, it is important to align the eye with the meniscus by placing it at the same height.
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Measurements given by two readings: When using a ruler, you need to consider the uncertainty at the point where the measurement is taken on the ruler, as well as the uncertainty at zero, i.e. the point where the ruler was placed to take the measurement. In these cases, it is preferable to double the uncertainty of the reading.
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Zero readings: There is some uncertainty about zero readings, since they have to be taken in the same way as if they were not at zero.
The relative uncertainty is the ratio between the absolute uncertainty and the measurement. This ratio is expressed as a percentage.
To calculate the relative uncertainty, it is important to determine the absolute uncertainty of the device. The advantage of calculating the relative uncertainty is that you can compare the accuracy of different measurements. The most accurate measurement is the one with the lowest relative uncertainty.
|\text {Relative uncertainty} = \displaystyle \frac{\text {Absolute uncertainty}}{\text {Measured value}}\times \text {100}|
What is the relative uncertainty of a measurement taken with a ruler, given that the length of the object to be measured is |21.3: \text {cm}|?
Since the smallest unit of measurement of a ruler is |0.1 \: \text {cm}|, the absolute uncertainty associated with this measuring instrument is |pm 0.05 \: \text {cm}|. The relative uncertainty is therefore expressed as follows:
|text {Relative uncertainty} = \displaystyle \frac{0.05 \: \text {cm}}{{21.3 \: \text {cm}}}\times \text {100}|
Uncertainty in addition or subtraction
To calculate the uncertainty of an addition or subtraction, the absolute uncertainties are added together to give the absolute uncertainty of the result of the sum or subtraction.
What is the total volume of water if we add |25,0 \: \text {ml} \pm 0,3 \: \text {ml} | of water into a 50.0 ml graduated cylinder containing |10.0 \: \text {ml} \0.4 \pm: \text {ml} |?
To find the total volume, add the volumes together: |25,0 + 10,0 = 35,0 \: \text {ml}|.
To find the uncertainty, add the uncertainties: |0.3 + 0.4 = \pm 0.7 \: \text {ml}|.
The final measurement is therefore |\left( 35,0 \pm 0,7 \right) \: \text {ml}|.
What is the total volume of acid remaining in a burette if it contained |50.00? \text {ml} \pm 0.05 \: \text {ml} | and that |18.50 \: \text {ml} \pm 0.05 \: \text {ml} | have been used in a neutralisation?
To find the remaining volume, subtract the volumes: |50.00 - 18.50 = 31.50 \: \text {ml}|.
To find the uncertainty, add the uncertainties: |0.05 + 0.05 = \pm 0.1 \: \text {ml}|.
The final measurement is therefore |\left( 31.5 \pm 0.1 \right) \: \text {ml}|.
Two methods of calculating uncertainties are proposed. These methods represent ways of calculating the uncertainty of data obtained following a mathematical calculation.
To calculate the uncertainty when multiplying or dividing, the relative uncertainties of the initial data must be added together and the sum multiplied by the final answer.
What is the area of a rectangle with a length of |20.0? \text {m} \pm 0,5 \: \text {m} | and whose width measures |12,0 \: \text {m} \pm 0,4 \: \text {m} |?
To find the total area, multiply the length and width: |20,0 \times 12,0 = 240,0 \: \text {m}^2|.
To find the uncertainty, use the relative uncertainties.
|\Delta \text {x} =left( \left( \displaystyle \frac{{0,5 \: \text {m}}}{{20,0 \: \text {m}} \right) + \left( \displaystyle \frac{{0,4 \: \text {m}}{{12,0 \: \text {m}} \right) \times 240,0 \: \text {m}^2 = 14\: \text {m}^2 = 1 \times 10^1 \: \text {m}^2|
The final measurement is therefore |\left( 24 \pm 1 \right) \times 10^1 \: \text {m}^2|.
What is the density of an object with a mass of |109.47? \text {g} \pm 0.05 \: \text {g} | and the volume is |12.3 \: \text {ml} \pm 0,3 \: \text {ml} |?
To find the density, divide the mass by the volume: |109.47 \div 12.3 = 8.90 \: \text {g/ml}|.
To find the uncertainty, use the relative uncertainties.
|\Delta \text {x} =left( \left( \displaystyle \frac{{0,05 \: \text {g}}}{{109,47 \: \text {g}} \right) + \left( \displaystyle \frac{{0,3 \: \text {ml}}{{12,3 \: \text {ml}} \right) \times 8,90 \: \text {g/ml} = 0,2\: \text {g/ml}|
The final measurement is therefore |\left( 8,9 \pm 0,2 \right) \: \text {g/ml}|.
To calculate the uncertainty when multiplying or dividing, divide the difference between the maximum value and the minimum value that can be obtained by the uncertainties by two.
What is the area of a rectangle with a length of |20.0? \text {m} \pm 0,5 \: \text {m} | and whose width measures |12,0 \: \text {m} \pm 0,4 \: \text {m} |?
To find the total area, multiply the length and width: |20,0 \times 12,0 = 240,0 \: \text {m}^2|.
To find the uncertainty, determine the minimum and maximum values.
\text{x}_{\text {min}} = (20,0 - 0,5)\times (12,0 - 0,4) = 226,2 \: \text{m}^2|
|\text{x}_{\text {max}} = (20,0 + 0,5)\times (12,0 + 0,4) = 254,2 \: \text {m}^2|
The uncertainty is then determined.
|\Delta \text {x} = \left( \displaystyle \frac{\text{x}_{\text {max}}-\text{x}_{\text {min}}{{2}} \right)|
|\Delta \text {x} = \left( \displaystyle \frac{254,2\: \text {m}^2-226,2\: \text {m}^2}{{2}} \right)|
|Delta \text {x} = 10\: \text {m}^2|
The final measurement is therefore |\left( 240 \pm 10 \right) \: \text {m}^2|.
When dividing, you need to pay attention to the values used to find the maximum and minimum values.
What is the density of an object with a mass of |109.47? \text {g} \pm 0.05 \: \text {g} | and the volume is |12.3 \: \text {ml} \pm 0,3 \: \text {ml} |?
To find the density, divide the mass by the volume: |109.47 \div 12.3 = 8.90 \: \text {g/ml}|.
To find the uncertainty, determine the minimum and maximum values.
\text{x}_{\text {min}} = (109.47 - 0.05)\div (12.3 + 0.3) = 8.68 \: \text {g/ml}|
|text{x}_{\text {max}} = (109.47 + 0.05)\div (12.3 - 0.3) = 9.13 \: \text {g/ml}|
The uncertainty is then determined.
|\Delta \text {x} = \left( \displaystyle \frac{\text{x}_{\text {max}}-\text{x}_{\text {min}}{{2}} \right)|
|displaystyle \frac{9,13 \: \text {g/ml}-8,68 \: \text {g/ml}{{2}} \right)|
|Delta \text {x} = 0.225 \: \text {g/ml} = 0,2 \: \text {g/ml}|
The final measurement is therefore |\left( 8,9 \pm 0,2 \right) \: \text {g/ml}|.
To validate your understanding of uncertainty and significant figures interactively, consult the following MiniRecup:
